(*<*)

theory Paper

imports "../ReStar" "~~/src/HOL/Library/LaTeXsugar"

begin

declare [[show_question_marks = false]]

abbreviation 

 "der_syn r c \<equiv> der c r"

abbreviation 

 "ders_syn r s \<equiv> ders s r"

notation (latex output)

  If  ("(\<^raw:\textrm{>if\<^raw:}> (_)/ \<^raw:\textrm{>then\<^raw:}> (_)/ \<^raw:\textrm{>else\<^raw:}> (_))" 10) and

  Cons ("_\<^raw:\mbox{$\,$}>::\<^raw:\mbox{$\,$}>_" [75,75] 73) and  

  ZERO ("\<^bold>0" 78) and 

  ONE ("\<^bold>1" 78) and 

  CHAR ("_" [1000] 80) and

  ALT ("_ + _" [77,77] 78) and

  SEQ ("_ \<cdot> _" [77,77] 78) and

  STAR ("_\<^sup>\<star>" [1000] 78) and

  val.Void ("'(')" 79) and

  val.Char ("Char _" [1000] 79) and

  val.Left ("Left _" [79] 78) and

  val.Right ("Right _" [79] 78) and

  val.Seq ("Seq _ _" [79,79] 78) and

  val.Stars ("Stars _" [79] 78) and

  L ("L'(_')" [10] 78) and

  der_syn ("_\\_" [79, 1000] 76) and  

  ders_syn ("_\\_" [79, 1000] 76) and

  flat ("|_|" [75] 73) and

  Sequ ("_ @ _" [78,77] 63) and

  injval ("inj _ _ _" [79,77,79] 76) and 

  mkeps ("mkeps _" [79] 76) and 

  projval ("proj _ _ _" [1000,77,1000] 77) and 

  length ("len _" [78] 73) and

  matcher ("lexer _ _" [78,78] 77) and

  Prf ("\<triangleright> _ : _" [75,75] 75) and  

  PMatch ("'(_, _') \<rightarrow> _" [63,75,75] 75)

  (* and ValOrd ("_ \<succeq>\<^bsub>_\<^esub> _" [78,77,77] 73) *)

definition 

  "match r s \<equiv> nullable (ders s r)"

(*>*)

section {* Introduction *}

text {*

Brzozowski \cite{Brzozowski1964} introduced the notion of the {\em

derivative} @{term "der c r"} of a regular expression @{text r} w.r.t.\ a

character~@{text c}, and showed that it gave 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}). The derivative has the property (which may be

regarded as its specification) that, for every string @{term s} and regular

expression @{term r} and character @{term c}, one has @{term "cs \<in> L(r)"} if

and only if \mbox{@{term "s \<in> L(der c r)"}}. The beauty of Brzozowski's

derivatives 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. A

completely formalised correctness proof of this matcher in for example HOL4

has been mentioned in~\cite{Owens2008}. Another one in Isabelle/HOL is

in \cite{Krauss2011}.

One limitation of Brzozowski's matcher is that it only generates a YES/NO

answer for whether a string is being matched by a regular expression.

Sulzmann and Lu \cite{Sulzmann2014} extended this matcher to allow

generation not just of a YES/NO answer but of an actual matching, called a

[lexical] {\em value}. They give a simple algorithm to calculate a value

that appears to be the value associated with POSIX matching

\cite{Kuklewicz,Vansummeren2006}. The challenge then is to specify that

value, in an algorithm-independent fashion, and to show that Sulzamann and

Lu's derivative-based algorithm does indeed calculate a value that is

correct according to the specification.

The answer given by Sulzmann and Lu \cite{Sulzmann2014} is to define a

relation (called an ``Order Relation'') on the set of values of @{term r},

and to show that (once a string to be matched is chosen) there is a maximum

element and that it is computed by their derivative-based algorithm. This

proof idea is inspired by work of Frisch and Cardelli \cite{Frisch2004} on a

GREEDY regular expression matching algorithm. Beginning with our

observations that, without evidence that it is transitive, it cannot be

called an ``order relation'', and that the relation is called a ``total

order'' despite being evidently not total\footnote{The relation @{text

"\<ge>\<^bsub>r\<^esub>"} defined in \cite{Sulzmann2014} is a relation on the

values for the regular expression @{term r}; but it only holds between

@{term v} and @{term "v'"} in cases where @{term v} and @{term "v'"} have

the same flattening (underlying string). So a counterexample to totality is

given by taking two values @{term v} and @{term "v'"} for @{term r} that

have different flattenings (see Section~\ref{posixsec}). A different

relation @{text "\<ge>\<^bsub>r,s\<^esub>"} on the set of values for @{term r}

with flattening @{term s} is definable by the same approach, and is indeed

total; but that is not what Proposition 1 of \cite{Sulzmann2014} does.}, we

identify problems with this approach (of which some of the proofs are not

published in \cite{Sulzmann2014}); perhaps more importantly, we give a

simple inductive (and algorithm-independent) definition of what we call

being a {\em POSIX value} for a regular expression @{term r} and a string

@{term s}; we show that the algorithm computes such a value and that such a

value is unique. Proofs are both done by hand and checked in Isabelle/HOL.

The experience of doing our proofs has been that this mechanical checking

was absolutely essential: this subject area has hidden snares. This was also

noted by Kuklewitz \cite{Kuklewicz} who found that nearly all POSIX matching

implementations are ``buggy'' \cite[Page 203]{Sulzmann2014}.

If a regular expression matches a string, then in general there is more than

one way of how the string is matched. There are two commonly used

disambiguation strategies to generate a unique answer: one is called GREEDY

matching \cite{Frisch2004} and the other is POSIX

matching~\cite{Kuklewicz,Sulzmann2014,Vansummeren2006}. For example consider

the string @{term xy} and the regular expression \mbox{@{term "STAR (ALT

(ALT x y) xy)"}}. Either the string can be matched in two `iterations' by

the single letter-regular expressions @{term x} and @{term y}, or directly

in one iteration by @{term xy}. The first case corresponds to GREEDY

matching, which first matches with the left-most symbol and only matches the

next symbol in case of a mismatch (this is greedy in the sense of preferring

instant gratification to delayed repletion). The second case is POSIX

matching, which prefers the longest match.

In the context of lexing, where an input string needs to be split up into a

sequence of tokens, POSIX is the more natural disambiguation strategy for

what programmers consider basic syntactic building blocks in their programs.

These building blocks are often specified by some regular expressions, say

@{text "r\<^bsub>key\<^esub>"} and @{text "r\<^bsub>id\<^esub>"} for recognising keywords and

identifiers, respectively. There are two underlying (informal) rules behind

tokenising a string in a POSIX fashion:

\begin{itemize} 

\item[$\bullet$] \underline{The Longest Match Rule (or ``maximal munch rule''):}

The longest initial substring matched by any regular expression is taken as

next token.\smallskip

For a particular longest initial substring, the first regular expression

that can match determines the token.

\end{itemize}

\noindent Consider for example @{text "r\<^bsub>key\<^esub>"} recognising keywords such as

@{text "if"}, @{text "then"} and so on; and @{text "r\<^bsub>id\<^esub>"} recognising

identifiers (say, a single character followed by characters or numbers). Then we

can form the regular expression @{text "(r\<^bsub>key\<^esub> + r\<^bsub>id\<^esub>)\<^sup>\<star>"} and use POSIX

matching to tokenise strings, say @{text "iffoo"} and @{text "if"}. In the

first case we obtain by the longest match rule a single identifier token,

not a keyword followed by an identifier. In the second case we obtain by rule

priority a keyword token, not an identifier token---even if @{text "r\<^bsub>id\<^esub>"}

matches also.\bigskip

\noindent {\bf Contributions:} (NOT DONE YET) We have implemented in Isabelle/HOL the

derivative-based regular expression matching algorithm as described by

Sulzmann and Lu \cite{Sulzmann2014}. We have proved the correctness of this

algorithm according to our specification of what a POSIX value is. The

informal correctness proof given in \cite{Sulzmann2014} is in final

form\footnote{} and to us contains unfillable gaps. Our specification of a

POSIX value consists of a simple inductive definition that given a string

and a regular expression uniquely determines this value. Derivatives as

calculated by Brzozowski's method are usually more complex regular

expressions than the initial one; various optimisations are possible, such

as the simplifications of @{term "ALT ZERO r"}, @{term "ALT r ZERO"}, @{term

"SEQ ONE r"} and @{term "SEQ r ONE"} to @{term r}. One of the advantages of

having a simple specification and correctness proof is that the latter can

be refined to allow for such optimisations and simple correctness proof.

An extended version of \cite{Sulzmann2014} is available at the website of

its first author; this includes some ``proofs'', claimed in

\cite{Sulzmann2014} to be ``rigorous''. Since these are evidently not in

final form, we make no comment thereon, preferring to give general reasons

for our belief that the approach of \cite{Sulzmann2014} is problematic

rather than to discuss details of unpublished work.

*}

section {* Preliminaries *}

text {* \noindent Strings in Isabelle/HOL are lists of characters with the

empty string being represented by the empty list, written @{term "[]"}, and

list-cons being written as @{term "DUMMY # DUMMY"}. Often we use the usual

bracket notation for lists also for strings; for example a string consisting

of just a single character @{term c} is written @{term "[c]"}. By using the

type @{type char} for characters we have a supply of finitely many

characters roughly corresponding to the ASCII character set. 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 "CHAR 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

  language of a regular expression is also defined as usual by the

  recursive function @{term L} with the clauses:

  \begin{center}

  \begin{tabular}{rcl}

  @{thm (lhs) L.simps(1)} & $\dn$ & @{thm (rhs) L.simps(1)}\\

  @{thm (lhs) L.simps(2)} & $\dn$ & @{thm (rhs) L.simps(2)}\\

  @{thm (lhs) L.simps(3)} & $\dn$ & @{thm (rhs) L.simps(3)}\\

  @{thm (lhs) L.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) L.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\

  @{thm (lhs) L.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) L.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\

  @{thm (lhs) L.simps(6)} & $\dn$ & @{thm (rhs) L.simps(6)}\\

  \end{tabular}

  \end{center}

  \noindent In the fourth clause we use the operation @{term "DUMMY ;;

  DUMMY"} for the concatenation of two languages (it is also list-append for

  strings). We use the star-notation for regular expressions and languages

  (in the last clause above). The star on languages is defined inductively

  by two clauses: @{text "(i)"} for the empty string being in the star of a

  language and @{text "(ii)"} if @{term "s\<^sub>1"} is in a language and @{term

  "s\<^sub>2"} in the star of this language, then also @{term "s\<^sub>1 @ s\<^sub>2"} is in

  the star of this language. It will also be convenient to use the following

  notion of a \emph{semantic derivative} (or \emph{left quotient}) of a

  language, say @{text A}, defined as:

  \begin{center}

  \begin{tabular}{lcl}

  @{thm (lhs) Der_def} & $\dn$ & @{thm (rhs) Der_def}\\

  \end{tabular}

  \end{center}

  \noindent 

  For semantic derivatives we have the following equations (for example

  mechanically proved in \cite{Krauss2011}):

  \begin{equation}\label{SemDer}

  \begin{array}{lcl}

  @{thm (lhs) Der_null}  & \dn & @{thm (rhs) Der_null}\\

  @{thm (lhs) Der_empty}  & \dn & @{thm (rhs) Der_empty}\\

  @{thm (lhs) Der_char}  & \dn & @{thm (rhs) Der_char}\\

  @{thm (lhs) Der_union}  & \dn & @{thm (rhs) Der_union}\\

  @{thm (lhs) Der_Sequ}  & \dn & @{thm (rhs) Der_Sequ}\\

  @{thm (lhs) Der_star}  & \dn & @{thm (rhs) Der_star}

  \end{array}

  \end{equation}

  \noindent \emph{\Brz's derivatives} of regular expressions

  \cite{Brzozowski1964} can be easily defined by two recursive functions:

  the first is from regular expressions to booleans (implementing a test

  when a regular expression can match the empty string), and the second

  takes a regular expression and a character to a (derivative) regular

  expression:

  \begin{center}

  \begin{tabular}{lcl}

  @{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\\

  @{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"]} & $\dn$ & @{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}

  \end{center}

  \noindent

  We may extend this definition to give derivatives w.r.t.~strings:

  \begin{center}

  \begin{tabular}{lcl}

  @{thm (lhs) ders.simps(1)} & $\dn$ & @{thm (rhs) ders.simps(1)}\\

  @{thm (lhs) ders.simps(2)} & $\dn$ & @{thm (rhs) ders.simps(2)}\\

  \end{tabular}

  \end{center}

  \noindent Given the equations in \eqref{SemDer}, it is a relatively easy

  exercise in mechanical reasoning to establish that

  \begin{proposition}\mbox{}\\ 

  \begin{tabular}{ll}

  @{text "(1)"} & @{thm (lhs) nullable_correctness} if and only if

  @{thm (rhs) nullable_correctness}, and \\ 

  @{text "(2)"} & @{thm[mode=IfThen] der_correctness}.

  \end{tabular}

  \end{proposition}

  \noindent With this in place it is also very routine to prove that the

  regular expression matcher defined as

  \begin{center}

  @{thm match_def}

  \end{center}

  \noindent gives a positive answer if and only if @{term "s \<in> L r"}.

  Consequently, this regular expression matching algorithm satisfies the

  usual specification. While the matcher above calculates a provably correct

  a YES/NO answer for whether a regular expression matches a string, the

  novel idea of Sulzmann and Lu \cite{Sulzmann2014} is to append another

  phase to this algorithm in order to calculate a [lexical] value. We will

  explain the details next.

*}

section {* POSIX Regular Expression Matching\label{posixsec} *}

text {* 

The clever idea in \cite{Sulzmann2014} is to introduce values for encoding

\emph{how} a regular expression matches a string and then define a function

on values that mirrors (but inverts) the construction of the derivative on

regular expressions. \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} standing for a list of values. (This is

  similar to the approach taken by Frisch and Cardelli for GREEDY matching

  \cite{Frisch2014}, and Sulzmann and Lu \cite{2014} for POSIX matching).

  The string underlying a value can be calculated by the @{const flat}

  function, written @{term "flat DUMMY"} and defined as:

  \begin{center}

  \begin{tabular}{lcl}

  @{thm (lhs) flat.simps(1)} & $\dn$ & @{thm (rhs) flat.simps(1)}\\

  @{thm (lhs) flat.simps(2)} & $\dn$ & @{thm (rhs) flat.simps(2)}\\

  @{thm (lhs) flat.simps(3)} & $\dn$ & @{thm (rhs) flat.simps(3)}\\

  @{thm (lhs) flat.simps(4)} & $\dn$ & @{thm (rhs) flat.simps(4)}\\

  @{thm (lhs) flat.simps(5)[of "v\<^sub>1" "v\<^sub>2"]} & $\dn$ & @{thm (rhs) flat.simps(5)[of "v\<^sub>1" "v\<^sub>2"]}\\

  @{thm (lhs) flat.simps(6)} & $\dn$ & @{thm (rhs) flat.simps(6)}\\

  @{thm (lhs) flat.simps(7)} & $\dn$ & @{thm (rhs) flat.simps(7)}\\

  \end{tabular}

  \end{center}

  \noindent Sulzmann and Lu also define inductively an inhabitation relation

  that associates values to regular expressions:

  \begin{center}

  \begin{tabular}{c}

  @{thm[mode=Axiom] Prf.intros(4)} \qquad

  @{thm[mode=Axiom] Prf.intros(5)[of "c"]}\medskip\\

  @{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"]}\medskip\\

  @{thm[mode=Rule] Prf.intros(1)[of "v\<^sub>1" "r\<^sub>1" "v\<^sub>2" "r\<^sub>2"]}\medskip\\ 

  @{thm[mode=Axiom] Prf.intros(6)[of "r"]} \qquad  

  @{thm[mode=Rule] Prf.intros(7)[of "v" "r" "vs"]}\medskip\\

  \end{tabular}

  \end{center}

  \noindent Note that no values are associated with the regular expression

  @{term ZERO}, and that the only value associated with the regular

  expression @{term ONE} is @{term Void}, pronounced (if one must) as {\em

  expression correspond to the language of a regular expression, namely

  \begin{proposition}

  @{thm L_flat_Prf}

  \end{proposition}

  In general there are more than one value associated with a regular

  expression. In case of POSIX matching the problem is to calculate the

  unique value that satisfies the (informal) POSIX constraints from the

  Introduction. Graphically the regular expression matching 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 @{const der}/@{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. This picture shows the steps required when a

  regular expression, say @{text "r\<^sub>1"}, matches the string @{term

  "[a,b,c]"}. We first build the three derivatives (according to @{term a},

  @{term b} and @{term c}). We then use @{const nullable} to find out

  whether the resulting derivative regular expression @{term "r\<^sub>4"}

  can match the empty string. If yes, we call the function @{const mkeps}

  that produces a value @{term "v\<^sub>4"} for how @{term "r\<^sub>4"} can

  match the empty string (taking into account the POSIX constraints in case

  there are several ways). This functions is defined by the clauses:

\begin{figure}[t]

\begin{center}

\begin{tikzpicture}[scale=2,node distance=1.3cm,

                    every node/.style={minimum size=7mm}]

\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] {@{text "inj r\<^sub>3 c"}};

\node (v2) [left=of v3]{@{term "v\<^sub>2"}};

\draw[->,line width=1mm](v3)--(v2) node[below,midway] {@{text "inj r\<^sub>2 b"}};

\node (v1) [left=of v2] {@{term "v\<^sub>1"}};

\draw[->,line width=1mm](v2)--(v1) node[below,midway] {@{text "inj r\<^sub>1 a"}};

\draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{@{term "mkeps"}}};

\end{tikzpicture}

\end{center}

\caption{The two phases of the 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 succesive derivatives. If at the 

last regular expression is @{term nullable}, then functions of the 

second phase are called: first @{term mkeps} calculates a value 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 arrows from right to left).

\label{Sulz}}

\end{figure} 

  \begin{center}

  \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}

  \end{center}

  \noindent Note that this function needs only to be partially defined,

  namely only for regular expressions that are nullable. In case @{const

  nullable} fails, the string @{term "[a,b,c]"} cannot be matched by @{term

  "r\<^sub>1"} and an error is raised. Note also how this function makes

  some subtle choices leading to a POSIX value: for example if the

  alternative, say @{term "ALT r\<^sub>1 r\<^sub>2"}, can match the empty

  string and furthermore @{term "r\<^sub>1"} can match the empty string,

  then we return a @{const Left}-value. The @{const Right}-value will only

  be returned if @{term "r\<^sub>1"} is not nullable.

  The most interesting novelty from Sulzmann and Lu \cite{Sulzmann2014} is

  the construction value for how @{term "r\<^sub>1"} can match the string

  @{term "[a,b,c]"} from the value how the last derivative, @{term

  "r\<^sub>4"} in Fig~\ref{Sulz}, can match the empty string. Sulzmann and

  Lu acchieve this by stepwise ``injecting back'' the characters into the

  values thus inverting the operation of building derivatives on the level

  of values. The corresponding function, called @{term inj}, takes three

  arguments, a regular expression, a character and a value. For example in

  the first @{term inj}-step in Fig~\ref{Sulz} the regular expression @{term

  "r\<^sub>3"}, the character @{term c} from the last derivative step and

  @{term "v\<^sub>4"}, which is the value corresponding to the derivative

  regular expression @{term "r\<^sub>4"}. The result is the new value @{term

  "v\<^sub>3"}. The final result of the algorithm is the value @{term

  "v\<^sub>1"} corresponding to the input regular expression. The @{term

  inj} function is by recursion on the regular expression and by analysing

  the shape of values.

  \begin{center}

  \begin{tabular}{llcl}

  (1) & @{thm (lhs) injval.simps(1)} & $\dn$ & @{thm (rhs) injval.simps(1)}\\

  (2) & @{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"]}\\

  (3) & @{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"]}\\

  (4) & @{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"]}\\

  (5) & @{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"]}\\

  (6) & @{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"]}\\

  (7) & @{thm (lhs) injval.simps(7)[of "r" "c" "v" "vs"]} & $\dn$ 

      & @{thm (rhs) injval.simps(7)[of "r" "c" "v" "vs"]}\\

  \end{tabular}

  \end{center}

  \noindent To better understand what is going on in this definition it

  might be instructive to look first at the three sequence cases (clauses

  (4)--(6)). In each case we need to construct an ``injected value'' for @{term