(*<*)
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{$\,$}>_" [75,73] 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] 74) 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 ("_ : _" [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 almost 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 part
of the work 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 Sulzmann 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

\item[$\bullet$] \underline{Priority Rule:}

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"}.
For @{text "iffoo"} we obtain by the longest match rule a single identifier
token, not a keyword followed by an identifier. For @{text "if"} we obtain by
the priority rule 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. Sulzmann and Lu sketch in \cite{Sulzmann2014} an informal
correctness proof: but to us it contains unfillable gaps.

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}{l@ {\hspace{5mm}}rcl}
  (1) & @{thm (lhs) L.simps(1)} & $\dn$ & @{thm (rhs) L.simps(1)}\\
  (2) & @{thm (lhs) L.simps(2)} & $\dn$ & @{thm (rhs) L.simps(2)}\\
  (3) & @{thm (lhs) L.simps(3)} & $\dn$ & @{thm (rhs) L.simps(3)}\\
  (4) & @{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"]}\\
  (5) & @{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"]}\\
  (6) & @{thm (lhs) L.simps(6)} & $\dn$ & @{thm (rhs) L.simps(6)}\\
  \end{tabular}
  \end{center}
  
  \noindent In clause (4) 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 for
  languages (in the last clause above). The star for languages is defined
  inductively by two clauses: @{text "(i)"} 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 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}\label{derprop}\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 for regular expression matching. While the matcher
  above calculates a provably correct YES/NO answer for whether a regular
  expression matches a string or not, 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} to stand for a list of values. (This is
  similar to the approach taken by Frisch and Cardelli for GREEDY matching
  \cite{Frisch2004}, and Sulzmann and Lu \cite{Sulzmann2014} 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 @{text
  "Void"}. 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 associated with a regular
  expression. In case of POSIX matching the problem is to calculate the
  unique value that satisfies the (informal) POSIX rules from the
  Introduction. Graphically the POSIX value calculation 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. 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 rules 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 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 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.
\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 instead. Note also how this function
  makes some subtle choices leading to a POSIX value: for example if an
  alternative regular expression, 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 @{text Left}-value. The @{text
  Right}-value will only be returned if @{term "r\<^sub>1"} cannot match the empty
  string.

  The most interesting idea from Sulzmann and Lu \cite{Sulzmann2014} is
  the construction of a 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 achieve 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 (or right-most) @{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
  expressions and by analysing the shape of values (corresponding to 
  the derivative regular expressions).

  \begin{center}
  \begin{tabular}{l@ {\hspace{5mm}}lcl}
  (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 "SEQ r\<^sub>1 r\<^sub>2"}. This must be a value of the form @{term
  "Seq DUMMY DUMMY"}. Recall the clause of the @{text derivative}-function
  for sequence regular expressions:

  \begin{center}
  @{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"]}
  \end{center}

  \noindent Consider first the else-branch where the derivative is @{term
  "SEQ (der c r\<^sub>1) r\<^sub>2"}. The corresponding value must therefore
  be the form @{term "Seq v\<^sub>1 v\<^sub>2"}, which matches the left-hand
  side in clause (4) of @{term inj}. In the if-branch the derivative is an
  alternative, namely @{term "ALT (SEQ (der c r\<^sub>1) r\<^sub>2) (der c
  r\<^sub>2)"}. This means we either have to consider a @{text Left}- or
  @{text Right}-value. In case of the @{text Left}-value we know further it
  must be a value for a sequence regular expression. Therefore the pattern
  we match in the clause (5) is @{term "Left (Seq v\<^sub>1 v\<^sub>2)"},
  while in (6) it is just @{term "Right v\<^sub>2"}. One more interesting
  point is in the right-hand side of clause (6): since in this case the
  regular expression @{text "r\<^sub>1"} does not ``contribute'' to
  matching the string, that means it only matches the empty string, we need to
  call @{const mkeps} in order to construct a value for how @{term "r\<^sub>1"}
  can match this empty string. A similar argument applies for why we can
  expect in the left-hand side of clause (7) that the value is of the form
  @{term "Seq v (Stars vs)"}---the derivative of a star is @{term "SEQ r
  (STAR r)"}. Finally, the reason for why we can ignore the second argument
  in clause (1) of @{term inj} is that it will only ever be called in cases
  where @{term "c=d"}, but the usual linearity restrictions in patterns do
  not allow is to build this constraint explicitly into our function
  definition.\footnote{Sulzmann and Lu state this clause as @{thm (lhs)
  injval.simps(1)[of "c" "c"]} $\dn$ @{thm (rhs) injval.simps(1)[of "c"]},
  but our deviation is harmless.}

  The idea of @{term inj} to ``inject back'' a character into a value can
  be made precise by the first part of the following lemma; the second
  part shows that the underlying string of an @{const mkeps}-value is
  always the empty string.

  \begin{lemma}\mbox{}\smallskip\\\label{Prf_injval_flat}
  \begin{tabular}{ll}
  (1) & @{thm[mode=IfThen] Prf_injval_flat}\\
  (2) & @{thm[mode=IfThen] mkeps_flat}
  \end{tabular}
  \end{lemma}

  \begin{proof}
  Both properties are by routine inductions: the first one, for example,
  by an induction over the definition of @{term derivatives}; the second by
  induction on @{term r}. There are no interesting cases.\qed
  \end{proof}

  Having defined the @{const mkeps} and @{text inj} function we can extend
  \Brz's matcher so that a [lexical] value is constructed (assuming the
  regular expression matches the string). The clauses of the lexer are

  \begin{center}
  \begin{tabular}{lcl}
  @{thm (lhs) matcher.simps(1)} & $\dn$ & @{thm (rhs) matcher.simps(1)}\\
  @{thm (lhs) matcher.simps(2)} & $\dn$ & @{text "case"} @{term "matcher (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}

  \noindent If the regular expression does not match the string, @{const None} is
  returned, indicating an error is raised. If the regular expression does
  match the string, then @{const Some} value is returned. One important
  virtue of this algorithm is that it can be implemented with ease in a
  functional programming language and also in Isabelle/HOL. In the remaining
  part of this section we prove that this algorithm is correct.

  The well-known idea of POSIX matching is informally defined by the longest
  match and priority rule; as correctly argued in \cite{Sulzmann2014}, this
  needs formal specification. Sulzmann and Lu define a \emph{dominance}
  relation\footnote{Sulzmann and Lu call it an ordering relation, but
  without giving evidence that it is transitive.} between values and argue
  that there is a maximum value, as given by the derivative-based algorithm.
  In contrast, we shall introduce a simple inductive definition that
  specifies directly what a \emph{POSIX value} is, incorporating the
  POSIX-specific choices into the side-conditions of our rules. Our
  definition is inspired by the matching relation given in
  \cite{Vansummeren2006}. The relation we define is ternary and written as
  \mbox{@{term "s \<in> r \<rightarrow> v"}}, relating strings, regular expressions and
  values.

  \begin{center}
  \begin{tabular}{c}
  @{thm[mode=Axiom] PMatch.intros(1)}@{text "P"}@{term "ONE"} \qquad
  @{thm[mode=Axiom] PMatch.intros(2)}@{text "P"}@{term "c"}\bigskip\\
  @{thm[mode=Rule] PMatch.intros(3)[of "s" "r\<^sub>1" "v" "r\<^sub>2"]}@{text "P+L"}\qquad
  @{thm[mode=Rule] PMatch.intros(4)[of "s" "r\<^sub>2" "v" "r\<^sub>1"]}@{text "P+R"}\bigskip\\
  $\mprset{flushleft}
   \inferrule
   {@{thm (prem 1) PMatch.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \qquad
    @{thm (prem 2) PMatch.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \\\\
    @{thm (prem 3) PMatch.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}
   {@{thm (concl) PMatch.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}$@{text "PS"}\\
  @{thm[mode=Axiom] PMatch.intros(7)}@{text "P[]"}\bigskip\\
  $\mprset{flushleft}
   \inferrule
   {@{thm (prem 1) PMatch.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
    @{thm (prem 2) PMatch.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
    @{thm (prem 3) PMatch.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \\\\
    @{thm (prem 4) PMatch.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}
   {@{thm (concl) PMatch.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}$@{text "P\<star>"}
  \end{tabular}
  \end{center}

  \noindent We claim that this relation captures the idea behind the two
  informal POSIX rules shown in the Introduction: Consider for example the
  rules @{text "P+L"} and @{text "P+R"} where the POSIX value for an
  alternative regular expression is specified---it is always a @{text
  "Left"}-value, \emph{except} when the string to be matched is not in the
  language of @{term "r\<^sub>1"}; only then it is a @{text Right}-value (see the
  side-condition in @{text "P+R"}). Interesting is also the rule for
  sequence regular expressions (@{text "PS"}). The first two premises state
  that @{term "v\<^sub>1"} and @{term "v\<^sub>2"} are the POSIX values for
  @{term "(s\<^sub>1, r\<^sub>1)"} and @{term "(s\<^sub>2, r\<^sub>2)"}
  respectively. Consider now the third premise and note that the POSIX value
  of this rule should match the string @{term "s\<^sub>1 @ s\<^sub>2"}. According to the
  longest match rule, we want that the @{term "s\<^sub>1"} is the longest initial
  split of @{term "s\<^sub>1 @ s\<^sub>2"} such that @{term "s\<^sub>2"} is still recognised
  by @{term "r\<^sub>2"}. Let us assume, contrary to the third premise, that there
  \emph{exists} an @{term "s\<^sub>3"} and @{term "s\<^sub>4"} such that @{term "s\<^sub>2"}
  can be split up into a non-empty @{term "s\<^sub>3"} and @{term "s\<^sub>4"}. Moreover
  the longer @{term "s\<^sub>1 @ s\<^sub>3"} can be matched by @{text "r\<^sub>1"} and the
  shorter @{term "s\<^sub>4"} can still be matched by @{term "r\<^sub>2"}. In this case
  @{term "s\<^sub>1"} would not be the longest initial split of @{term "s\<^sub>1 @
  s\<^sub>2"} and therefore @{term "Seq v\<^sub>1 v\<^sub>2"} cannot be a POSIX value
  for @{term "(s\<^sub>1 @ s\<^sub>2, SEQ r\<^sub>1 r\<^sub>2)"}. A similar condition is imposed
  onto the POSIX value in the @{text "P\<star>"}-rule. Also there we want that
   @{term "s\<^sub>1"} is the longest initial split of @{term "s\<^sub>1 @ s\<^sub>2"} and
   furthermore the corresponding value @{term v} cannot be flatten to
   the empty string. In effect, we require that in each ``iteration''
   of the star, some parts of the string need to be ``nibbled'' away; only
   in case of the empty string weBy accept @{term "Stars []"} as the 
   POSIX value.

   We can prove that given a string @{term s} and regular expression @{term
   r}, the POSIX value @{term v} is uniquely determined by @{term "s \<in> r \<rightarrow>
   v"}.

  \begin{theorem}
  @{thm[mode=IfThen] PMatch_determ(1)[of _ _ "v\<^sub>1" "v\<^sub>2"]}
  \end{theorem}

  \begin{proof}
  By induction on the definition of @{term "s \<in> r \<rightarrow> v\<^sub>1"} and a case
  analysis of @{term "s \<in> r \<rightarrow> v\<^sub>2"}.\qed
  \end{proof}

  \begin{lemma}\label{lemmkeps}
  @{thm[mode=IfThen] PMatch_mkeps}
  \end{lemma}

  \begin{proof}
  By routine induction on @{term r}.\qed 
  \end{proof}

  \noindent
  The central lemma for our POSIX relation is that the @{text inj}-function
  preserves POSIX values.

  \begin{lemma}\label{PMatch2}
  @{thm[mode=IfThen] PMatch2_roy_version}
  \end{lemma}

  \begin{proof}

  By induction on @{text r}. Suppose @{term "r = ALT r\<^sub>1 r\<^sub>2"}. There are
  two subcases, namely @{text "(a)"} \mbox{@{term "v = Left v'"}} and @{term
  "s \<in> der c r\<^sub>1 \<rightarrow> v'"}; and @{text "(b)"} @{term "v = Right v'"}, @{term
  "s \<notin> L (der c r\<^sub>1)"} and @{term "s \<in> der c r\<^sub>2 \<rightarrow> v'"}. In @{text "(a)"} we
  know @{term "s \<in> der c r\<^sub>1 \<rightarrow> v'"}, from which we can infer @{term "(c # s)
  \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v'"} by induction hypothesis and hence @{term "(c #
  s) \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> injval (ALT r\<^sub>1 r\<^sub>2) c (Left v')"} as needed. Similarly
  in subcase @{text "(b)"} where, however, in addition we have to use
  Prop.~\ref{derprop}(2) in order to infer @{term "c # s \<notin> L r\<^sub>1"} from @{term
  "s \<notin> L (der c r\<^sub>1)"}.

  Suppose @{term "r = SEQ r\<^sub>1 r\<^sub>2"}. There are three subcases:
  
  \begin{quote}
  \begin{description}
  \item[@{text "(a)"}] @{term "v = Left (Seq v\<^sub>1 v\<^sub>2)"} and @{term "nullable r\<^sub>1"} 
  \item[@{text "(b)"}] @{term "v = Right v\<^sub>1"} and @{term "nullable r\<^sub>1"} 
  \item[@{text "(c)"}] @{term "v = Seq v\<^sub>1 v\<^sub>2"} and @{term "\<not> nullable r\<^sub>1"} 
  \end{description}
  \end{quote}

  \noindent For @{text "(a)"} we know @{term "s\<^sub>1 \<in> der c r\<^sub>1 \<rightarrow> v\<^sub>1"} and
  @{term "s\<^sub>2 \<in> r\<^sub>2 \<rightarrow> v\<^sub>2"} as well as
  
  \[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> s\<^sub>1 @ s\<^sub>3 \<in> L (der c r\<^sub>1) \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]

  \noindent From the latter we can infer by Prop.~\ref{derprop}(2):

  \[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> (c # s\<^sub>1) @ s\<^sub>3 \<in> L r\<^sub>1 \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]

  \noindent We can use the induction hypothesis for @{text "r\<^sub>1"} to obtain
  @{term "(c # s\<^sub>1) \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v\<^sub>1"}. This allows us to infer
  @{term "((c # s\<^sub>1) @ s\<^sub>2) \<in> SEQ r\<^sub>1 r\<^sub>2 \<rightarrow> Seq (injval r\<^sub>1 c v\<^sub>1) v\<^sub>2"}. The case @{text "(c)"}
  is similarly.

  For @{text "(b)"} we know @{term "s \<in> der c r\<^sub>2 \<rightarrow> v\<^sub>1"} and 
  @{term "s\<^sub>1 @ s\<^sub>2 \<notin> L (SEQ (der c r\<^sub>1) r\<^sub>2)"}. From the former
  we have @{term "(c # s) \<in> r\<^sub>2 \<rightarrow> (injval r\<^sub>2 c v\<^sub>1)"} by induction hypothesis
  for @{term "r\<^sub>2"}. From the latter we can infer

  \[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> s\<^sub>3 \<in> L r\<^sub>1 \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]

  \noindent By Lem.~\ref{lemmkeps} we know @{term "[] \<in> r\<^sub>1 \<rightarrow> (mkeps r\<^sub>1)"}
  holds. Putting this all together, we can conclude with @{term "(c #
  s) \<in> SEQ r\<^sub>1 r\<^sub>2 \<rightarrow> Seq (mkeps r\<^sub>1) (injval r\<^sub>2 c v\<^sub>1)"}.

  Finally suppose @{term "r = STAR r\<^sub>1"}. This case is very similar to the
  sequence case, except that we need to ensure that @{term "flat (injval r\<^sub>1
  c v\<^sub>1) \<noteq> []"}. This follows by Lem.~\ref{posixbasic} from @{term "(c # s\<^sub>1)
  \<in> r' \<rightarrow> injval r\<^sub>1 c v\<^sub>1"} (which in turn follows from @{term "s\<^sub>1 \<in> der c
  r\<^sub>1 \<rightarrow> v\<^sub>1"} and the induction hypothesis).\qed
  \end{proof}

  \noindent
  With Lem.~\ref{PMatch2} in place, it is completely routine to establish
  that the Sulzmann and Lu lexer satisfies its specification (returning
  an ``error'' iff the string is not in the language of the regular expression,
  and returning a unique POSIX value iff the string \emph{is} in the language):

  \begin{theorem}\mbox{}\smallskip\\
  \begin{tabular}{ll}
  (1) & @{thm (lhs) lex_correct1a} if and only if @{thm (rhs) lex_correct1a}\\
  (2) & @{thm (lhs) lex_correct3b} if and only if @{thm (rhs) lex_correct3b}\\
  \end{tabular}
  \end{theorem}

  \begin{proof}
  By induction on @{term s}.\qed  
  \end{proof}

  This concludes our correctness proof. Note that we have not changed the
  algorithm by Sulzmann and Lu, but introduced our own specification for
  what a correct result---a POSIX value---should be.
*}

section {* Extensions *}

text {*

  Derivatives as calculated by \Brz's method are usually more complex
  regular expressions than the initial one; the result is that the matching
  and lexing algorithms are often absymally slow. 


  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.


*}

section {* The Argument by Sulzmmann and Lu *}

section {* Conclusion *}

text {*

  Nipkow lexer from 2000

*}


text {*


  \noindent
  We have also introduced a slightly restricted version of this relation
  where the last rule is restricted so that @{term "flat v \<noteq> []"}.
  \bigskip


  \noindent
  

  \noindent
  Our version of Sulzmann's ordering relation


*} 

text {* 
  \noindent
  Some lemmas we have proved:\bigskip
  
  @{thm L_flat_Prf}

  @{thm[mode=IfThen] mkeps_nullable}

  @{thm[mode=IfThen] mkeps_flat}

  @{thm[mode=IfThen] Prf_injval}

  @{thm[mode=IfThen] Prf_injval_flat}
  
  @{thm[mode=IfThen] PMatch_mkeps}
  
  @{thm[mode=IfThen] PMatch1(2)}

  @{thm[mode=IfThen] PMatch_determ(1)[of "s" "r" "v\<^sub>1" "v\<^sub>2"]}

  
  \noindent {\bf Proof} The proof is by induction on the definition of
  @{const der}. Other inductions would go through as well. The
  interesting case is for @{term "SEQ r\<^sub>1 r\<^sub>2"}. First we analyse the
  case where @{term "nullable r\<^sub>1"}. We have by induction hypothesis

  \[
  \begin{array}{l}
  (IH1)\quad @{text "\<forall>s v."} \text{\;if\;} @{term "s \<in> der c r\<^sub>1 \<rightarrow> v"} 
  \text{\;then\;} @{term "(c # s) \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v"}\\
  (IH2)\quad @{text "\<forall>s v."} \text{\;if\;} @{term "s \<in> der c r\<^sub>2 \<rightarrow> v"} 
  \text{\;then\;} @{term "(c # s) \<in> r\<^sub>2 \<rightarrow> injval r\<^sub>2 c v"}
  \end{array}
  \]
  
  \noindent
  and have 

  \[
  @{term "s \<in> ALT (SEQ (der c r\<^sub>1) r\<^sub>2) (der c r\<^sub>2) \<rightarrow> v"}
  \]
  
  \noindent
  There are two cases what @{term v} can be: (1) @{term "Left v'"} and (2) @{term "Right v'"}.

  \begin{itemize}
  \item[(1)] We know @{term "s \<in> SEQ (der c r\<^sub>1) r\<^sub>2 \<rightarrow> v'"} holds, from which we
  can infer that there are @{text "s\<^sub>1"}, @{term "s\<^sub>2"}, @{text "v\<^sub>1"}, @{term "v\<^sub>2"}
  with

  \[
  @{term "s\<^sub>1 \<in> der c r\<^sub>1 \<rightarrow> v\<^sub>1"} \qquad\text{and}\qquad @{term "s\<^sub>2 \<in> r\<^sub>2 \<rightarrow> v\<^sub>2"}
  \]

  and also

  \[
  @{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> s\<^sub>1 @ s\<^sub>3 \<in> L (der c r\<^sub>1) \<and> s\<^sub>4 \<in> L r\<^sub>2)"}
  \]

  \noindent
  and have to prove
  
  \[
  @{term "((c # s\<^sub>1) @ s\<^sub>2) \<in> SEQ r\<^sub>1 r\<^sub>2 \<rightarrow> Seq (injval r\<^sub>1 c v\<^sub>1) v\<^sub>2"}
  \]

  \noindent
  The two requirements @{term "(c # s\<^sub>1) \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v\<^sub>1"} and 
  @{term "s\<^sub>2 \<in> r\<^sub>2 \<rightarrow> v\<^sub>2"} can be proved by the induction hypothese (IH1) and the
  fact above.

  \noindent
  This leaves to prove
  
  \[
  @{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> (c # s\<^sub>1) @ s\<^sub>3 \<in> L r\<^sub>1 \<and> s\<^sub>4 \<in> L r\<^sub>2)"}
  \]
  
  \noindent
  which holds because @{term "(c # s\<^sub>1) @ s\<^sub>3 \<in> L r\<^sub>1 "} implies @{term "s\<^sub>1 @ s\<^sub>3 \<in> L (der c r\<^sub>1) "}

  \item[(2)] This case is similar.
  \end{itemize}

  \noindent 
  The final case is that @{term " \<not> nullable r\<^sub>1"} holds. This case again similar
  to the cases above.
*}


text {*
  %\noindent
  %{\bf Acknowledgements:}
  %We are grateful for the comments we received from anonymous
  %referees.

  \bibliographystyle{plain}
  \bibliography{root}

*}


(*<*)
end
(*>*)