(*<*)
theory Paper
imports 
   "../Lexer"
   "../Simplifying" 
   "../Positions"
   "../SizeBound4" 
   "HOL-Library.LaTeXsugar"
begin

declare [[show_question_marks = false]]

notation (latex output)
  If  ("(\<^latex>\<open>\\textrm{\<close>if\<^latex>\<open>}\<close> (_)/ \<^latex>\<open>\\textrm{\<close>then\<^latex>\<open>}\<close> (_)/ \<^latex>\<open>\\textrm{\<close>else\<^latex>\<open>}\<close> (_))" 10) and
  Cons ("_\<^latex>\<open>\\mbox{$\\,$}\<close>::\<^latex>\<open>\\mbox{$\\,$}\<close>_" [75,73] 73) 


abbreviation 
  "der_syn r c \<equiv> der c r"
abbreviation 
 "ders_syn r s \<equiv> ders s r"  
abbreviation 
  "bder_syn r c \<equiv> bder c r"  

notation (latex output)
  der_syn ("_\\_" [79, 1000] 76) and
  ders_syn ("_\\_" [79, 1000] 76) and
  bder_syn ("_\\_" [79, 1000] 76) and
  bders ("_\\_" [79, 1000] 76) and
  bders_simp ("_\\\<^sub>b\<^sub>s\<^sub>i\<^sub>m\<^sub>p _" [79, 1000] 76) and

  ZERO ("\<^bold>0" 81) and 
  ONE ("\<^bold>1" 81) and 
  CH ("_" [1000] 80) and
  ALT ("_ + _" [77,77] 78) and
  SEQ ("_ \<cdot> _" [77,77] 78) and
  STAR ("_\<^sup>*" [79] 78) and

  val.Void ("Empty" 78) and
  val.Char ("Char _" [1000] 78) and
  val.Left ("Left _" [79] 78) and
  val.Right ("Right _" [1000] 78) and
  val.Seq ("Seq _ _" [79,79] 78) and
  val.Stars ("Stars _" [79] 78) and

  Prf ("\<turnstile> _ : _" [75,75] 75) and  
  Posix ("'(_, _') \<rightarrow> _" [63,75,75] 75) and

  flat ("|_|" [75] 74) and
  flats ("|_|" [72] 74) and
  injval ("inj _ _ _" [79,77,79] 76) and 
  mkeps ("mkeps _" [79] 76) and 
  length ("len _" [73] 73) and
  set ("_" [73] 73) and

  AZERO ("ZERO" 81) and 
  AONE ("ONE _" [79] 78) and 
  ACHAR ("CHAR _ _" [79, 79] 80) and
  AALTs ("ALTs _ _" [77,77] 78) and
  ASEQ ("SEQ _ _ _" [79, 79,79] 78) and
  ASTAR ("STAR _ _" [79, 79] 78) and

  code ("code _" [79] 74) and
  intern ("_\<^latex>\<open>\\mbox{$^\\uparrow$}\<close>" [900] 80) and
  erase ("_\<^latex>\<open>\\mbox{$^\\downarrow$}\<close>" [1000] 74) and
  bnullable ("bnullable _" [1000] 80) and
  bsimp_AALTs ("bsimpALT _ _" [10,1000] 80) and
  bsimp_ASEQ ("bsimpSEQ _ _ _" [10,1000,1000] 80) and
  bmkeps ("bmkeps _" [1000] 80) and

  srewrite ("_\<^latex>\<open>\\mbox{$\\,\\stackrel{s}{\\leadsto}$}\<close> _" [71, 71] 80) and
  rrewrites ("_ \<^latex>\<open>\\mbox{$\\,\\leadsto^*$}\<close> _" [71, 71] 80) and
  srewrites ("_ \<^latex>\<open>\\mbox{$\\,\\stackrel{s}{\\leadsto}^*$}\<close> _" [71, 71] 80) and
  blexer_simp ("blexer\<^sup>+" 1000) 


lemma better_retrieve:
   shows "rs \<noteq> Nil ==> retrieve (AALTs bs (r#rs)) (Left v) = bs @ retrieve r v"
   and   "rs \<noteq> Nil ==> retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v"
  apply (metis list.exhaust retrieve.simps(4))
  by (metis list.exhaust retrieve.simps(5))

(*>*)

section {* Introduction *}

text {*

In the last fifteen or so years, Brzozowski's derivatives of regular
expressions have sparked quite a bit of interest in the functional
programming and theorem prover communities.  The beauty of
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
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
succession) all the characters of the string matches the empty string,
then @{term r} matches @{term s} (and {\em vice versa}).  We are aware
of a mechanised correctness proof of Brzozowski's derivative-based matcher in HOL4 by
Owens and Slind~\cite{Owens2008}. Another one in Isabelle/HOL is part
of the work by Krauss and Nipkow \cite{Krauss2011}.  And another one
in Coq is given by Coquand and Siles \cite{Coquand2012}.
Also Ribeiro and Du Bois give one in Agda \cite{RibeiroAgda2017}.


However, there are two difficulties with derivative-based matchers:
First, Brzozowski's original matcher only generates a yes/no answer
for whether a regular expression matches a string or not.  This is too
little information in the context of lexing where separate tokens must
be identified and also classified (for example as keywords
or identifiers).  Sulzmann and Lu~\cite{Sulzmann2014} overcome this
difficulty by cleverly extending Brzozowski's matching
algorithm. Their extended version generates additional information on
\emph{how} a regular expression matches a string following the POSIX
rules for regular expression matching. They achieve this by adding a
second ``phase'' to Brzozowski's algorithm involving an injection
function.  In our own earlier work we provided the formal
specification of what POSIX matching means and proved in Isabelle/HOL
the correctness
of Sulzmann and Lu's extended algorithm accordingly
\cite{AusafDyckhoffUrban2016}.

The second difficulty is that Brzozowski's derivatives can 
grow to arbitrarily big sizes. For example if we start with the
regular expression \mbox{@{text "(a + aa)\<^sup>*"}} and take
successive derivatives according to the character $a$, we end up with
a sequence of ever-growing derivatives like 

\def\ll{\stackrel{\_\backslash{} a}{\longrightarrow}}
\begin{center}
\begin{tabular}{rll}
$(a + aa)^*$ & $\ll$ & $(\ONE + \ONE{}a) \cdot (a + aa)^*$\\
& $\ll$ & $(\ZERO + \ZERO{}a + \ONE) \cdot (a + aa)^* \;+\; (\ONE + \ONE{}a) \cdot (a + aa)^*$\\
& $\ll$ & $(\ZERO + \ZERO{}a + \ZERO) \cdot (a + aa)^* + (\ONE + \ONE{}a) \cdot (a + aa)^* \;+\; $\\
& & $\qquad(\ZERO + \ZERO{}a + \ONE) \cdot (a + aa)^* + (\ONE + \ONE{}a) \cdot (a + aa)^*$\\
& $\ll$ & \ldots \hspace{15mm}(regular expressions of sizes 98, 169, 283, 468, 767, \ldots)
\end{tabular}
\end{center}
 
\noindent where after around 35 steps we run out of memory on a
typical computer (we shall define shortly the precise details of our
regular expressions and the derivative operation).  Clearly, the
notation involving $\ZERO$s and $\ONE$s already suggests
simplification rules that can be applied to regular regular
expressions, for example $\ZERO{}\,r \Rightarrow \ZERO$, $\ONE{}\,r
\Rightarrow r$, $\ZERO{} + r \Rightarrow r$ and $r + r \Rightarrow
r$. While such simple-minded simplifications have been proved in our
earlier work to preserve the correctness of Sulzmann and Lu's
algorithm \cite{AusafDyckhoffUrban2016}, they unfortunately do
\emph{not} help with limiting the growth of the derivatives shown
above: the growth is slowed, but the derivatives can still grow rather
quickly beyond any finite bound.


Sulzmann and Lu overcome this ``growth problem'' in a second algorithm
\cite{Sulzmann2014} where they introduce bitcoded
regular expressions. In this version, POSIX values are
represented as bitsequences and such sequences are incrementally generated
when derivatives are calculated. The compact representation
of bitsequences and regular expressions allows them to define a more
``aggressive'' simplification method that keeps the size of the
derivatives finite no matter what the length of the string is.
They make some informal claims about the correctness and linear behaviour
of this version, but do not provide any supporting proof arguments, not
even ``pencil-and-paper'' arguments. They write about their bitcoded
\emph{incremental parsing method} (that is the algorithm to be formalised
in this paper):

\begin{quote}\it
  ``Correctness Claim: We further claim that the incremental parsing
  method [..] in combination with the simplification steps [..]
  yields POSIX parse trees. We have tested this claim
  extensively [..] but yet
  have to work out all proof details.'' \cite[Page 14]{Sulzmann2014}
\end{quote}  

\noindent{}\textbf{Contributions:} We have implemented in Isabelle/HOL
the derivative-based lexing algorithm of Sulzmann and Lu
\cite{Sulzmann2014} where regular expressions are annotated with
bitsequences. We define the crucial simplification function as a
recursive function, without the need of a fix-point operation. One objective of
the simplification function is to remove duplicates of regular
expressions.  For this Sulzmann and Lu use in their paper the standard
@{text nub} function from Haskell's list library, but this function
does not achieve the intended objective with bitcoded regular expressions.  The
reason is that in the bitcoded setting, each copy generally has a
different bitcode annotation---so @{text nub} would never ``fire''.
Inspired by Scala's library for lists, we shall instead use a @{text
distinctBy} function that finds duplicates under an erasing function
which deletes bitcodes.
We shall also introduce our own argument and definitions for
establishing the correctness of the bitcoded algorithm when 
simplifications are included.\medskip

\noindent In this paper, we shall first briefly introduce the basic notions
of regular expressions and describe the basic definitions
of POSIX lexing from our earlier work \cite{AusafDyckhoffUrban2016}. This serves
as a reference point for what correctness means in our Isabelle/HOL proofs. We shall then prove
the correctness for the bitcoded algorithm without simplification, and
after that extend the proof to include simplification. 

*}

section {* Background *}

text {*
  In our Isabelle/HOL formalisation strings are lists of characters with
  the empty string being represented by the empty list, written $[]$,
  and list-cons being written as $\_\!::\!\_\,$; string
  concatenation is $\_ \,@\, \_\,$. We often use the usual
  bracket notation for lists also for strings; for example a string
  consisting of just a single character $c$ is written $[c]$.   
  Our regular expressions are defined as usual as the elements of the following inductive
  datatype:

  \begin{center}
  @{text "r ::="} \;
  @{const "ZERO"} $\mid$
  @{const "ONE"} $\mid$
  @{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"} 
  \end{center}

  \noindent where @{const ZERO} stands for the regular expression that does
  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.
  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}).

  Central to Brzozowski's regular expression matcher are two functions
  called @{text nullable} and \emph{derivative}. The latter is written
  $r\backslash c$ for the derivative of the regular expression $r$
  w.r.t.~the character $c$. Both functions are defined by recursion over
  regular expressions.  

\begin{center}
\begin{tabular}{cc}
  \begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
  @{thm (lhs) der.simps(1)} & $\dn$ & @{thm (rhs) der.simps(1)}\\
  @{thm (lhs) der.simps(2)} & $\dn$ & @{thm (rhs) der.simps(2)}\\
  @{thm (lhs) der.simps(3)} & $\dn$ & @{thm (rhs) der.simps(3)}\\
  @{thm (lhs) der.simps(4)[of c "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) der.simps(4)[of c "r\<^sub>1" "r\<^sub>2"]}\\
  @{thm (lhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{text "if"} @{term "nullable(r\<^sub>1)"}\\
  & & @{text "then"} @{term "ALT (SEQ (der c r\<^sub>1) r\<^sub>2) (der c r\<^sub>2)"}\\
  & & @{text "else"} @{term "SEQ (der c r\<^sub>1) r\<^sub>2"}\\
  % & & @{thm (rhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]}\\
  @{thm (lhs) der.simps(6)} & $\dn$ & @{thm (rhs) der.simps(6)}
  \end{tabular}
  &
  \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  @{thm (lhs) nullable.simps(1)} & $\dn$ & @{thm (rhs) nullable.simps(1)}\\
  @{thm (lhs) nullable.simps(2)} & $\dn$ & @{thm (rhs) nullable.simps(2)}\\
  @{thm (lhs) nullable.simps(3)} & $\dn$ & @{thm (rhs) nullable.simps(3)}\\
  @{thm (lhs) nullable.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) nullable.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
  @{thm (lhs) nullable.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) nullable.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
  @{thm (lhs) nullable.simps(6)} & $\dn$ & @{thm (rhs) nullable.simps(6)}\medskip\\
  \end{tabular}  
\end{tabular}  
\end{center}

  \noindent
  We can extend this definition to give derivatives w.r.t.~strings:

  \begin{center}
  \begin{tabular}{cc}
  \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  @{thm (lhs) ders.simps(1)} & $\dn$ & @{thm (rhs) ders.simps(1)}
  \end{tabular}
  &
  \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  @{thm (lhs) ders.simps(2)} & $\dn$ & @{thm (rhs) ders.simps(2)}
  \end{tabular}
  \end{tabular}
  \end{center}

\noindent
Using @{text nullable} and the derivative operation, we can
define the following simple regular expression matcher:
%
\[
@{text "match s r"} \;\dn\; @{term nullable}(r\backslash s)
\]

\noindent This is essentially Brzozowski's algorithm from 1964. Its
main virtue is that the algorithm can be easily implemented as a
functional program (either in a functional programming language or in
a theorem prover). The correctness proof for @{text match} amounts to
establishing the property
%
\begin{proposition}\label{matchcorr} 
@{text "match s r"} \;\;\text{if and only if}\;\; $s \in L(r)$
\end{proposition}

\noindent
It is a fun exercise to formally prove this property in a theorem prover.

The novel idea of Sulzmann and Lu is to extend this algorithm for 
lexing, where it is important to find out which part of the string
is matched by which part of the regular expression.
For this Sulzmann and Lu presented two lexing algorithms in their paper
  \cite{Sulzmann2014}. The first algorithm consists of two phases: first a
  matching phase (which is Brzozowski's algorithm) and then a value
  construction phase. The values encode \emph{how} a regular expression
  matches a string. \emph{Values} are defined as the inductive datatype

  \begin{center}
  @{text "v :="}
  @{const "Void"} $\mid$
  @{term "val.Char c"} $\mid$
  @{term "Left v"} $\mid$
  @{term "Right v"} $\mid$
  @{term "Seq v\<^sub>1 v\<^sub>2"} $\mid$ 
  @{term "Stars vs"} 
  \end{center}  

  \noindent where we use @{term vs} to stand for a list of values. The
  string underlying a value can be calculated by a @{const flat}
  function, written @{term "flat DUMMY"}. It traverses a value and
  collects the characters contained in it. Sulzmann and Lu also define inductively an
  inhabitation relation that associates values to regular expressions:

  \begin{center}
  \begin{tabular}{c}
  \\[-8mm]
  @{thm[mode=Axiom] Prf.intros(4)} \qquad
  @{thm[mode=Axiom] Prf.intros(5)[of "c"]}\\[4mm]
  @{thm[mode=Rule] Prf.intros(2)[of "v\<^sub>1" "r\<^sub>1" "r\<^sub>2"]} \qquad 
  @{thm[mode=Rule] Prf.intros(3)[of "v\<^sub>2" "r\<^sub>1" "r\<^sub>2"]}\\[4mm]
  @{thm[mode=Rule] Prf.intros(1)[of "v\<^sub>1" "r\<^sub>1" "v\<^sub>2" "r\<^sub>2"]} \qquad
  @{thm[mode=Rule] Prf.intros(6)[of "vs" "r"]}
  \end{tabular}
  \end{center}

  \noindent Note that no values are associated with the regular expression
  @{term ZERO}, since it cannot match any string.
  It is routine to establish how values ``inhabiting'' a regular
  expression correspond to the language of a regular expression, namely

  \begin{proposition}
  @{thm L_flat_Prf}
  \end{proposition}

  In general there is more than one value inhabited by a regular
  expression (meaning regular expressions can typically match more
  than one string). But even when fixing a string from the language of the
  regular expression, there are generally more than one way of how the
  regular expression can match this string. POSIX lexing is about
  identifying the unique value for a given regular expression and a
  string that satisfies the informal POSIX rules (see
  \cite{POSIX,Kuklewicz,OkuiSuzuki2010,Sulzmann2014,Vansummeren2006}).\footnote{POSIX
	lexing acquired its name from the fact that the corresponding
	rules were described as part of the POSIX specification for
	Unix-like operating systems \cite{POSIX}.} Sometimes these
  informal rules are called \emph{maximal much rule} and \emph{rule priority}.
  One contribution of our earlier paper is to give a convenient
 specification for what POSIX values are (the inductive rules are shown in
  Figure~\ref{POSIXrules}).

\begin{figure}[t]
  \begin{center}
  \begin{tabular}{c}
  @{thm[mode=Axiom] Posix.intros(1)}\<open>P\<close>@{term "ONE"} \qquad
  @{thm[mode=Axiom] Posix.intros(2)}\<open>P\<close>@{term "c"}\medskip\\
  @{thm[mode=Rule] Posix.intros(3)[of "s" "r\<^sub>1" "v" "r\<^sub>2"]}\<open>P+L\<close>\qquad
  @{thm[mode=Rule] Posix.intros(4)[of "s" "r\<^sub>2" "v" "r\<^sub>1"]}\<open>P+R\<close>\medskip\\
  $\mprset{flushleft}
   \inferrule
   {@{thm (prem 1) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \qquad
    @{thm (prem 2) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \\\\
    @{thm (prem 3) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}
   {@{thm (concl) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}$\<open>PS\<close>\medskip\smallskip\\
  @{thm[mode=Axiom] Posix.intros(7)}\<open>P[]\<close>\qquad
  $\mprset{flushleft}
   \inferrule
   {@{thm (prem 1) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
    @{thm (prem 2) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
    @{thm (prem 3) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \\\\
    @{thm (prem 4) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}
   {@{thm (concl) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}$\<open>P\<star>\<close>\\[-4mm]
  \end{tabular}
  \end{center}
  \caption{The inductive definition of POSIX values taken from our earlier paper \cite{AusafDyckhoffUrban2016}. The ternary relation, written $(s, r) \rightarrow v$, formalises the notion
  of given a string $s$ and a regular
  expression $r$ what is the unique value $v$ that satisfies the informal POSIX constraints for
  regular expression matching.}\label{POSIXrules}
  \end{figure}

  The clever idea by Sulzmann and Lu \cite{Sulzmann2014} in their first algorithm is to define
  an injection function on values that mirrors (but inverts) the
  construction of the derivative on regular expressions. Essentially it
  injects back a character into a value.
  For this they define two functions called @{text mkeps} and @{text inj}:
 
  \begin{center}
  \begin{tabular}{l}
  \begin{tabular}{lcl}
  @{thm (lhs) mkeps.simps(1)} & $\dn$ & @{thm (rhs) mkeps.simps(1)}\\
  @{thm (lhs) mkeps.simps(2)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) mkeps.simps(2)[of "r\<^sub>1" "r\<^sub>2"]}\\
  @{thm (lhs) mkeps.simps(3)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) mkeps.simps(3)[of "r\<^sub>1" "r\<^sub>2"]}\\
  @{thm (lhs) mkeps.simps(4)} & $\dn$ & @{thm (rhs) mkeps.simps(4)}\\
  \end{tabular}\smallskip\\

  \begin{tabular}{lcl}
  @{thm (lhs) injval.simps(1)} & $\dn$ & @{thm (rhs) injval.simps(1)}\\
  @{thm (lhs) injval.simps(2)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1"]} & $\dn$ & 
      @{thm (rhs) injval.simps(2)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1"]}\\
  @{thm (lhs) injval.simps(3)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]} & $\dn$ & 
      @{thm (rhs) injval.simps(3)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]}\\
  @{thm (lhs) injval.simps(4)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]} & $\dn$ 
      & @{thm (rhs) injval.simps(4)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]}\\
  @{thm (lhs) injval.simps(5)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]} & $\dn$ 
      & @{thm (rhs) injval.simps(5)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]}\\
  @{thm (lhs) injval.simps(6)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]} & $\dn$ 
      & @{thm (rhs) injval.simps(6)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]}\\
  @{thm (lhs) injval.simps(7)[of "r" "c" "v" "vs"]} & $\dn$ 
      & @{thm (rhs) injval.simps(7)[of "r" "c" "v" "vs"]}
  \end{tabular}
  \end{tabular}
  \end{center}

  \noindent
  The function @{text mkeps} is run when the last derivative is nullable, that is
  the string to be matched is in the language of the regular expression. It generates
  a value for how the last derivative can match the empty string. The injection function
  then calculates the corresponding value for each intermediate derivative until
  a value for the original regular expression is generated.
  Graphically the algorithm by
  Sulzmann and Lu can be illustrated by the picture in Figure~\ref{Sulz}
  where the path from the left to the right involving @{term derivatives}/@{const
  nullable} is the first phase of the algorithm (calculating successive
  \Brz's derivatives) and @{const mkeps}/@{text inj}, the path from right to
  left, the second phase. The picture above shows the steps required when a
  regular expression, say @{text "r\<^sub>1"}, matches the string @{term
  "[a,b,c]"}. The first lexing algorithm by Sulzmann and Lu can be defined as:

  \begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=2,node distance=1.3cm,
                    every node/.style={minimum size=6mm}]
\node (r1)  {@{term "r\<^sub>1"}};
\node (r2) [right=of r1]{@{term "r\<^sub>2"}};
\draw[->,line width=1mm](r1)--(r2) node[above,midway] {@{term "der a DUMMY"}};
\node (r3) [right=of r2]{@{term "r\<^sub>3"}};
\draw[->,line width=1mm](r2)--(r3) node[above,midway] {@{term "der b DUMMY"}};
\node (r4) [right=of r3]{@{term "r\<^sub>4"}};
\draw[->,line width=1mm](r3)--(r4) node[above,midway] {@{term "der c DUMMY"}};
\draw (r4) node[anchor=west] {\;\raisebox{3mm}{@{term nullable}}};
\node (v4) [below=of r4]{@{term "v\<^sub>4"}};
\draw[->,line width=1mm](r4) -- (v4);
\node (v3) [left=of v4] {@{term "v\<^sub>3"}};
\draw[->,line width=1mm](v4)--(v3) node[below,midway] {\<open>inj r\<^sub>3 c\<close>};
\node (v2) [left=of v3]{@{term "v\<^sub>2"}};
\draw[->,line width=1mm](v3)--(v2) node[below,midway] {\<open>inj r\<^sub>2 b\<close>};
\node (v1) [left=of v2] {@{term "v\<^sub>1"}};
\draw[->,line width=1mm](v2)--(v1) node[below,midway] {\<open>inj r\<^sub>1 a\<close>};
\draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{@{term "mkeps"}}};
\end{tikzpicture}
\end{center}
\mbox{}\\[-13mm]

\caption{The two phases of the first algorithm by Sulzmann \& Lu \cite{Sulzmann2014},
matching the string @{term "[a,b,c]"}. The first phase (the arrows from 
left to right) is \Brz's matcher building successive derivatives. If the 
last regular expression is @{term nullable}, then the functions of the 
second phase are called (the top-down and right-to-left arrows): first 
@{term mkeps} calculates a value @{term "v\<^sub>4"} witnessing
how the empty string has been recognised by @{term "r\<^sub>4"}. After
that the function @{term inj} ``injects back'' the characters of the string into
the values. The value @{term "v\<^sub>1"} is the result of the algorithm representing
the POSIX value for this string and
regular expression.
\label{Sulz}}
\end{figure} 



  \begin{center}
  \begin{tabular}{lcl}
  @{thm (lhs) lexer.simps(1)} & $\dn$ & @{thm (rhs) lexer.simps(1)}\\
  @{thm (lhs) lexer.simps(2)} & $\dn$ & @{text "case"} @{term "lexer (der c r) s"} @{text of}\\
                     & & \phantom{$|$} @{term "None"}  @{text "\<Rightarrow>"} @{term None}\\
                     & & $|$ @{term "Some v"} @{text "\<Rightarrow>"} @{term "Some (injval r c v)"}                          
  \end{tabular}
  \end{center}


We have shown in our earlier paper \cite{AusafDyckhoffUrban2016} that
this algorithm is correct, that is it generates POSIX values. The
central property we established relates the derivative operation to the
injection function.

  \begin{proposition}\label{Posix2}
	\textit{If} $(s,\; r\backslash c) \rightarrow v$ \textit{then} $(c :: s,\; r) \rightarrow$ \textit{inj} $r\; c\; v$. 
\end{proposition}

  \noindent
  With this in place we were able to prove:


  \begin{proposition}\mbox{}\smallskip\\\label{lexercorrect}
  \begin{tabular}{ll}
  (1) & @{thm (lhs) lexer_correct_None} if and only if @{thm (rhs) lexer_correct_None}\\
  (2) & @{thm (lhs) lexer_correct_Some} if and only if @{thm (rhs) lexer_correct_Some}\\
  \end{tabular}
  \end{proposition}

  \noindent
  In fact we have shown that in the success case the generated POSIX value $v$ is
  unique and in the failure case that there is no POSIX value $v$ that satisfies
  $(s, r) \rightarrow v$. While the algorithm is correct, it is excruciatingly
  slow in cases where the derivatives grow arbitrarily (recall the example from the
  Introduction). However it can be used as a convenient reference point for the correctness
  proof of the second algorithm by Sulzmann and Lu, which we shall describe next.
  
*}

section {* Bitcoded Regular Expressions and Derivatives *}

text {*

  In the second part of their paper \cite{Sulzmann2014},
  Sulzmann and Lu describe another algorithm that also generates POSIX
  values but dispenses with the second phase where characters are
  injected ``back'' into values. For this they annotate bitcodes to
  regular expressions, which we define in Isabelle/HOL as the datatype

  \begin{center}
  \begin{tabular}{lcl}
  @{term breg} & $::=$ & @{term "AZERO"} $\quad\mid\quad$ @{term "AONE bs"}\\
               & $\mid$ & @{term "ACHAR bs c"}\\
               & $\mid$ & @{term "AALTs bs rs"}\\
               & $\mid$ & @{term "ASEQ bs r\<^sub>1 r\<^sub>2"}\\
               & $\mid$ & @{term "ASTAR bs r"}
  \end{tabular}
  \end{center}

  \noindent where @{text bs} stands for bitsequences; @{text r},
  @{text "r\<^sub>1"} and @{text "r\<^sub>2"} for bitcoded regular
  expressions; and @{text rs} for lists of bitcoded regular
  expressions. The binary alternative @{text "ALT bs r\<^sub>1 r\<^sub>2"}
  is just an abbreviation for \mbox{@{text "ALTs bs [r\<^sub>1, r\<^sub>2]"}}. 
  For bitsequences we use lists made up of the
  constants @{text Z} and @{text S}.  The idea with bitcoded regular
  expressions is to incrementally generate the value information (for
  example @{text Left} and @{text Right}) as bitsequences. For this 
  Sulzmann and Lu define a coding
  function for how values can be coded into bitsequences.

  \begin{center}
  \begin{tabular}{cc}
  \begin{tabular}{lcl}
  @{thm (lhs) code.simps(1)} & $\dn$ & @{thm (rhs) code.simps(1)}\\
  @{thm (lhs) code.simps(2)} & $\dn$ & @{thm (rhs) code.simps(2)}\\
  @{thm (lhs) code.simps(3)} & $\dn$ & @{thm (rhs) code.simps(3)}\\
  @{thm (lhs) code.simps(4)} & $\dn$ & @{thm (rhs) code.simps(4)}
  \end{tabular}
  &
  \begin{tabular}{lcl}
  @{thm (lhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]} & $\dn$ & @{thm (rhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]}\\
  @{thm (lhs) code.simps(6)} & $\dn$ & @{thm (rhs) code.simps(6)}\\
  @{thm (lhs) code.simps(7)} & $\dn$ & @{thm (rhs) code.simps(7)}\\
  \mbox{\phantom{XX}}\\
  \end{tabular}
  \end{tabular}
  \end{center}
   
  \noindent
  As can be seen, this coding is ``lossy'' in the sense that we do not
  record explicitly character values and also not sequence values (for
  them we just append two bitsequences). However, the
  different alternatives for @{text Left}, respectively @{text Right}, are recorded as @{text Z} and
  @{text S} followed by some bitsequence. Similarly, we use @{text Z} to indicate
  if there is still a value coming in the list of @{text Stars}, whereas @{text S}
  indicates the end of the list. The lossiness makes the process of
  decoding a bit more involved, but the point is that if we have a
  regular expression \emph{and} a bitsequence of a corresponding value,
  then we can always decode the value accurately. The decoding can be
  defined by using two functions called $\textit{decode}'$ and
  \textit{decode}:

  \begin{center}
  \begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}}
  $\textit{decode}'\,bs\,(\ONE)$ & $\dn$ & $(\Empty, bs)$\\
  $\textit{decode}'\,bs\,(c)$ & $\dn$ & $(\Char\,c, bs)$\\
  $\textit{decode}'\,(\Z\!::\!bs)\;(r_1 + r_2)$ & $\dn$ &
     $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r_1\;\textit{in}\;
       (\Left\,v, bs_1)$\\
  $\textit{decode}'\,(\S\!::\!bs)\;(r_1 + r_2)$ & $\dn$ &
     $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r_2\;\textit{in}\;
       (\Right\,v, bs_1)$\\                           
  $\textit{decode}'\,bs\;(r_1\cdot r_2)$ & $\dn$ &
        $\textit{let}\,(v_1, bs_1) = \textit{decode}'\,bs\,r_1\;\textit{in}$\\
  & &   $\textit{let}\,(v_2, bs_2) = \textit{decode}'\,bs_1\,r_2$
        \hspace{2mm}$\textit{in}\;(\Seq\,v_1\,v_2, bs_2)$\\
  $\textit{decode}'\,(\Z\!::\!bs)\,(r^*)$ & $\dn$ & $(\Stars\,[], bs)$\\
  $\textit{decode}'\,(\S\!::\!bs)\,(r^*)$ & $\dn$ & 
         $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r\;\textit{in}$\\
  & &   $\textit{let}\,(\Stars\,vs, bs_2) = \textit{decode}'\,bs_1\,r^*$
        \hspace{2mm}$\textit{in}\;(\Stars\,v\!::\!vs, bs_2)$\bigskip\\
  $\textit{decode}\,bs\,r$ & $\dn$ &
     $\textit{let}\,(v, bs') = \textit{decode}'\,bs\,r\;\textit{in}$\\
  & & \hspace{7mm}$\textit{if}\;bs' = []\;\textit{then}\;\textit{Some}\,v\;
       \textit{else}\;\textit{None}$   
  \end{tabular}    
  \end{center}

  \noindent
  The function \textit{decode} checks whether all of the bitsequence is
  consumed and returns the corresponding value as @{term "Some v"}; otherwise
  it fails with @{text "None"}. We can establish that for a value $v$
  inhabited by a regular expression $r$, the decoding of its
  bitsequence never fails.

\begin{lemma}\label{codedecode}\it
  If $\;\vdash v : r$ then
  $\;\textit{decode}\,(\textit{code}\, v)\,r = \textit{Some}\, v$.
\end{lemma}

\begin{proof}
  This follows from the property that
  $\textit{decode}'\,((\textit{code}\,v) \,@\, bs)\,r = (v, bs)$ holds
  for any bit-sequence $bs$ and $\vdash v : r$. This property can be
  easily proved by induction on $\vdash v : r$.
\end{proof}  

  Sulzmann and Lu define the function \emph{internalise}
  in order to transform (standard) regular expressions into annotated
  regular expressions. We write this operation as $r^\uparrow$.
  This internalisation uses the following
  \emph{fuse} function.	     

  \begin{center}
  \begin{tabular}{lcl}
  $\textit{fuse}\,bs\,(\textit{ZERO})$ & $\dn$ & $\textit{ZERO}$\\
  $\textit{fuse}\,bs\,(\textit{ONE}\,bs')$ & $\dn$ &
     $\textit{ONE}\,(bs\,@\,bs')$\\
  $\textit{fuse}\,bs\,(\textit{CHAR}\,bs'\,c)$ & $\dn$ &
     $\textit{CHAR}\,(bs\,@\,bs')\,c$\\
  $\textit{fuse}\,bs\,(\textit{ALTs}\,bs'\,rs)$ & $\dn$ &
     $\textit{ALTs}\,(bs\,@\,bs')\,rs$\\
  $\textit{fuse}\,bs\,(\textit{SEQ}\,bs'\,r_1\,r_2)$ & $\dn$ &
     $\textit{SEQ}\,(bs\,@\,bs')\,r_1\,r_2$\\
  $\textit{fuse}\,bs\,(\textit{STAR}\,bs'\,r)$ & $\dn$ &
     $\textit{STAR}\,(bs\,@\,bs')\,r$
  \end{tabular}    
  \end{center}    

  \noindent
  A regular expression can then be \emph{internalised} into a bitcoded
  regular expression as follows:

  \begin{center}
  \begin{tabular}{lcl}
  $(\ZERO)^\uparrow$ & $\dn$ & $\textit{ZERO}$\\
  $(\ONE)^\uparrow$ & $\dn$ & $\textit{ONE}\,[]$\\
  $(c)^\uparrow$ & $\dn$ & $\textit{CHAR}\,[]\,c$\\
  $(r_1 + r_2)^\uparrow$ & $\dn$ &
         $\textit{ALT}\;[]\,(\textit{fuse}\,[\Z]\,r_1^\uparrow)\,
                            (\textit{fuse}\,[\S]\,r_2^\uparrow)$\\
  $(r_1\cdot r_2)^\uparrow$ & $\dn$ &
         $\textit{SEQ}\;[]\,r_1^\uparrow\,r_2^\uparrow$\\
  $(r^*)^\uparrow$ & $\dn$ &
         $\textit{STAR}\;[]\,r^\uparrow$\\
  \end{tabular}    
  \end{center}    

  \noindent
  There is also an \emph{erase}-function, written $r^\downarrow$, which
  transforms a bitcoded regular expression into a (standard) regular
  expression by just erasing the annotated bitsequences. We omit the
  straightforward definition. For defining the algorithm, we also need
  the functions \textit{bnullable} and \textit{bmkeps}(\textit{s}), which are the
  ``lifted'' versions of \textit{nullable} and \textit{mkeps} acting on
  bitcoded regular expressions.
  %
  \begin{center}
  \begin{tabular}{@ {}c@ {}c@ {}}
  \begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  $\textit{bnullable}\,(\textit{ZERO})$ & $\dn$ & $\textit{false}$\\
  $\textit{bnullable}\,(\textit{ONE}\,bs)$ & $\dn$ & $\textit{true}$\\
  $\textit{bnullable}\,(\textit{CHAR}\,bs\,c)$ & $\dn$ & $\textit{false}$\\
  $\textit{bnullable}\,(\textit{ALTs}\,bs\,\rs)$ & $\dn$ &
     $\exists\, r \in \rs. \,\textit{bnullable}\,r$\\
  $\textit{bnullable}\,(\textit{SEQ}\,bs\,r_1\,r_2)$ & $\dn$ &
     $\textit{bnullable}\,r_1\wedge \textit{bnullable}\,r_2$\\
  $\textit{bnullable}\,(\textit{STAR}\,bs\,r)$ & $\dn$ &
     $\textit{true}$
  \end{tabular}
  &
  \begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}}
  $\textit{bmkeps}\,(\textit{ONE}\,bs)$ & $\dn$ & $bs$\\
  $\textit{bmkeps}\,(\textit{ALTs}\,bs\,\rs)$ & $\dn$ &
  $bs\,@\,\textit{bmkepss}\,\rs$\\
  $\textit{bmkeps}\,(\textit{SEQ}\,bs\,r_1\,r_2)$ & $\dn$ &\\
  \multicolumn{3}{r}{$bs \,@\,\textit{bmkeps}\,r_1\,@\, \textit{bmkeps}\,r_2$}\\
  $\textit{bmkeps}\,(\textit{STAR}\,bs\,r)$ & $\dn$ &
     $bs \,@\, [\S]$\\
  $\textit{bmkepss}\,(r\!::\!\rs)$ & $\dn$ &
     $\textit{if}\;\textit{bnullable}\,r$\\
  & &$\textit{then}\;\textit{bmkeps}\,r$\\
  & &$\textit{else}\;\textit{bmkepss}\,\rs$
  \end{tabular}
  \end{tabular}
  \end{center}    
 

  \noindent
  The key function in the bitcoded algorithm is the derivative of a
  bitcoded regular expression. This derivative calculates the
  derivative but at the same time also the incremental part of the bitsequences
  that contribute to constructing a POSIX value.	

  \begin{center}
  \begin{tabular}{@ {}lcl@ {}}
  $(\textit{ZERO})\backslash c$ & $\dn$ & $\textit{ZERO}$ \\  
  $(\textit{ONE}\;bs)\backslash c$ & $\dn$ & $\textit{ZERO}$\\  
  $(\textit{CHAR}\;bs\,d)\backslash c$ & $\dn$ &
        $\textit{if}\;c=d\; \;\textit{then}\;
         \textit{ONE}\;bs\;\textit{else}\;\textit{ZERO}$\\  
  $(\textit{ALTs}\;bs\,\rs)\backslash c$ & $\dn$ &
        $\textit{ALTs}\,bs\,(\mathit{map}\,(\_\backslash c)\,\rs)$\\
  $(\textit{SEQ}\;bs\,r_1\,r_2)\backslash c$ & $\dn$ &
     $\textit{if}\;\textit{bnullable}\,r_1$\\
  & &$\textit{then}\;\textit{ALT}\,bs\,(\textit{SEQ}\,[]\,(r_1\backslash c)\,r_2)$\\
  & &$\phantom{\textit{then}\;\textit{ALT}\,bs\,}(\textit{fuse}\,(\textit{bmkeps}\,r_1)\,(r_2\backslash c))$\\
  & &$\textit{else}\;\textit{SEQ}\,bs\,(r_1\backslash c)\,r_2$\\
  $(\textit{STAR}\,bs\,r)\backslash c$ & $\dn$ &
      $\textit{SEQ}\;bs\,(\textit{fuse}\, [\Z] (r\backslash c))\,
       (\textit{STAR}\,[]\,r)$
  \end{tabular}    
  \end{center}


  \noindent
  This function can also be extended to strings, written $r\backslash s$,
  just like the standard derivative.  We omit the details. Finally we
  can define Sulzmann and Lu's bitcoded lexer, which we call \textit{blexer}:

  \begin{center}
\begin{tabular}{lcl}
  $\textit{blexer}\;r\,s$ & $\dn$ &
      $\textit{let}\;r_{der} = (r^\uparrow)\backslash s\;\textit{in}$\\                
  & & $\;\;\;\;\textit{if}\; \textit{bnullable}(r_{der}) \;\;\textit{then}\;\textit{decode}\,(\textit{bmkeps}\,r_{der})\,r
       \;\;\textit{else}\;\textit{None}$
\end{tabular}
\end{center}

  \noindent
This bitcoded lexer first internalises the regular expression $r$ and then
builds the bitcoded derivative according to $s$. If the derivative is
(b)nullable the string is in the language of $r$ and it extracts the bitsequence using the
$\textit{bmkeps}$ function. Finally it decodes the bitsequence into a value.  If
the derivative is \emph{not} nullable, then $\textit{None}$ is
returned. We can show that this way of calculating a value
generates the same result as \textit{lexer}.

Before we can proceed we need to define a helper function, called
\textit{retrieve}, which Sulzmann and Lu introduced for the correctness proof.

\begin{center}
  \begin{tabular}{lcl}
  @{thm (lhs) retrieve.simps(1)} & $\dn$ & @{thm (rhs) retrieve.simps(1)}\\
  @{thm (lhs) retrieve.simps(2)} & $\dn$ & @{thm (rhs) retrieve.simps(2)}\\
  @{thm (lhs) retrieve.simps(3)} & $\dn$ & @{thm (rhs) retrieve.simps(3)}\\
  @{thm (lhs) better_retrieve(1)} & $\dn$ & @{thm (rhs) better_retrieve(1)}\\
  @{thm (lhs) better_retrieve(2)} & $\dn$ & @{thm (rhs) better_retrieve(2)}\\
  @{thm (lhs) retrieve.simps(6)[of _ "r\<^sub>1" "r\<^sub>2" "v\<^sub>1" "v\<^sub>2"]}
      & $\dn$ & @{thm (rhs) retrieve.simps(6)[of _ "r\<^sub>1" "r\<^sub>2" "v\<^sub>1" "v\<^sub>2"]}\\
  @{thm (lhs) retrieve.simps(7)} & $\dn$ & @{thm (rhs) retrieve.simps(7)}\\
  @{thm (lhs) retrieve.simps(8)} & $\dn$ & @{thm (rhs) retrieve.simps(8)}
  \end{tabular}
  \end{center}

\noindent
The idea behind this function is to retrieve a possibly partial
bitsequence from a bitcoded regular expression, where the retrieval is
guided by a value.  For example if the value is $\Left$ then we
descend into the left-hand side of an alternative in order to
assemble the bitcode. Similarly for
$\Right$. The property we can show is that for a given $v$ and $r$
with $\vdash v : r$, the retrieved bitsequence from the internalised
regular expression is equal to the bitcoded version of $v$.

\begin{lemma}\label{retrievecode}
If $\vdash v : r$ then $\textit{code}\, v = \textit{retrieve}\,(r^\uparrow)\,v$.
\end{lemma}

\noindent
We also need some auxiliary facts about how the bitcoded operations
relate to the ``standard'' operations on regular expressions. For
example if we build a bitcoded derivative and erase the result, this
is the same as if we first erase the bitcoded regular expression and
then perform the ``standard'' derivative operation.

\begin{lemma}\label{bnullable}\mbox{}\smallskip\\
  \begin{tabular}{ll}
\textit{(1)} & $(a\backslash s)^\downarrow = (a^\downarrow)\backslash s$\\    
\textit{(2)} & $\textit{bnullable}(a)$ iff $\textit{nullable}(a^\downarrow)$\\
\textit{(3)} & $\textit{bmkeps}(a) = \textit{retrieve}\,a\,(\textit{mkeps}\,(a^\downarrow))$ provided $\textit{nullable}(a^\downarrow)$.
\end{tabular}  
\end{lemma}

\begin{proof}
  All properties are by induction on annotated regular expressions. There are no
  interesting cases.
\end{proof}

\noindent
The only difficulty left for the correctness proof is that the bitcoded algorithm
has only a ``forward phase'' where POSIX values are generated incrementally.
We can achieve the same effect with @{text lexer} (which has two phases) by stacking up injection
functions during the forward phase. An auxiliary function, called $\textit{flex}$,
allows us to recast the rules of $\lexer$ in terms of a single
phase and stacked up injection functions.

\begin{center}
\begin{tabular}{lcl}
  $\textit{flex}\;r\,f\,[]$ & $\dn$ & $f$\\
  $\textit{flex}\;r\,f\,(c\!::\!s)$ & $\dn$ &
  $\textit{flex}\,(r\backslash c)\,(\lambda v.\,f\,(\inj\,r\,c\,v))\,s$\\
\end{tabular}    
\end{center}    

\noindent
The point of this function is that when
reaching the end of the string, we just need to apply the stacked up
injection functions to the value generated by @{text mkeps}.
Using this function we can recast the success case in @{text lexer} 
as follows:

\begin{proposition}\label{flex}
If @{text "lexer r s = Some v"} \;then\; @{text "v = "}$\,\textit{flex}\,r\,id\,s\,
      (\mkeps (r\backslash s))$.
\end{proposition}

\noindent
Note we did not redefine \textit{lexer}, we just established that the
value generated by \textit{lexer} can also be obtained by a different
method. While this different method is not efficient (we essentially
need to traverse the string $s$ twice, once for building the
derivative $r\backslash s$ and another time for stacking up injection
functions using \textit{flex}), it helps us with proving
that incrementally building up values in @{text blexer} generates the same result.

This brings us to our main lemma in this section: if we calculate a
derivative, say $r\backslash s$, and have a value, say $v$, inhabited
by this derivative, then we can produce the result @{text lexer} generates
by applying this value to the stacked-up injection functions
that $\textit{flex}$ assembles. The lemma establishes that this is the same
value as if we build the annotated derivative $r^\uparrow\backslash s$
and then retrieve the corresponding bitcoded version, followed by a
decoding step.

\begin{lemma}[Main Lemma]\label{mainlemma}\it
If $\vdash v : r\backslash s$ then 
\[\textit{Some}\,(\textit{flex}\,r\,\textit{id}\,s\,v) =
  \textit{decode}(\textit{retrieve}\,(r^\uparrow \backslash s)\,v)\,r\]
\end{lemma}  

\begin{proof}
  This can be proved by induction on $s$ and generalising over
  $v$. The interesting point is that we need to prove this in the
  reverse direction for $s$. This means instead of cases $[]$ and
  $c\!::\!s$, we have cases $[]$ and $s\,@\,[c]$ where we unravel the
  string from the back.\footnote{Isabelle/HOL provides an induction principle
    for this way of performing the induction.}

  The case for $[]$ is routine using Lemmas~\ref{codedecode}
  and~\ref{retrievecode}. In the case $s\,@\,[c]$, we can infer from
  the assumption that $\vdash v : (r\backslash s)\backslash c$
  holds. Hence by Prop.~\ref{Posix2} we know that 
  (*) $\vdash \inj\,(r\backslash s)\,c\,v : r\backslash s$ holds too.
  By definition of $\textit{flex}$ we can unfold the left-hand side
  to be
  \[
    \textit{Some}\,(\textit{flex}\;r\,\textit{id}\,(s\,@\,[c])\,v) =
    \textit{Some}\,(\textit{flex}\;r\,\textit{id}\,s\,(\inj\,(r\backslash s)\,c\,v))  
  \]  

  \noindent
  By induction hypothesis and (*) we can rewrite the right-hand side to
  %
  \[
    \textit{decode}\,(\textit{retrieve}\,(r^\uparrow\backslash s)\;
    (\inj\,(r\backslash s)\,c\,\,v))\,r
  \]

  \noindent
  which is equal to
  $\textit{decode}\,(\textit{retrieve}\, (r^\uparrow\backslash
  (s\,@\,[c]))\,v)\,r$ as required. The last rewrite step is possible
  because we generalised over $v$ in our induction.
\end{proof}  

\noindent
With this lemma in place, we can prove the correctness of \textit{blexer}---it indeed
produces the same result as \textit{lexer}.


\begin{theorem}\label{thmone}
$\textit{lexer}\,r\,s = \textit{blexer}\,r\,s$
\end{theorem}  

\begin{proof}
  We can first expand both sides using Prop.~\ref{flex} and the
  definition of \textit{blexer}. This gives us two
  \textit{if}-statements, which we need to show to be equal. By 
  Lemma~\ref{bnullable}\textit{(2)} we know the \textit{if}-tests coincide:
  \[
    \textit{bnullable}(r^\uparrow\backslash s) \;\textit{iff}\;
    \nullable(r\backslash s)
  \]

  \noindent
  For the \textit{if}-branch suppose $r_d \dn r^\uparrow\backslash s$ and
  $d \dn r\backslash s$. We have (*) @{text "nullable d"}. We can then show
  by Lemma~\ref{bnullable}\textit{(3)} that
  %
  \[
    \textit{decode}(\textit{bmkeps}\,r_d)\,r =
    \textit{decode}(\textit{retrieve}\,a\,(\textit{mkeps}\,d))\,r
  \]

  \noindent
  where the right-hand side is equal to
  $\textit{Some}\,(\textit{flex}\,r\,\textit{id}\,s\,(\textit{mkeps}\,
  d))$ by Lemma~\ref{mainlemma} (we know
  $\vdash \textit{mkeps}\,d : d$ by (*)).  This shows the
  \textit{if}-branches return the same value. In the
  \textit{else}-branches both \textit{lexer} and \textit{blexer} return
  \textit{None}. Therefore we can conclude the proof.
\end{proof}  

\noindent This establishes that the bitcoded algorithm by Sulzmann and
Lu \emph{without} simplification produces correct results. This was
only conjectured by Sulzmann and Lu in their paper
\cite{Sulzmann2014}. The next step is to add simplifications.

*}


section {* Simplification *}

text {*

     Derivatives as calculated by Brzozowski’s method are usually more
     complex regular expressions than the initial one; the result is
     that derivative-based matching and lexing algorithms are
     often abysmally slow if the ``growth problem'' is not addressed. As Sulzmann and Lu wrote, various
     optimisations are possible, such as the simplifications
     $\ZERO{}\,r \Rightarrow \ZERO$, $\ONE{}\,r \Rightarrow r$,
     $\ZERO{} + r \Rightarrow r$ and $r + r \Rightarrow r$. While these
     simplifications can considerably speed up the two algorithms  in many
     cases, they do not solve fundamentally the growth problem with
     derivatives. To see this let us return to the example from the
     Introduction that shows the derivatives for \mbox{@{text "(a + aa)\<^sup>*"}}.
     If we delete in the 3rd step all $\ZERO{}s$ and $\ONE$s according to
     the simplification rules shown above we obtain
     %
     \def\xll{\xrightarrow{\_\backslash{} [a, a, a]}}%%
     %
     \begin{equation}\label{derivex}
     (a + aa)^* \quad\xll\quad
      \underbrace{\mbox{$(\ONE + \ONE{}a) \cdot (a + aa)^*$}}_{r} \;+\;
     ((a + aa)^* + \underbrace{\mbox{$(\ONE + \ONE{}a) \cdot (a + aa)^*$}}_{r})
     \end{equation}

     \noindent This is a simpler derivative, but unfortunately we
     cannot make any further simplifications. This is a problem because
     the outermost alternatives contains two copies of the same
     regular expression (underlined with $r$). These copies will
     spawn new copies in later derivative steps and they in turn even more copies. This
     destroys any hope of taming the size of the derivatives.  But the
     second copy of $r$ in \eqref{derivex} will never contribute to a
     value, because POSIX lexing will always prefer matching a string
     with the first copy. So it could be safely removed without affecting the correctness of the algorithm.
     The dilemma with the simple-minded
     simplification rules above is that the rule $r + r \Rightarrow r$
     will never be applicable because as can be seen in this example the
     regular expressions are not next to each other but separated by another regular expression.

     But here is where Sulzmann and Lu's representation of generalised
     alternatives in the bitcoded algorithm shines: in @{term
     "ALTs bs rs"} we can define a more aggressive simplification by
     recursively simplifying all regular expressions in @{text rs} and
     then analyse the resulting list and remove any duplicates.
     Another advantage with the bitsequences in  bitcoded regular
     expressions is that they can be easily modified such that simplification does not
     interfere with the value constructions. For example we can ``flatten'', or
     de-nest, @{text ALTs} as follows
     %
     \[
     @{term "ALTs bs\<^sub>1 ((ALTs bs\<^sub>2 rs\<^sub>2) # rs\<^sub>1)"}
     \quad\xrightarrow{bsimp}\quad
     @{term "ALTs bs\<^sub>1 ((map (fuse bs\<^sub>2) rs\<^sub>2) # rs\<^sub>1)"}
     \]

     \noindent
     where we just need to fuse the bitsequence that has accumulated in @{text "bs\<^sub>2"}
     to the alternatives in @{text "rs\<^sub>2"}. As we shall show below this will
     ensure that the correct value corresponding to the original (unsimplified)
     regular expression can still be extracted. %In this way the value construction
     %is not affected by simplification. 

     However there is one problem with the definition for the more
     aggressive simplification rules described by Sulzmann and Lu. Recasting
     their definition with our syntax they define the step of removing
     duplicates as
     %
     \[ @{text "bsimp (ALTs bs rs)"} \dn @{text "ALTs
     bs (nup (map bsimp rs))"}
     \]

     \noindent where they first recursively simplify the regular
     expressions in @{text rs} (using @{text map}) and then use
     Haskell's @{text nub}-function to remove potential
     duplicates. While this makes sense when considering the example
     shown in \eqref{derivex}, @{text nub} is the inappropriate
     function in the case of bitcoded regular expressions. The reason
     is that in general the elements in @{text rs} will have a
     different annotated bitsequence and in this way @{text nub}
     will never find a duplicate to be removed. The correct way to
     handle this situation is to first \emph{erase} the regular
     expressions when comparing potential duplicates. This is inspired
     by Scala's list functions of the form \mbox{@{text "distinctBy rs f
     acc"}} where a function is applied first before two elements
     are compared. We define this function in Isabelle/HOL as

     \begin{center}
     \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
     @{thm (lhs) distinctBy.simps(1)} & $\dn$ & @{thm (rhs) distinctBy.simps(1)}\\
     @{thm (lhs) distinctBy.simps(2)} & $\dn$ & @{thm (rhs) distinctBy.simps(2)}
     \end{tabular}
     \end{center}

     \noindent where we scan the list from left to right (because we
     have to remove later copies). In @{text distinctBy}, @{text f} is a
     function and @{text acc} is an accumulator for regular
     expressions---essentially a set of regular expressions that we have already seen
     while scanning the list. Therefore we delete an element, say @{text x},
     from the list provided @{text "f x"} is already in the accumulator;
     otherwise we keep @{text x} and scan the rest of the list but 
     add @{text "f x"} as another ``seen'' element to @{text acc}. We will use
     @{term distinctBy} where @{text f} is the erase functions, @{term "erase (DUMMY)"},
     that deletes bitsequences from bitcoded regular expressions.
     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.

     Our simplification function depends on three helper functions, one is called
     @{text flts} and analyses lists of regular expressions coming from alternatives.
     It is defined as follows:

     \begin{center}
     \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
     @{thm (lhs) flts.simps(1)} & $\dn$ & @{thm (rhs) flts.simps(1)}\\
     @{thm (lhs) flts.simps(2)} & $\dn$ & @{thm (rhs) flts.simps(2)}\\
     @{thm (lhs) flts.simps(3)[of "bs'" "rs'"]} & $\dn$ & @{thm (rhs) flts.simps(3)[of "bs'" "rs'"]}\\
     \end{tabular}
     \end{center}

     \noindent
     The second clause of @{text flts} removes all instances of @{text ZERO} in alternatives and
     the second ``spills'' out nested alternatives (but retaining the
     bitsequence @{text "bs'"} accumulated in the inner alternative). There are
     some corner cases to be considered when the resulting list inside an alternative is
     empty or a singleton list. We take care of those cases in the
     @{text "bsimpALTs"} function; similarly we define a helper function that simplifies
     sequences according to the usual rules about @{text ZERO}s and @{text ONE}s:

     \begin{center}
     \begin{tabular}{c@ {\hspace{5mm}}c}
     \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
     @{text "bsimpALTs bs []"}  & $\dn$ & @{text "ZERO"}\\
     @{text "bsimpALTs bs [r]"} & $\dn$ & @{text "fuse bs r"}\\
     @{text "bsimpALTs bs rs"}  & $\dn$ & @{text "ALTs bs rs"}\\
     \mbox{}\\
     \end{tabular}
     &
     \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
     @{text "bsimpSEQ bs _ ZERO"}  & $\dn$ & @{text "ZERO"}\\
     @{text "bsimpSEQ bs ZERO _"} & $\dn$ & @{text "ZERO"}\\
     @{text "bsimpSEQ bs\<^sub>1 (ONE bs\<^sub>2) r\<^sub>2"}
         & $\dn$ & @{text "fuse (bs\<^sub>1 @ bs\<^sub>2) r\<^sub>2"}\\
     @{text "bsimpSEQ bs r\<^sub>1 r\<^sub>2"} & $\dn$ &  @{text "SEQ bs r\<^sub>1 r\<^sub>2"}
     \end{tabular}
     \end{tabular}
     \end{center}

     \noindent
     With this in place we can define our simplification function as

     \begin{center}
     \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
     @{thm (lhs) bsimp.simps(1)[of "bs" "r\<^sub>1" "r\<^sub>2"]} & $\dn$ &
         @{thm (rhs) bsimp.simps(1)[of "bs" "r\<^sub>1" "r\<^sub>2"]}\\
     @{thm (lhs) bsimp.simps(2)[of "bs" _]} & $\dn$ & @{thm (rhs) bsimp.simps(2)[of "bs" _]}\\
     @{text "bsimp r"} & $\dn$ & @{text r}
     \end{tabular}
     \end{center}

     \noindent
     As far as we can see, our recursive function @{term bsimp} simplifies regular
     expressions as intended by Sulzmann and Lu. There is no point in applying the
     @{text bsimp} function repeatedly (like the simplification in their paper which needs to be
     applied until a fixpoint is reached) because we can show that @{term bsimp} is idempotent,
     that is

     \begin{proposition}
     @{term "bsimp (bsimp r) = bsimp r"}
     \end{proposition}

     \noindent
     This can be proved by induction on @{text r} but requires a detailed analysis
     that the de-nesting of alternatives always results in a flat list of regular
     expressions. We omit the details since it does not concern the correctness proof.
     
     Next we can include simplification after each derivative step leading to the
     following notion of bitcoded derivatives:
     
     \begin{center}
      \begin{tabular}{cc}
      \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
      @{thm (lhs) bders_simp.simps(1)} & $\dn$ & @{thm (rhs) bders_simp.simps(1)}
      \end{tabular}
      &
      \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
      @{thm (lhs) bders_simp.simps(2)} & $\dn$ & @{thm (rhs) bders_simp.simps(2)}
      \end{tabular}
      \end{tabular}
      \end{center}

      \noindent
      and use it in the improved lexing algorithm defined as

     \begin{center}
\begin{tabular}{lcl}
  $\textit{blexer}^+\;r\,s$ & $\dn$ &
      $\textit{let}\;r_{der} = (r^\uparrow)\backslash_{bsimp}\, s\;\textit{in}$\\                
  & & $\;\;\;\;\textit{if}\; \textit{bnullable}(r_{der}) \;\;\textit{then}\;\textit{decode}\,(\textit{bmkeps}\,r_{der})\,r
       \;\;\textit{else}\;\textit{None}$
\end{tabular}
\end{center}

       \noindent The remaining task is to show that @{term blexer} and
       @{term "blexer_simp"} generate the same answers.

       When we first
       attempted this proof we encountered a problem with the idea
       in Sulzmann and Lu's paper where the argument seems to be to appeal
       again to the @{text retrieve}-function defined for the unsimplified version
       of the algorithm. But
       this does not work, because desirable properties such as
     %
     \[
     @{text "retrieve r v = retrieve (bsimp r) v"}
     \]

     \noindent do not hold under simplification---this property
     essentially purports that we can retrieve the same value from a
     simplified version of the regular expression. To start with @{text retrieve}
     depends on the fact that the value @{text v} correspond to the
     structure of the regular expressions---but the whole point of simplification
     is to ``destroy'' this structure by making the regular expression simpler.
     To see this consider the regular expression @{text "r = r' + 0"} and a corresponding
     value @{text "v = Left v'"}. If we annotate bitcodes to @{text "r"}, then 
     we can use @{text retrieve} and @{text v} in order to extract a corresponding
     bitsequence. The reason that this works is that @{text r} is an alternative
     regular expression and @{text v} a corresponding value. However, if we simplify
     @{text r}, then @{text v} does not correspond to the shape of the regular 
     expression anymore. So unless one can somehow
     synchronise the change in the simplified regular expressions with
     the original POSIX value, there is no hope of appealing to @{text retrieve} in the
     correctness argument for @{term blexer_simp}.

     We found it more helpful to introduce the rewriting systems shown in
     Figure~\ref{SimpRewrites}. The idea is to generate 
     simplified regular expressions in small steps (unlike the @{text bsimp}-function which
     does the same in a big step), and show that each of
     the small steps preserves the bitcodes that lead to the final POSIX value.
     The rewrite system is organised such that $\leadsto$ is for bitcoded regular
     expressions and $\stackrel{s}{\leadsto}$ for lists of bitcoded regular
     expressions. The former essentially implements the simplifications of
     @{text "bsimpSEQ"} and @{text flts}; while the latter implements the
     simplifications in @{text "bsimpALTs"}. We can show that any bitcoded
     regular expression reduces in zero or more steps to the simplified
     regular expression generated by @{text bsimp}:

     \begin{lemma}\label{lemone}
     @{thm[mode=IfThen] rewrites_to_bsimp}
     \end{lemma}

     \begin{proof}
     By induction on @{text r}. For this we can use the properties
     @{thm fltsfrewrites} and @{thm ss6_stronger}. The latter uses
     repeated applications of the $LD$ rule which allows the removal
     of duplicates that can recognise the same strings. 
     \end{proof}

     \noindent
     We can show that this rewrite system preserves @{term bnullable}, that 
     is simplification, essentially, does not affect nullability:

     \begin{lemma}
     @{thm[mode=IfThen] bnullable0(1)[of "r\<^sub>1" "r\<^sub>2"]}
     \end{lemma}

     \begin{proof}
     Straightforward mutual induction on the definition of $\leadsto$ and $\stackrel{s}{\leadsto}$.
     The only interesting case is the rule $LD$ where the property holds since by the side-conditions of that rule the empty string will
     be in both @{text "L (rs\<^sub>a @ [r\<^sub>1] @ rs\<^sub>b @ [r\<^sub>2] @ rs\<^sub>c)"} and
     @{text "L (rs\<^sub>a @ [r\<^sub>1] @ rs\<^sub>b @ rs\<^sub>c)"}.
     \end{proof}

     \noindent
     From this, we can show that @{text bmkeps} will produce the same bitsequence
     as long as one of the bitcoded regular expressions in $\leadsto$ is nullable (this lemma
     establishes the missing fact we were not able to establish using @{text retrieve}, as suggested
     in the paper by Sulzmannn and Lu). 


     \begin{lemma}\label{lemthree}
     @{thm[mode=IfThen] rewrite_bmkeps_aux(1)[of "r\<^sub>1" "r\<^sub>2"]}
     \end{lemma}

     \begin{proof}
     By straightforward mutual induction on the definition of $\leadsto$ and $\stackrel{s}{\leadsto}$.
     Again the only interesting case is the rule $LD$ where we need to ensure that
     \[
     @{text "bmkeps (rs\<^sub>a @ [r\<^sub>1] @ rs\<^sub>b @ [r\<^sub>2] @ rs\<^sub>c) =
        bmkeps (rs\<^sub>a @ [r\<^sub>1] @ rs\<^sub>b @ rs\<^sub>c)"}	
     \]

     \noindent holds. This is indeed the case because according to the POSIX rules the
     generated bitsequence is determined by the first alternative that can match the
     string (in this case being nullable).
     \end{proof}

     \noindent
     Crucial is also the fact that derivative steps and simplification steps can be interleaved,
     which is shown by the fact that $\leadsto$ is preserved under derivatives.

     \begin{lemma}
     @{thm[mode=IfThen] rewrite_preserves_bder(1)[of "r\<^sub>1" "r\<^sub>2"]}
     \end{lemma}

     \begin{proof}
     By straightforward mutual induction on the definition of $\leadsto$ and $\stackrel{s}{\leadsto}$.
     The case for $LD$ holds because @{term "L (erase (bder c r\<^sub>2)) \<subseteq> L (erase (bder c r\<^sub>1))"}
     if and only if @{term "L (erase (r\<^sub>2)) \<subseteq> L (erase (r\<^sub>1))"}.
     \end{proof}


     \noindent
     Using this fact together with Lemma~\ref{lemone} allows us to prove the central lemma
     that the unsimplified
     derivative (with a string @{term s}) reduces to the simplified derivative (with the same string).
     
     \begin{lemma}\label{lemtwo}
     @{thm[mode=IfThen] central}
     \end{lemma}

     \begin{proof}
     By reverse induction on @{term s} generalising over @{text r}.
     \end{proof}

     \noindent
     With these lemmas in place we can finally establish that @{term "blexer_simp"} and @{term "blexer"}
     generate the same value, and using Theorem~\ref{thmone} from the previous section that this value
     is indeed the POSIX value.
     
     \begin{theorem}
     @{thm[mode=IfThen] main_blexer_simp}
     \end{theorem}

     \begin{proof}
     By unfolding the definitions and using Lemmas~\ref{lemtwo} and \ref{lemthree}. 	
     \end{proof}
     
     \noindent
     This completes the correctness proof for the second POSIX lexing algorithm by Sulzmann and Lu.
     The interesting point of this algorithm is that the sizes of derivatives do not grow arbitrarily, which
     we shall show next.

   \begin{figure}[t]
  \begin{center}
  \begin{tabular}{c}
  @{thm[mode=Axiom] bs1[of _ "r\<^sub>2"]}$S\ZERO{}_l$\qquad
  @{thm[mode=Axiom] bs2[of _ "r\<^sub>1"]}$S\ZERO{}_r$\\
  @{thm[mode=Axiom] bs3[of "bs\<^sub>1" "bs\<^sub>2"]}$S\ONE$\\
  @{thm[mode=Rule] bs4[of "r\<^sub>1" "r\<^sub>2" _ "r\<^sub>3"]}SL\qquad
  @{thm[mode=Rule] bs5[of "r\<^sub>3" "r\<^sub>4" _ "r\<^sub>1"]}SR\\
  @{thm[mode=Axiom] bs6}$A0$\qquad
  @{thm[mode=Axiom] bs7}$A1$\\
  @{thm[mode=Rule] bs8[of "rs\<^sub>1" "rs\<^sub>2"]}$AL$\\
  @{thm[mode=Rule] ss2[of "rs\<^sub>1" "rs\<^sub>2"]}$LH$\qquad
  @{thm[mode=Rule] ss3[of "r\<^sub>1" "r\<^sub>2"]}$LT$\\
  @{thm[mode=Axiom] ss4}$L\ZERO$\qquad
  @{thm[mode=Axiom] ss5[of "bs" "rs\<^sub>1" "rs\<^sub>2"]}$LS$\medskip\\
  @{thm[mode=Rule] ss6[of "r\<^sub>2" "r\<^sub>1" "rs\<^sub>1" "rs\<^sub>2" "rs\<^sub>3"]}$LD$\\
  \end{tabular}
  \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
  regular expressions. Interesting is the $LD$ rule that allows copies of regular
  expressions be removed provided a regular expression earlier in the list can
  match the same strings.}\label{SimpRewrites}
  \end{figure}
*}

section {* Finiteness of Derivatives *}

text {*

In this section let us sketch our argument for why the size of the simplified
derivatives with the aggressive simplification function is finite. Suppose
we have a size function for bitcoded regular expressions, written
@{text "|r|"}, which counts the number of nodes if we regard $r$ as a tree
(we omit the precise definition). For this we show that for every $r$
there exists a bound $N$
such that 

\begin{center}
$\forall s. \; |@{term "bders_simp r s"}| < N$
\end{center}

\noindent
We prove this by induction on $r$. The base cases for @{term AZERO},
@{term "AONE bs"} and @{term "ACHAR bs c"} are straightforward. The interesting case is
for sequences of the form @{term "ASEQ bs r\<^sub>1 r\<^sub>2"}. In this case our induction
hypotheses state $\forall s. \; |@{term "bders_simp r\<^sub>1 s"}| < N_1$ and
$\forall s. \; |@{term "bders_simp r\<^sub>2 s"}| < N_2$. We can reason as follows

\begin{center}
\begin{tabular}{lcll}
& & $ |@{term "bders_simp (ASEQ bs r\<^sub>1 r\<^sub>2) s"}|$\\
& $ = $ & $|bsimp(ALTs\;bs\;((@{term "bders_simp r\<^sub>1 s"}) \cdot r_2) ::
    [@{term "bders_simp r\<^sub>2 s'"} \;|\; s' \in Suf\!fix(s)])| $ & (1) \\
& $\leq$ &
    $|distinctBy\,(flts\,((@{term "bders_simp r\<^sub>1 s "}) \cdot r_2) ::
    [@{term "bders_simp r\<^sub>2 s'"} \;|\; s' \in Suf\!fix(s)])| + 1 $ & (2) \\
& $\leq$ & $|(@{term "bders_simp r\<^sub>1 s"}) \cdot r_2| +
             |distinctBy\,(flts\,   [@{term "bders_simp r\<^sub>2 s'"} \;|\; s' \in Suf\!fix(s)])| + 1 $ & (3) \\
& $\leq$ & $N_1 + |r_2| + 2 + |distinctBy\,(flts\,   [@{term "bders_simp r\<^sub>2 s'"} \;|\; s' \in Suf\!fix(s)])|$ & (4)\\
& $\leq$ & $N_1 + |r_2| + 2 + l_{N_{2}} * N_{2}$ & (5)
\end{tabular}
\end{center}

% tell Chengsong about Indian paper of closed forms of derivatives

\noindent
where in (1) the $Suf\!fix(s')$ are the suffixes where @{term "bders_simp r\<^sub>1 s''"} is nullable for
@{text "s = s'' @ s'"}. In (3) we know that  $|(@{term "bders_simp r\<^sub>1 s"}) \cdot r_2|$ is 
bounded by $N_1 + |r_2|$. In (5) we know the list comprehension contains only regular expressions of size smaller
than $N_2$. The list length after @{text distinctBy} is bounded by a number, which we call $l_{N_2}$. It stands
for the number of distinct regular expressions with a maximum size $N_2$ (there can only be finitely many of them).
We reason similarly in the @{text Star}-case.\medskip

\noindent
Clearly we give in this finiteness argument (Step (5)) a very loose bound that is
far from the actual bound we can expect. We can do better than this, but this does not improve
the finiteness property we are proving. If we are interested in a polynomial bound,
one would hope to obtain a similar tight bound as for partial
derivatives introduced by Antimirov \cite{Antimirov95}. After all the idea with
@{text distinctBy} is to maintain a ``set'' of alternatives (like the sets in
partial derivatives). Unfortunately to obtain the exact same bound would mean
we need to introduce simplifications such as
%
\[ (r_1 + r_2) \cdot r_3 \longrightarrow (r_1 \cdot r_3) + (r_2 \cdot r_3)
\]

\noindent
which exist for partial derivatives. However, if we introduce them in our
setting we would lose the POSIX property of our calculated values. We leave better
bounds for future work.

*}


section {* Conclusion *}

text {*

   We set out in this work to prove in Isabelle/HOL the correctness of
   the second POSIX lexing algorithm by Sulzmann and Lu
   \cite{Sulzmann2014}. This follows earlier work where we established
   the correctness of the first algorithm
   \cite{AusafDyckhoffUrban2016}. In the earlier work we needed to
   introduce our own specification about what POSIX values are,
   because the informal definition given by Sulzmann and Lu did not
   stand up to a formal proof. Also for the second algorithm we needed
   to introduce our own definitions and proof ideas in order to establish the
   correctness.  Our interest in the second algorithm 
   lies in the fact that by using bitcoded regular expressions and an aggressive
   simplification method there is a chance that the the derivatives
   can be kept universally small  (we established in this paper that
   they can be kept finite for any string). This is important if one is after
   an efficient POSIX lexing algorithm.

   Having proved the correctness of the POSIX lexing algorithm, which
   lessons have we learned? Well, we feel this is a very good example
   where formal proofs give further insight into the matter at
   hand. For example it is very hard to see a problem with @{text nub}
   vs @{text distinctBy} with only experimental data---one would still
   see the correct result but find that simplification does not simplify in well-chosen, but not
   obscure, examples. We found that from an implementation
   point-of-view it is really important to have the formal proofs of
   the corresponding properties at hand. We have also developed a
   healthy suspicion when experimental data is used to back up
   efficiency claims. For example Sulzmann and Lu write about their
   equivalent of @{term blexer_simp} ``...we can incrementally compute
   bitcoded parse trees in linear time in the size of the input''
   \cite[Page 14]{Sulzmann2014}. 
   Given the growth of the
   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 POSIX matches and is based on
   derivatives \cite{verbatim}. In this case derivatives are not simplified.
   Clearly our result of having finite
   derivatives is rather weak in this context but we think such effeciency claims
   require further scrutiny.\medskip

   \noindent
   Our Isabelle/HOL code is available under \url{https://github.com/urbanchr/posix}.


%%\bibliographystyle{plain}
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*}

(*<*)
end
(*>*)