diff -r c80720289645 -r 0efa7ffd96ff thys2/Journal/Paper.thy~
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/Journal/Paper.thy~	Tue Jan 11 23:58:39 2022 +0000
@@ -0,0 +1,2313 @@
+(*<*)
+theory Paper
+imports 
+   "../Lexer"
+   "../Simplifying" 
+   "../Positions"
+
+   "../SizeBound" 
+   "HOL-Library.LaTeXsugar"
+begin
+
+lemma Suc_0_fold:
+  "Suc 0 = 1"
+by simp
+
+
+
+declare [[show_question_marks = false]]
+
+syntax (latex output)
+  "_Collect" :: "pttrn => bool => 'a set"              ("(1{_ \<^latex>\<open>\\mbox{\\boldmath$\\mid$}\<close> _})")
+  "_CollectIn" :: "pttrn => 'a set => bool => 'a set"   ("(1{_ \<in> _ |e _})")
+
+syntax
+  "_Not_Ex" :: "idts \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<nexists>_.a./ _)" [0, 10] 10)
+  "_Not_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<nexists>!_.a./ _)" [0, 10] 10)
+
+
+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"
+
+abbreviation 
+  "bders_syn r s \<equiv> bders r s"
+
+
+abbreviation
+  "nprec v1 v2 \<equiv> \<not>(v1 :\<sqsubset>val v2)"
+
+
+
+
+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) 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>\<star>" [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
+
+  L ("L'(_')" [10] 78) and
+  LV ("LV _ _" [80,73] 78) and
+  der_syn ("_\\_" [79, 1000] 76) and  
+  ders_syn ("_\\_" [79, 1000] 76) and
+  flat ("|_|" [75] 74) and
+  flats ("|_|" [72] 74) and
+  Sequ ("_ @ _" [78,77] 63) and
+  injval ("inj _ _ _" [79,77,79] 76) and 
+  mkeps ("mkeps _" [79] 76) and 
+  length ("len _" [73] 73) and
+  intlen ("len _" [73] 73) and
+  set ("_" [73] 73) and
+ 
+  Prf ("_ : _" [75,75] 75) and
+  Posix ("'(_, _') \<rightarrow> _" [63,75,75] 75) and
+ 
+  lexer ("lexer _ _" [78,78] 77) and 
+  F_RIGHT ("F\<^bsub>Right\<^esub> _") and
+  F_LEFT ("F\<^bsub>Left\<^esub> _") and  
+  F_ALT ("F\<^bsub>Alt\<^esub> _ _") and
+  F_SEQ1 ("F\<^bsub>Seq1\<^esub> _ _") and
+  F_SEQ2 ("F\<^bsub>Seq2\<^esub> _ _") and
+  F_SEQ ("F\<^bsub>Seq\<^esub> _ _") and
+  simp_SEQ ("simp\<^bsub>Seq\<^esub> _ _" [1000, 1000] 1) and
+  simp_ALT ("simp\<^bsub>Alt\<^esub> _ _" [1000, 1000] 1) and
+  slexer ("lexer\<^sup>+" 1000) and
+
+  at ("_\<^latex>\<open>\\mbox{$\\downharpoonleft$}\<close>\<^bsub>_\<^esub>") and
+  lex_list ("_ \<prec>\<^bsub>lex\<^esub> _") and
+  PosOrd ("_ \<prec>\<^bsub>_\<^esub> _" [77,77,77] 77) and
+  PosOrd_ex ("_ \<prec> _" [77,77] 77) and
+  PosOrd_ex_eq ("_ \<^latex>\<open>\\mbox{$\\preccurlyeq$}\<close> _" [77,77] 77) and
+  pflat_len ("\<parallel>_\<parallel>\<^bsub>_\<^esub>") and
+  nprec ("_ \<^latex>\<open>\\mbox{$\\not\\prec$}\<close> _" [77,77] 77) and
+
+  bder_syn ("_\<^latex>\<open>\\mbox{$\\bbslash$}\<close>_" [79, 1000] 76) and  
+  bders_syn ("_\<^latex>\<open>\\mbox{$\\bbslash$}\<close>_" [79, 1000] 76) and
+  intern ("_\<^latex>\<open>\\mbox{$^\\uparrow$}\<close>" [900] 80) and
+  erase ("_\<^latex>\<open>\\mbox{$^\\downarrow$}\<close>" [1000] 74) and
+  bnullable ("nullable\<^latex>\<open>\\mbox{$_b$}\<close> _" [1000] 80) and
+  bmkeps ("mkeps\<^latex>\<open>\\mbox{$_b$}\<close> _" [1000] 80) and
+  blexer ("lexer\<^latex>\<open>\\mbox{$_b$}\<close> _ _" [77, 77] 80) and
+  code ("code _" [79] 74) and
+
+  DUMMY ("\<^latex>\<open>\\underline{\\hspace{2mm}}\<close>")
+
+
+definition 
+  "match r s \<equiv> nullable (ders s r)"
+
+
+lemma LV_STAR_ONE_empty: 
+  shows "LV (STAR ONE) [] = {Stars []}"
+by(auto simp add: LV_def elim: Prf.cases intro: Prf.intros)
+
+
+
+(*
+comments not implemented
+
+p9. The condition "not exists s3 s4..." appears often enough (in particular in
+the proof of Lemma 3) to warrant a definition.
+
+*)
+
+
+(*>*)
+
+section\<open>Core of the proof\<close>
+text \<open>
+This paper builds on previous work by Ausaf and Urban using 
+regular expression'd bit-coded derivatives to do lexing that 
+is both fast and satisfies the POSIX specification.
+In their work, a bit-coded algorithm introduced by Sulzmann and Lu
+was formally verified in Isabelle, by a very clever use of
+flex function and retrieve to carefully mimic the way a value is 
+built up by the injection funciton.
+
+In the previous work, Ausaf and Urban established the below equality:
+\begin{lemma}
+@{thm [mode=IfThen] MAIN_decode}
+\end{lemma}
+
+This lemma establishes a link with the lexer without bit-codes.
+
+With it we get the correctness of bit-coded algorithm.
+\begin{lemma}
+@{thm [mode=IfThen] blexer_correctness}
+\end{lemma}
+
+However what is not certain is whether we can add simplification
+to the bit-coded algorithm, without breaking the correct lexing output.
+
+
+The reason that we do need to add a simplification phase
+after each derivative step of  $\textit{blexer}$ is
+because it produces intermediate
+regular expressions that can grow exponentially.
+For example, the regular expression $(a+aa)^*$ after taking
+derivative against just 10 $a$s will have size 8192.
+
+%TODO: add figure for this?
+
+
+Therefore, we insert a simplification phase
+after each derivation step, as defined below:
+\begin{lemma}
+@{thm blexer_simp_def}
+\end{lemma}
+
+The simplification function is given as follows:
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{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)} & $\dn$ & @{thm (rhs) bsimp.simps(2)}\\
+  @{thm (lhs) bsimp.simps(3)} & $\dn$ & @{thm (rhs) bsimp.simps(3)}\\
+
+\end{tabular}
+\end{center}
+
+And the two helper functions are:
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) bsimp_AALTs.simps(2)[of  "bs\<^sub>1" "r" ]} & $\dn$ & @{thm (rhs) bsimp.simps(1)[of  "bs\<^sub>1" "r" ]}\\
+  @{thm (lhs) bsimp_AALTs.simps(2)} & $\dn$ & @{thm (rhs) bsimp.simps(2)}\\
+  @{thm (lhs) bsimp_AALTs.simps(3)} & $\dn$ & @{thm (rhs) bsimp.simps(3)}\\
+
+\end{tabular}
+\end{center}
+
+
+This might sound trivial in the case of producing a YES/NO answer,
+but once we require a lexing output to be produced (which is required
+in applications like compiler front-end, malicious attack domain extraction, 
+etc.), it is not straightforward if we still extract what is needed according
+to the POSIX standard.
+
+
+
+
+
+By simplification, we mean specifically the following rules:
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm[mode=Axiom] rrewrite.intros(1)[of "bs" "r\<^sub>2"]}\\
+  @{thm[mode=Axiom] rrewrite.intros(2)[of "bs" "r\<^sub>1"]}\\
+  @{thm[mode=Axiom] rrewrite.intros(3)[of  "bs" "bs\<^sub>1" "r\<^sub>1"]}\\
+  @{thm[mode=Rule] rrewrite.intros(4)[of  "r\<^sub>1" "r\<^sub>2" "bs" "r\<^sub>3"]}\\
+  @{thm[mode=Rule] rrewrite.intros(5)[of "r\<^sub>3" "r\<^sub>4" "bs" "r\<^sub>1"]}\\
+  @{thm[mode=Rule] rrewrite.intros(6)[of "r" "r'" "bs" "rs\<^sub>1" "rs\<^sub>2"]}\\
+  @{thm[mode=Axiom] rrewrite.intros(7)[of "bs" "rs\<^sub>a" "rs\<^sub>b"]}\\
+  @{thm[mode=Axiom] rrewrite.intros(8)[of "bs" "rs\<^sub>a" "bs\<^sub>1" "rs\<^sub>1" "rs\<^sub>b"]}\\
+  @{thm[mode=Axiom] rrewrite.intros(9)[of "bs" "bs\<^sub>1" "rs"]}\\
+  @{thm[mode=Axiom] rrewrite.intros(10)[of "bs" ]}\\
+  @{thm[mode=Axiom] rrewrite.intros(11)[of "bs" "r\<^sub>1"]}\\
+  @{thm[mode=Rule] rrewrite.intros(12)[of "a\<^sub>1" "a\<^sub>2" "bs" "rs\<^sub>a" "rs\<^sub>b" "rs\<^sub>c"]}\\
+
+  \end{tabular}
+\end{center}
+
+
+And these can be made compact by the following simplification function:
+
+where the function $\mathit{bsimp_AALTs}$
+
+The core idea of the proof is that two regular expressions,
+if "isomorphic" up to a finite number of rewrite steps, will
+remain "isomorphic" when we take the same sequence of
+derivatives on both of them.
+This can be expressed by the following rewrite relation lemma:
+\begin{lemma}
+@{thm [mode=IfThen] central}
+\end{lemma}
+
+This isomorphic relation implies a property that leads to the 
+correctness result: 
+if two (nullable) regular expressions are "rewritable" in many steps
+from one another, 
+then a call to function $\textit{bmkeps}$ gives the same
+bit-sequence :
+\begin{lemma}
+@{thm [mode=IfThen] rewrites_bmkeps}
+\end{lemma}
+
+Given the same bit-sequence, the decode function
+will give out the same value, which is the output
+of both lexers:
+\begin{lemma}
+@{thm blexer_def}
+\end{lemma}
+
+\begin{lemma}
+@{thm blexer_simp_def}
+\end{lemma}
+
+And that yields the correctness result:
+\begin{lemma}
+@{thm blexersimp_correctness}
+\end{lemma}
+
+The nice thing about the above
+\<close>
+
+
+section \<open> Additional Simp Rules?\<close>
+
+
+text  \<open>
+One question someone would ask is:
+can we add more "atomic" simplification/rewriting rules,
+so the simplification is even more aggressive, making
+the intermediate results smaller, and therefore more space-efficient? 
+For example, one might want to do open up alternatives who is a child
+of a sequence:
+
+\begin{center}
+  \begin{tabular}{lcl}
+    @{thm[mode=Rule] aggressive.intros(1)[of "bs" "bs1" "rs" "r"]}\\
+  \end{tabular}
+\end{center}
+
+This rule allows us to simplify \mbox{@{term "(ALT (SEQ (ALT a b) c)  (SEQ a c))"}}
+ into  \mbox{@{term "ALT (SEQ a c)  (SEQ b c)"}},
+which is cannot be done under the \mbox{ \<leadsto>} rule because only alternatives which are
+children of another alternative can be spilled out.
+\<close>
+
+(*This rule allows us to simplify \mbox{@{term "ALT (SEQ (ALT a b) c) (SEQ a c)"}}
+ into  \mbox{@{term "ALT (SEQ a c)  (SEQ b c)"}},
+which is cannot be done under the  \<leadsto> rule because only alternatives which are 
+children of another alternative can be spilled out.*)
+section \<open>Introduction\<close>
+
+
+text \<open>
+
+
+Brzozowski \cite{Brzozowski1964} introduced the notion of the {\em
+derivative} @{term "der c r"} of a regular expression \<open>r\<close> w.r.t.\
+a character~\<open>c\<close>, 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
+mechanised correctness proof of Brzozowski's matcher in for example HOL4
+has been mentioned 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}.
+
+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{POSIX,Kuklewicz,OkuiSuzuki2010,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 \<open>r\<^bsub>key\<^esub>\<close> and \<open>r\<^bsub>id\<^esub>\<close> for recognising keywords and identifiers,
+respectively. There are a few underlying (informal) rules behind
+tokenising a string in a POSIX \cite{POSIX} fashion:
+
+\begin{itemize} 
+\item[$\bullet$] \emph{The Longest Match Rule} (or \emph{``{M}aximal {M}unch {R}ule''}):
+The longest initial substring matched by any regular expression is taken as
+next token.\smallskip
+
+\item[$\bullet$] \emph{Priority Rule:}
+For a particular longest initial substring, the first (leftmost) regular expression
+that can match determines the token.\smallskip
+
+\item[$\bullet$] \emph{Star Rule:} A subexpression repeated by ${}^\star$ shall 
+not match an empty string unless this is the only match for the repetition.\smallskip
+
+\item[$\bullet$] \emph{Empty String Rule:} An empty string shall be considered to 
+be longer than no match at all.
+\end{itemize}
+
+\noindent Consider for example a regular expression \<open>r\<^bsub>key\<^esub>\<close> for recognising keywords such as \<open>if\<close>,
+\<open>then\<close> and so on; and \<open>r\<^bsub>id\<^esub>\<close>
+recognising identifiers (say, a single character followed by
+characters or numbers).  Then we can form the regular expression
+\<open>(r\<^bsub>key\<^esub> + r\<^bsub>id\<^esub>)\<^sup>\<star>\<close>
+and use POSIX matching to tokenise strings, say \<open>iffoo\<close> and
+\<open>if\<close>.  For \<open>iffoo\<close> we obtain by the Longest Match Rule
+a single identifier token, not a keyword followed by an
+identifier. For \<open>if\<close> we obtain by the Priority Rule a keyword
+token, not an identifier token---even if \<open>r\<^bsub>id\<^esub>\<close>
+matches also. By the Star Rule we know \<open>(r\<^bsub>key\<^esub> +
+r\<^bsub>id\<^esub>)\<^sup>\<star>\<close> matches \<open>iffoo\<close>,
+respectively \<open>if\<close>, in exactly one `iteration' of the star. The
+Empty String Rule is for cases where, for example, the regular expression 
+\<open>(a\<^sup>\<star>)\<^sup>\<star>\<close> matches against the
+string \<open>bc\<close>. Then the longest initial matched substring is the
+empty string, which is matched by both the whole regular expression
+and the parenthesised subexpression.
+
+
+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}. Assuming a regular
+expression matches a string, values encode the information of
+\emph{how} the string is matched by the regular expression---that is,
+which part of the string is matched by which part of the regular
+expression. For this consider again the string \<open>xy\<close> and
+the regular expression \mbox{\<open>(x + (y + xy))\<^sup>\<star>\<close>}
+(this time fully parenthesised). We can view this regular expression
+as tree and if the string \<open>xy\<close> is matched by two Star
+`iterations', then the \<open>x\<close> is matched by the left-most
+alternative in this tree and the \<open>y\<close> by the right-left alternative. This
+suggests to record this matching as
+
+\begin{center}
+@{term "Stars [Left(Char x), Right(Left(Char y))]"}
+\end{center}
+
+\noindent where @{const Stars}, \<open>Left\<close>, \<open>Right\<close> and \<open>Char\<close> are constructors for values. \<open>Stars\<close> records how many
+iterations were used; \<open>Left\<close>, respectively \<open>Right\<close>, which
+alternative is used. This `tree view' leads naturally to the idea that
+regular expressions act as types and values as inhabiting those types
+(see, for example, \cite{HosoyaVouillonPierce2005}).  The value for
+matching \<open>xy\<close> in a single `iteration', i.e.~the POSIX value,
+would look as follows
+
+\begin{center}
+@{term "Stars [Seq (Char x) (Char y)]"}
+\end{center}
+
+\noindent where @{const Stars} has only a single-element list for the
+single iteration and @{const Seq} indicates that @{term xy} is matched 
+by a sequence regular expression.
+
+%, which we will in what follows 
+%write more formally as @{term "SEQ x y"}.
+
+
+Sulzmann and Lu give a simple algorithm to calculate a value that
+appears to be the value associated with POSIX matching.  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. However, we were not able to
+establish transitivity and totality for the ``order relation'' by
+Sulzmann and Lu.  There are some inherent problems with their approach
+(of which some of the proofs are not published in
+\cite{Sulzmann2014}); perhaps more importantly, we give in this paper
+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 by Sulzmann and Lu
+computes such a value and that such a value is unique. Our 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
+Kuklewicz \cite{Kuklewicz} who found that nearly all POSIX matching
+implementations are ``buggy'' \cite[Page 203]{Sulzmann2014} and by
+Grathwohl et al \cite[Page 36]{CrashCourse2014} who wrote:
+
+\begin{quote}
+\it{}``The POSIX strategy is more complicated than the greedy because of 
+the dependence on information about the length of matched strings in the 
+various subexpressions.''
+\end{quote}
+
+
+
+\noindent {\bf Contributions:} We have implemented in Isabelle/HOL the
+derivative-based regular expression matching algorithm of
+Sulzmann and Lu \cite{Sulzmann2014}. We have proved the correctness of this
+algorithm according to our specification of what a POSIX value is (inspired
+by work of Vansummeren \cite{Vansummeren2006}). Sulzmann
+and Lu sketch in \cite{Sulzmann2014} an informal correctness proof: but to
+us it contains unfillable gaps.\footnote{An extended version of
+\cite{Sulzmann2014} is available at the website of its first author; this
+extended version already includes remarks in the appendix that their
+informal proof contains gaps, and possible fixes are not fully worked out.}
+Our specification of a POSIX value consists of a simple inductive definition
+that given a string and a regular expression uniquely determines this value.
+We also show that our definition is equivalent to an ordering 
+of values based on positions by Okui and Suzuki \cite{OkuiSuzuki2010}.
+
+%Derivatives as calculated by Brzozowski's method are usually more complex
+%regular expressions than the initial one; various optimisations are
+%possible. We prove the correctness when simplifications of @{term "ALT ZERO r"}, 
+%@{term "ALT r ZERO"}, @{term "SEQ ONE r"} and @{term "SEQ r ONE"} to
+%@{term r} are applied. 
+
+We extend our results to ??? Bitcoded version??
+
+\<close>
+
+
+
+
+section \<open>Preliminaries\<close>
+
+text \<open>\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]"}. We use the usual definitions for 
+\emph{prefixes} and \emph{strict prefixes} of strings.  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}
+  \<open>r :=\<close>
+  @{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
+  language of a regular expression is also defined as usual by the
+  recursive function @{term L} with the six clauses:
+
+  \begin{center}
+  \begin{tabular}{l@ {\hspace{4mm}}rcl}
+  \textit{(1)} & @{thm (lhs) L.simps(1)} & $\dn$ & @{thm (rhs) L.simps(1)}\\
+  \textit{(2)} & @{thm (lhs) L.simps(2)} & $\dn$ & @{thm (rhs) L.simps(2)}\\
+  \textit{(3)} & @{thm (lhs) L.simps(3)} & $\dn$ & @{thm (rhs) L.simps(3)}\\
+  \textit{(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"]}\\
+  \textit{(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"]}\\
+  \textit{(6)} & @{thm (lhs) L.simps(6)} & $\dn$ & @{thm (rhs) L.simps(6)}\\
+  \end{tabular}
+  \end{center}
+  
+  \noindent In clause \textit{(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: \<open>(i)\<close> the empty string being in
+  the star of a language and \<open>(ii)\<close> 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}
+  @{thm Der_def}\;.
+  \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\\
+
+%  \end{tabular}
+%  \end{center}
+
+%  \begin{center}
+%  \begin{tabular}{lcl}
+
+  @{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}
+  \textit{(1)} & @{thm (lhs) nullable_correctness} if and only if
+  @{thm (rhs) nullable_correctness}, and \\ 
+  \textit{(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.
+
+\<close>
+
+section \<open>POSIX Regular Expression Matching\label{posixsec}\<close>
+
+text \<open>
+
+  There have been many previous works that use values for encoding 
+  \emph{how} a regular expression matches a string.
+  The clever idea by Sulzmann and Lu \cite{Sulzmann2014} is to 
+  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}
+  \<open>v :=\<close>
+  @{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
+  for POSIX matching \cite{Sulzmann2014}). The string underlying a
+  value can be calculated by the @{const flat} function, written
+  @{term "flat DUMMY"} and defined as:
+
+  \begin{center}
+  \begin{tabular}[t]{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)}
+  \end{tabular}\hspace{14mm}
+  \begin{tabular}[t]{lcl}
+  @{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 We will sometimes refer to the underlying string of a
+  value as \emph{flattened value}.  We will also overload our notation and 
+  use @{term "flats vs"} for flattening a list of values and concatenating
+  the resulting strings.
+  
+  Sulzmann and Lu define
+  inductively an \emph{inhabitation relation} that associates values to
+  regular expressions. We define this relation as
+  follows:\footnote{Note that the rule for @{term Stars} differs from
+  our earlier paper \cite{AusafDyckhoffUrban2016}. There we used the
+  original definition by Sulzmann and Lu which does not require that
+  the values @{term "v \<in> set vs"} flatten to a non-empty
+  string. The reason for introducing the more restricted version of
+  lexical values is convenience later on when reasoning about an
+  ordering relation for values.}
+
+  \begin{center}
+  \begin{tabular}{c@ {\hspace{12mm}}c}\label{prfintros}
+  \\[-8mm]
+  @{thm[mode=Axiom] Prf.intros(4)} & 
+  @{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"]} &
+  @{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"]}  &
+  @{thm[mode=Rule] Prf.intros(6)[of "vs"]}
+  \end{tabular}
+  \end{center}
+
+  \noindent where in the clause for @{const "Stars"} we use the
+  notation @{term "v \<in> set vs"} for indicating that \<open>v\<close> is a
+  member in the list \<open>vs\<close>.  We require in this rule that every
+  value in @{term vs} flattens to a non-empty string. The idea is that
+  @{term "Stars"}-values satisfy the informal Star Rule (see Introduction)
+  where the $^\star$ does not match the empty string unless this is
+  the only match for the repetition.  Note also 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}.  It is routine to establish how values ``inhabiting''
+  a regular expression correspond to the language of a regular
+  expression, namely
+
+  \begin{proposition}\label{inhabs}
+  @{thm L_flat_Prf}
+  \end{proposition}
+
+  \noindent
+  Given a regular expression \<open>r\<close> and a string \<open>s\<close>, we define the 
+  set of all \emph{Lexical Values} inhabited by \<open>r\<close> with the underlying string 
+  being \<open>s\<close>:\footnote{Okui and Suzuki refer to our lexical values 
+  as \emph{canonical values} in \cite{OkuiSuzuki2010}. The notion of \emph{non-problematic
+  values} by Cardelli and Frisch \cite{Frisch2004} is related, but not identical
+  to our lexical values.}
+  
+  \begin{center}
+  @{thm LV_def}
+  \end{center}
+
+  \noindent The main property of @{term "LV r s"} is that it is alway finite.
+
+  \begin{proposition}
+  @{thm LV_finite}
+  \end{proposition}
+
+  \noindent This finiteness property does not hold in general if we
+  remove the side-condition about @{term "flat v \<noteq> []"} in the
+  @{term Stars}-rule above. For example using Sulzmann and Lu's
+  less restrictive definition, @{term "LV (STAR ONE) []"} would contain
+  infinitely many values, but according to our more restricted
+  definition only a single value, namely @{thm LV_STAR_ONE_empty}.
+
+  If a regular expression \<open>r\<close> matches a string \<open>s\<close>, then
+  generally the set @{term "LV r s"} is not just a singleton set.  In
+  case of POSIX matching the problem is to calculate the unique lexical 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}/\<open>inj\<close>, the path from right to left, the second
+  phase. This picture shows the steps required when a regular
+  expression, say \<open>r\<^sub>1\<close>, 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
+  function is defined by the clauses:
+
+\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 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.
+\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 the null value @{term "None"} is returned. 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 \<open>Left\<close>-value. The \<open>Right\<close>-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, but 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"}. The @{term inj} function is defined by recursion on regular
+  expressions and by analysing the shape of values (corresponding to 
+  the derivative regular expressions).
+  %
+  \begin{center}
+  \begin{tabular}{l@ {\hspace{5mm}}lcl}
+  \textit{(1)} & @{thm (lhs) injval.simps(1)} & $\dn$ & @{thm (rhs) injval.simps(1)}\\
+  \textit{(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"]}\\
+  \textit{(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"]}\\
+  \textit{(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"]}\\
+  \textit{(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"]}\\
+  \textit{(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"]}\\
+  \textit{(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
+  \textit{(4)} -- \textit{(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 \<open>derivative\<close>-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 \<open>else\<close>-branch where the derivative is @{term
+  "SEQ (der c r\<^sub>1) r\<^sub>2"}. The corresponding value must therefore
+  be of the form @{term "Seq v\<^sub>1 v\<^sub>2"}, which matches the left-hand
+  side in clause~\textit{(4)} of @{term inj}. In the \<open>if\<close>-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 \<open>Left\<close>- or
+  \<open>Right\<close>-value. In case of the \<open>Left\<close>-value we know further it
+  must be a value for a sequence regular expression. Therefore the pattern
+  we match in the clause \textit{(5)} is @{term "Left (Seq v\<^sub>1 v\<^sub>2)"},
+  while in \textit{(6)} it is just @{term "Right v\<^sub>2"}. One more interesting
+  point is in the right-hand side of clause \textit{(6)}: since in this case the
+  regular expression \<open>r\<^sub>1\<close> 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 \textit{(7)} that the value is of the form
+  @{term "Seq v (Stars vs)"}---the derivative of a star is @{term "SEQ (der c r)
+  (STAR r)"}. Finally, the reason for why we can ignore the second argument
+  in clause \textit{(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 us 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 the @{term inj}-function to ``inject'' a character, say
+  @{term c}, into a value can be made precise by the first part of the
+  following lemma, which shows that the underlying string of an injected
+  value has a prepended character @{term c}; the second part shows that
+  the underlying string of an @{const mkeps}-value is always the empty
+  string (given the regular expression is nullable since otherwise
+  \<open>mkeps\<close> might not be defined).
+
+  \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 can, for example,
+  be proved by induction over the definition of @{term derivatives}; the second by
+  an induction on @{term r}. There are no interesting cases.\qed
+  \end{proof}
+
+  Having defined the @{const mkeps} and \<open>inj\<close> function we can extend
+  \Brz's matcher so that a value is constructed (assuming the
+  regular expression matches the string). The clauses of the Sulzmann and Lu lexer are
+
+  \begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) lexer.simps(1)} & $\dn$ & @{thm (rhs) lexer.simps(1)}\\
+  @{thm (lhs) lexer.simps(2)} & $\dn$ & \<open>case\<close> @{term "lexer (der c r) s"} \<open>of\<close>\\
+                     & & \phantom{$|$} @{term "None"}  \<open>\<Rightarrow>\<close> @{term None}\\
+                     & & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> @{term "Some (injval r c v)"}                          
+  \end{tabular}
+  \end{center}
+
+  \noindent If the regular expression does not match the string, @{const None} is
+  returned. If the regular expression \emph{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 any
+  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 some
+  rules such as the Longest Match and Priority Rules (see
+  Introduction); as correctly argued in \cite{Sulzmann2014}, this
+  needs formal specification. Sulzmann and Lu define an ``ordering
+  relation'' 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 by Vansummeren~\cite{Vansummeren2006}. 
+  The relation we define is ternary and
+  written as \mbox{@{term "s \<in> r \<rightarrow> v"}}, relating
+  strings, regular expressions and values; the inductive rules are given in 
+  Figure~\ref{POSIXrules}.
+  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{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>\\
+  @{thm[mode=Axiom] Posix.intros(7)}\<open>P[]\<close>\medskip\\
+  $\mprset{flushleft}
+   \inferrule
+   {@{thm (prem 1) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
+    @{thm (prem 2) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
+    @{thm (prem 3) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \\\\
+    @{thm (prem 4) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}
+   {@{thm (concl) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}$\<open>P\<star>\<close>
+  \end{tabular}
+  \end{center}
+  \caption{Our inductive definition of POSIX values.}\label{POSIXrules}
+  \end{figure}
+
+   
+
+  \begin{theorem}\mbox{}\smallskip\\\label{posixdeterm}
+  \begin{tabular}{ll}
+  (1) & If @{thm (prem 1) Posix1(1)} then @{thm (concl)
+  Posix1(1)} and @{thm (concl) Posix1(2)}.\\
+  (2) & @{thm[mode=IfThen] Posix_determ(1)[of _ _ "v" "v'"]}
+  \end{tabular}
+  \end{theorem}
+
+  \begin{proof} Both by induction on the definition of @{term "s \<in> r \<rightarrow> v"}. 
+  The second parts follows by a case analysis of @{term "s \<in> r \<rightarrow> v'"} and
+  the first part.\qed
+  \end{proof}
+
+  \noindent
+  We claim that our @{term "s \<in> r \<rightarrow> v"} relation captures the idea behind the four
+  informal POSIX rules shown in the Introduction: Consider for example the
+  rules \<open>P+L\<close> and \<open>P+R\<close> where the POSIX value for a string
+  and an alternative regular expression, that is @{term "(s, ALT r\<^sub>1 r\<^sub>2)"},
+  is specified---it is always a \<open>Left\<close>-value, \emph{except} when the
+  string to be matched is not in the language of @{term "r\<^sub>1"}; only then it
+  is a \<open>Right\<close>-value (see the side-condition in \<open>P+R\<close>).
+  Interesting is also the rule for sequence regular expressions (\<open>PS\<close>). 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 \mbox{@{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 \mbox{@{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{exist} an @{term "s\<^sub>3"} and @{term "s\<^sub>4"} such that @{term "s\<^sub>2"}
+  can be split up into a non-empty string @{term "s\<^sub>3"} and a possibly empty
+  string @{term "s\<^sub>4"}. Moreover the longer string @{term "s\<^sub>1 @ s\<^sub>3"} can be
+  matched by \<open>r\<^sub>1\<close> and the shorter @{term "s\<^sub>4"} can still be
+  matched by @{term "r\<^sub>2"}. In this case @{term "s\<^sub>1"} would \emph{not} be the
+  longest initial split of \mbox{@{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)"}. 
+  The main point is that our side-condition ensures the Longest 
+  Match Rule is satisfied.
+
+  A similar condition is imposed on the POSIX value in the \<open>P\<star>\<close>-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 flattened to the empty string. In effect, we require
+  that in each ``iteration'' of the star, some non-empty substring needs to
+  be ``chipped'' away; only in case of the empty string we accept @{term
+  "Stars []"} as the POSIX value. Indeed we can show that our POSIX values
+  are lexical values which exclude those \<open>Stars\<close> that contain subvalues 
+  that flatten to the empty string.
+
+  \begin{lemma}\label{LVposix}
+  @{thm [mode=IfThen] Posix_LV}
+  \end{lemma}
+
+  \begin{proof}
+  By routine induction on @{thm (prem 1) Posix_LV}.\qed 
+  \end{proof}
+
+  \noindent
+  Next is the lemma that shows the function @{term "mkeps"} calculates
+  the POSIX value for the empty string and a nullable regular expression.
+
+  \begin{lemma}\label{lemmkeps}
+  @{thm[mode=IfThen] Posix_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 \<open>inj\<close>-function
+  preserves POSIX values.
+
+  \begin{lemma}\label{Posix2}
+  @{thm[mode=IfThen] Posix_injval}
+  \end{lemma}
+
+  \begin{proof}
+  By induction on \<open>r\<close>. We explain two cases.
+
+  \begin{itemize}
+  \item[$\bullet$] Case @{term "r = ALT r\<^sub>1 r\<^sub>2"}. There are
+  two subcases, namely \<open>(a)\<close> \mbox{@{term "v = Left v'"}} and @{term
+  "s \<in> der c r\<^sub>1 \<rightarrow> v'"}; and \<open>(b)\<close> @{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 \<open>(a)\<close> 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 \<open>(b)\<close> where, however, in addition we have to use
+  Proposition~\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)"}.\smallskip
+
+  \item[$\bullet$] Case @{term "r = SEQ r\<^sub>1 r\<^sub>2"}. There are three subcases:
+  
+  \begin{quote}
+  \begin{description}
+  \item[\<open>(a)\<close>] @{term "v = Left (Seq v\<^sub>1 v\<^sub>2)"} and @{term "nullable r\<^sub>1"} 
+  \item[\<open>(b)\<close>] @{term "v = Right v\<^sub>1"} and @{term "nullable r\<^sub>1"} 
+  \item[\<open>(c)\<close>] @{term "v = Seq v\<^sub>1 v\<^sub>2"} and @{term "\<not> nullable r\<^sub>1"} 
+  \end{description}
+  \end{quote}
+
+  \noindent For \<open>(a)\<close> 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 Proposition~\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 \<open>r\<^sub>1\<close> to obtain
+  @{term "(c # s\<^sub>1) \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v\<^sub>1"}. Putting this all together 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 \<open>(c)\<close>
+  is similar.
+
+  For \<open>(b)\<close> 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 Lemma~\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)"}, as required.
+
+  Finally suppose @{term "r = STAR r\<^sub>1"}. This case is very similar to the
+  sequence case, except that we need to also ensure that @{term "flat (injval r\<^sub>1
+  c v\<^sub>1) \<noteq> []"}. This follows from @{term "(c # s\<^sub>1)
+  \<in> r\<^sub>1 \<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{itemize}
+  \end{proof}
+
+  \noindent
+  With Lemma~\ref{Posix2} in place, it is completely routine to establish
+  that the Sulzmann and Lu lexer satisfies our specification (returning
+  the null value @{term "None"} 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\\\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{theorem}
+
+  \begin{proof}
+  By induction on @{term s} using Lemma~\ref{lemmkeps} and \ref{Posix2}.\qed  
+  \end{proof}
+
+  \noindent In \textit{(2)} we further know by Theorem~\ref{posixdeterm} that the
+  value returned by the lexer must be unique.   A simple corollary 
+  of our two theorems is:
+
+  \begin{corollary}\mbox{}\smallskip\\\label{lexercorrectcor}
+  \begin{tabular}{ll}
+  (1) & @{thm (lhs) lexer_correctness(2)} if and only if @{thm (rhs) lexer_correctness(2)}\\ 
+  (2) & @{thm (lhs) lexer_correctness(1)} if and only if @{thm (rhs) lexer_correctness(1)}\\
+  \end{tabular}
+  \end{corollary}
+
+  \noindent This concludes our correctness proof. Note that we have
+  not changed the algorithm of Sulzmann and Lu,\footnote{All
+  deviations we introduced are harmless.} but introduced our own
+  specification for what a correct result---a POSIX value---should be.
+  In the next section we show that our specification coincides with
+  another one given by Okui and Suzuki using a different technique.
+
+\<close>
+
+section \<open>Ordering of Values according to Okui and Suzuki\<close>
+
+text \<open>
+  
+  While in the previous section we have defined POSIX values directly
+  in terms of a ternary relation (see inference rules in Figure~\ref{POSIXrules}),
+  Sulzmann and Lu took a different approach in \cite{Sulzmann2014}:
+  they introduced an ordering for values and identified POSIX values
+  as the maximal elements.  An extended version of \cite{Sulzmann2014}
+  is available at the website of its first author; this includes more
+  details of their proofs, but which are evidently not in final form
+  yet. Unfortunately, we were not able to verify claims that their
+  ordering has properties such as being transitive or having maximal
+  elements. 
+ 
+  Okui and Suzuki \cite{OkuiSuzuki2010,OkuiSuzukiTech} described
+  another ordering of values, which they use to establish the
+  correctness of their automata-based algorithm for POSIX matching.
+  Their ordering resembles some aspects of the one given by Sulzmann
+  and Lu, but overall is quite different. To begin with, Okui and
+  Suzuki identify POSIX values as minimal, rather than maximal,
+  elements in their ordering. A more substantial difference is that
+  the ordering by Okui and Suzuki uses \emph{positions} in order to
+  identify and compare subvalues. Positions are lists of natural
+  numbers. This allows them to quite naturally formalise the Longest
+  Match and Priority rules of the informal POSIX standard.  Consider
+  for example the value @{term v}
+
+  \begin{center}
+  @{term "v == Stars [Seq (Char x) (Char y), Char z]"}
+  \end{center}
+
+  \noindent
+  At position \<open>[0,1]\<close> of this value is the
+  subvalue \<open>Char y\<close> and at position \<open>[1]\<close> the
+  subvalue @{term "Char z"}.  At the `root' position, or empty list
+  @{term "[]"}, is the whole value @{term v}. Positions such as \<open>[0,1,0]\<close> or \<open>[2]\<close> are outside of \<open>v\<close>. If it exists, the subvalue of @{term v} at a position \<open>p\<close>, written @{term "at v p"}, can be recursively defined by
+  
+  \begin{center}
+  \begin{tabular}{r@ {\hspace{0mm}}lcl}
+  @{term v} &  \<open>\<downharpoonleft>\<^bsub>[]\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(1)}\\
+  @{term "Left v"} & \<open>\<downharpoonleft>\<^bsub>0::ps\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(2)}\\
+  @{term "Right v"} & \<open>\<downharpoonleft>\<^bsub>1::ps\<^esub>\<close> & \<open>\<equiv>\<close> & 
+  @{thm (rhs) at.simps(3)[simplified Suc_0_fold]}\\
+  @{term "Seq v\<^sub>1 v\<^sub>2"} & \<open>\<downharpoonleft>\<^bsub>0::ps\<^esub>\<close> & \<open>\<equiv>\<close> & 
+  @{thm (rhs) at.simps(4)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \\
+  @{term "Seq v\<^sub>1 v\<^sub>2"} & \<open>\<downharpoonleft>\<^bsub>1::ps\<^esub>\<close>
+  & \<open>\<equiv>\<close> & 
+  @{thm (rhs) at.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2", simplified Suc_0_fold]} \\
+  @{term "Stars vs"} & \<open>\<downharpoonleft>\<^bsub>n::ps\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(6)}\\
+  \end{tabular} 
+  \end{center}
+
+  \noindent In the last clause we use Isabelle's notation @{term "vs ! n"} for the
+  \<open>n\<close>th element in a list.  The set of positions inside a value \<open>v\<close>,
+  written @{term "Pos v"}, is given by 
+
+  \begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) Pos.simps(1)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(1)}\\
+  @{thm (lhs) Pos.simps(2)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(2)}\\
+  @{thm (lhs) Pos.simps(3)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(3)}\\
+  @{thm (lhs) Pos.simps(4)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(4)}\\
+  @{thm (lhs) Pos.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
+  & \<open>\<equiv>\<close> 
+  & @{thm (rhs) Pos.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+  @{thm (lhs) Pos_stars} & \<open>\<equiv>\<close> & @{thm (rhs) Pos_stars}\\
+  \end{tabular}
+  \end{center}
+
+  \noindent 
+  whereby \<open>len\<close> in the last clause stands for the length of a list. Clearly
+  for every position inside a value there exists a subvalue at that position.
+ 
+
+  To help understanding the ordering of Okui and Suzuki, consider again 
+  the earlier value
+  \<open>v\<close> and compare it with the following \<open>w\<close>:
+
+  \begin{center}
+  \begin{tabular}{l}
+  @{term "v == Stars [Seq (Char x) (Char y), Char z]"}\\
+  @{term "w == Stars [Char x, Char y, Char z]"}  
+  \end{tabular}
+  \end{center}
+
+  \noindent Both values match the string \<open>xyz\<close>, that means if
+  we flatten these values at their respective root position, we obtain
+  \<open>xyz\<close>. However, at position \<open>[0]\<close>, \<open>v\<close> matches
+  \<open>xy\<close> whereas \<open>w\<close> matches only the shorter \<open>x\<close>. So
+  according to the Longest Match Rule, we should prefer \<open>v\<close>,
+  rather than \<open>w\<close> as POSIX value for string \<open>xyz\<close> (and
+  corresponding regular expression). In order to
+  formalise this idea, Okui and Suzuki introduce a measure for
+  subvalues at position \<open>p\<close>, called the \emph{norm} of \<open>v\<close>
+  at position \<open>p\<close>. We can define this measure in Isabelle as an
+  integer as follows
+  
+  \begin{center}
+  @{thm pflat_len_def}
+  \end{center}
+
+  \noindent where we take the length of the flattened value at
+  position \<open>p\<close>, provided the position is inside \<open>v\<close>; if
+  not, then the norm is \<open>-1\<close>. The default for outside
+  positions is crucial for the POSIX requirement of preferring a
+  \<open>Left\<close>-value over a \<open>Right\<close>-value (if they can match the
+  same string---see the Priority Rule from the Introduction). For this
+  consider
+
+  \begin{center}
+  @{term "v == Left (Char x)"} \qquad and \qquad @{term "w == Right (Char x)"}
+  \end{center}
+
+  \noindent Both values match \<open>x\<close>. At position \<open>[0]\<close>
+  the norm of @{term v} is \<open>1\<close> (the subvalue matches \<open>x\<close>),
+  but the norm of \<open>w\<close> is \<open>-1\<close> (the position is outside
+  \<open>w\<close> according to how we defined the `inside' positions of
+  \<open>Left\<close>- and \<open>Right\<close>-values).  Of course at position
+  \<open>[1]\<close>, the norms @{term "pflat_len v [1]"} and @{term
+  "pflat_len w [1]"} are reversed, but the point is that subvalues
+  will be analysed according to lexicographically ordered
+  positions. According to this ordering, the position \<open>[0]\<close>
+  takes precedence over \<open>[1]\<close> and thus also \<open>v\<close> will be 
+  preferred over \<open>w\<close>.  The lexicographic ordering of positions, written
+  @{term "DUMMY \<sqsubset>lex DUMMY"}, can be conveniently formalised
+  by three inference rules
+
+  \begin{center}
+  \begin{tabular}{ccc}
+  @{thm [mode=Axiom] lex_list.intros(1)}\hspace{1cm} &
+  @{thm [mode=Rule] lex_list.intros(3)[where ?p1.0="p\<^sub>1" and ?p2.0="p\<^sub>2" and
+                                            ?ps1.0="ps\<^sub>1" and ?ps2.0="ps\<^sub>2"]}\hspace{1cm} &
+  @{thm [mode=Rule] lex_list.intros(2)[where ?ps1.0="ps\<^sub>1" and ?ps2.0="ps\<^sub>2"]}
+  \end{tabular}
+  \end{center}
+
+  With the norm and lexicographic order in place,
+  we can state the key definition of Okui and Suzuki
+  \cite{OkuiSuzuki2010}: a value @{term "v\<^sub>1"} is \emph{smaller at position \<open>p\<close>} than
+  @{term "v\<^sub>2"}, written @{term "v\<^sub>1 \<sqsubset>val p v\<^sub>2"}, 
+  if and only if  $(i)$ the norm at position \<open>p\<close> is
+  greater in @{term "v\<^sub>1"} (that is the string @{term "flat (at v\<^sub>1 p)"} is longer 
+  than @{term "flat (at v\<^sub>2 p)"}) and $(ii)$ all subvalues at 
+  positions that are inside @{term "v\<^sub>1"} or @{term "v\<^sub>2"} and that are
+  lexicographically smaller than \<open>p\<close>, we have the same norm, namely
+
+ \begin{center}
+ \begin{tabular}{c}
+ @{thm (lhs) PosOrd_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} 
+ \<open>\<equiv>\<close> 
+ $\begin{cases}
+ (i) & @{term "pflat_len v\<^sub>1 p > pflat_len v\<^sub>2 p"}   \quad\text{and}\smallskip \\
+ (ii) & @{term "(\<forall>q \<in> Pos v\<^sub>1 \<union> Pos v\<^sub>2. q \<sqsubset>lex p --> pflat_len v\<^sub>1 q = pflat_len v\<^sub>2 q)"}
+ \end{cases}$
+ \end{tabular}
+ \end{center}
+
+ \noindent The position \<open>p\<close> in this definition acts as the
+  \emph{first distinct position} of \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close>, where both values match strings of different length
+  \cite{OkuiSuzuki2010}.  Since at \<open>p\<close> the values \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> match different strings, the
+  ordering is irreflexive. Derived from the definition above
+  are the following two orderings:
+  
+  \begin{center}
+  \begin{tabular}{l}
+  @{thm PosOrd_ex_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+  @{thm PosOrd_ex_eq_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
+  \end{tabular}
+  \end{center}
+
+ While we encountered a number of obstacles for establishing properties like
+ transitivity for the ordering of Sulzmann and Lu (and which we failed
+ to overcome), it is relatively straightforward to establish this
+ property for the orderings
+ @{term "DUMMY :\<sqsubset>val DUMMY"} and @{term "DUMMY :\<sqsubseteq>val DUMMY"}  
+ by Okui and Suzuki.
+
+ \begin{lemma}[Transitivity]\label{transitivity}
+ @{thm [mode=IfThen] PosOrd_trans[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and ?v3.0="v\<^sub>3"]} 
+ \end{lemma}
+
+ \begin{proof} From the assumption we obtain two positions \<open>p\<close>
+ and \<open>q\<close>, where the values \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> (respectively \<open>v\<^sub>2\<close> and \<open>v\<^sub>3\<close>) are `distinct'.  Since \<open>\<prec>\<^bsub>lex\<^esub>\<close> is trichotomous, we need to consider
+ three cases, namely @{term "p = q"}, @{term "p \<sqsubset>lex q"} and
+ @{term "q \<sqsubset>lex p"}. Let us look at the first case.  Clearly
+ @{term "pflat_len v\<^sub>2 p < pflat_len v\<^sub>1 p"} and @{term
+ "pflat_len v\<^sub>3 p < pflat_len v\<^sub>2 p"} imply @{term
+ "pflat_len v\<^sub>3 p < pflat_len v\<^sub>1 p"}.  It remains to show
+ that for a @{term "p' \<in> Pos v\<^sub>1 \<union> Pos v\<^sub>3"}
+ with @{term "p' \<sqsubset>lex p"} that @{term "pflat_len v\<^sub>1
+ p' = pflat_len v\<^sub>3 p'"} holds.  Suppose @{term "p' \<in> Pos
+ v\<^sub>1"}, then we can infer from the first assumption that @{term
+ "pflat_len v\<^sub>1 p' = pflat_len v\<^sub>2 p'"}.  But this means
+ that @{term "p'"} must be in @{term "Pos v\<^sub>2"} too (the norm
+ cannot be \<open>-1\<close> given @{term "p' \<in> Pos v\<^sub>1"}).  
+ Hence we can use the second assumption and
+ infer @{term "pflat_len v\<^sub>2 p' = pflat_len v\<^sub>3 p'"},
+ which concludes this case with @{term "v\<^sub>1 :\<sqsubset>val
+ v\<^sub>3"}.  The reasoning in the other cases is similar.\qed
+ \end{proof}
+
+ \noindent 
+ The proof for $\preccurlyeq$ is similar and omitted.
+ It is also straightforward to show that \<open>\<prec>\<close> and
+ $\preccurlyeq$ are partial orders.  Okui and Suzuki furthermore show that they
+ are linear orderings for lexical values \cite{OkuiSuzuki2010} of a given
+ regular expression and given string, but we have not formalised this in Isabelle. It is
+ not essential for our results. What we are going to show below is
+ that for a given \<open>r\<close> and \<open>s\<close>, the orderings have a unique
+ minimal element on the set @{term "LV r s"}, which is the POSIX value
+ we defined in the previous section. We start with two properties that
+ show how the length of a flattened value relates to the \<open>\<prec>\<close>-ordering.
+
+ \begin{proposition}\mbox{}\smallskip\\\label{ordlen}
+ \begin{tabular}{@ {}ll}
+ (1) &
+ @{thm [mode=IfThen] PosOrd_shorterE[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+ (2) &
+ @{thm [mode=IfThen] PosOrd_shorterI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} 
+ \end{tabular} 
+ \end{proposition}
+ 
+ \noindent Both properties follow from the definition of the ordering. Note that
+ \textit{(2)} entails that a value, say @{term "v\<^sub>2"}, whose underlying 
+ string is a strict prefix of another flattened value, say @{term "v\<^sub>1"}, then
+ @{term "v\<^sub>1"} must be smaller than @{term "v\<^sub>2"}. For our proofs it
+ will be useful to have the following properties---in each case the underlying strings 
+ of the compared values are the same: 
+
+  \begin{proposition}\mbox{}\smallskip\\\label{ordintros}
+  \begin{tabular}{ll}
+  \textit{(1)} & 
+  @{thm [mode=IfThen] PosOrd_Left_Right[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+  \textit{(2)} & If
+  @{thm (prem 1) PosOrd_Left_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;then\;
+  @{thm (lhs) PosOrd_Left_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;iff\;
+  @{thm (rhs) PosOrd_Left_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+  \textit{(3)} & If
+  @{thm (prem 1) PosOrd_Right_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;then\;
+  @{thm (lhs) PosOrd_Right_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;iff\;
+  @{thm (rhs) PosOrd_Right_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+  \textit{(4)} & If
+  @{thm (prem 1) PosOrd_Seq_eq[where ?v2.0="v\<^sub>2" and ?w2.0="w\<^sub>2"]} \;then\;
+  @{thm (lhs) PosOrd_Seq_eq[where ?v2.0="v\<^sub>2" and ?w2.0="w\<^sub>2"]} \;iff\;
+  @{thm (rhs) PosOrd_Seq_eq[where ?v2.0="v\<^sub>2" and ?w2.0="w\<^sub>2"]}\\
+  \textit{(5)} & If
+  @{thm (prem 2) PosOrd_SeqI1[simplified, where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+                                    ?w1.0="w\<^sub>1" and ?w2.0="w\<^sub>2"]} \;and\;
+  @{thm (prem 1) PosOrd_SeqI1[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+                                    ?w1.0="w\<^sub>1" and ?w2.0="w\<^sub>2"]} \;then\;
+  @{thm (concl) PosOrd_SeqI1[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+                                   ?w1.0="w\<^sub>1" and ?w2.0="w\<^sub>2"]}\\
+  \textit{(6)} & If
+  @{thm (prem 1) PosOrd_Stars_append_eq[where ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;then\;
+  @{thm (lhs) PosOrd_Stars_append_eq[where ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;iff\;
+  @{thm (rhs) PosOrd_Stars_append_eq[where ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]}\\  
+  
+  \textit{(7)} & If
+  @{thm (prem 2) PosOrd_StarsI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+                            ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;and\;
+  @{thm (prem 1) PosOrd_StarsI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+                            ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;then\;
+   @{thm (concl) PosOrd_StarsI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+                            ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]}\\
+  \end{tabular} 
+  \end{proposition}
+
+  \noindent One might prefer that statements \textit{(4)} and \textit{(5)} 
+  (respectively \textit{(6)} and \textit{(7)})
+  are combined into a single \textit{iff}-statement (like the ones for \<open>Left\<close> and \<open>Right\<close>). Unfortunately this cannot be done easily: such
+  a single statement would require an additional assumption about the
+  two values @{term "Seq v\<^sub>1 v\<^sub>2"} and @{term "Seq w\<^sub>1 w\<^sub>2"}
+  being inhabited by the same regular expression. The
+  complexity of the proofs involved seems to not justify such a
+  `cleaner' single statement. The statements given are just the properties that
+  allow us to establish our theorems without any difficulty. The proofs 
+  for Proposition~\ref{ordintros} are routine.
+ 
+
+  Next we establish how Okui and Suzuki's orderings relate to our
+  definition of POSIX values.  Given a \<open>POSIX\<close> value \<open>v\<^sub>1\<close>
+  for \<open>r\<close> and \<open>s\<close>, then any other lexical value \<open>v\<^sub>2\<close> in @{term "LV r s"} is greater or equal than \<open>v\<^sub>1\<close>, namely:
+
+
+  \begin{theorem}\label{orderone}
+  @{thm [mode=IfThen] Posix_PosOrd[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
+  \end{theorem}
+
+  \begin{proof} By induction on our POSIX rules. By
+  Theorem~\ref{posixdeterm} and the definition of @{const LV}, it is clear
+  that \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> have the same
+  underlying string @{term s}.  The three base cases are
+  straightforward: for example for @{term "v\<^sub>1 = Void"}, we have
+  that @{term "v\<^sub>2 \<in> LV ONE []"} must also be of the form
+  \mbox{@{term "v\<^sub>2 = Void"}}. Therefore we have @{term
+  "v\<^sub>1 :\<sqsubseteq>val v\<^sub>2"}.  The inductive cases for
+  \<open>r\<close> being of the form @{term "ALT r\<^sub>1 r\<^sub>2"} and
+  @{term "SEQ r\<^sub>1 r\<^sub>2"} are as follows:
+
+
+  \begin{itemize} 
+
+  \item[$\bullet$] Case \<open>P+L\<close> with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
+  \<rightarrow> (Left w\<^sub>1)"}: In this case the value 
+  @{term "v\<^sub>2"} is either of the
+  form @{term "Left w\<^sub>2"} or @{term "Right w\<^sub>2"}. In the
+  latter case we can immediately conclude with \mbox{@{term "v\<^sub>1
+  :\<sqsubseteq>val v\<^sub>2"}} since a \<open>Left\<close>-value with the
+  same underlying string \<open>s\<close> is always smaller than a
+  \<open>Right\<close>-value by Proposition~\ref{ordintros}\textit{(1)}.  
+  In the former case we have @{term "w\<^sub>2
+  \<in> LV r\<^sub>1 s"} and can use the induction hypothesis to infer
+  @{term "w\<^sub>1 :\<sqsubseteq>val w\<^sub>2"}. Because @{term
+  "w\<^sub>1"} and @{term "w\<^sub>2"} have the same underlying string
+  \<open>s\<close>, we can conclude with @{term "Left w\<^sub>1
+  :\<sqsubseteq>val Left w\<^sub>2"} using
+  Proposition~\ref{ordintros}\textit{(2)}.\smallskip
+
+  \item[$\bullet$] Case \<open>P+R\<close> with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
+  \<rightarrow> (Right w\<^sub>1)"}: This case similar to the previous
+  case, except that we additionally know @{term "s \<notin> L
+  r\<^sub>1"}. This is needed when @{term "v\<^sub>2"} is of the form
+  \mbox{@{term "Left w\<^sub>2"}}. Since \mbox{@{term "flat v\<^sub>2 = flat
+  w\<^sub>2"} \<open>= s\<close>} and @{term "\<Turnstile> w\<^sub>2 :
+  r\<^sub>1"}, we can derive a contradiction for \mbox{@{term "s \<notin> L
+  r\<^sub>1"}} using
+  Proposition~\ref{inhabs}. So also in this case \mbox{@{term "v\<^sub>1
+  :\<sqsubseteq>val v\<^sub>2"}}.\smallskip
+
+  \item[$\bullet$] Case \<open>PS\<close> with @{term "(s\<^sub>1 @
+  s\<^sub>2) \<in> (SEQ r\<^sub>1 r\<^sub>2) \<rightarrow> (Seq
+  w\<^sub>1 w\<^sub>2)"}: We can assume @{term "v\<^sub>2 = Seq
+  (u\<^sub>1) (u\<^sub>2)"} with @{term "\<Turnstile> u\<^sub>1 :
+  r\<^sub>1"} and \mbox{@{term "\<Turnstile> u\<^sub>2 :
+  r\<^sub>2"}}. We have @{term "s\<^sub>1 @ s\<^sub>2 = (flat
+  u\<^sub>1) @ (flat u\<^sub>2)"}.  By the side-condition of the
+  \<open>PS\<close>-rule we know that either @{term "s\<^sub>1 = flat
+  u\<^sub>1"} or that @{term "flat u\<^sub>1"} is a strict prefix of
+  @{term "s\<^sub>1"}. In the latter case we can infer @{term
+  "w\<^sub>1 :\<sqsubset>val u\<^sub>1"} by
+  Proposition~\ref{ordlen}\textit{(2)} and from this @{term "v\<^sub>1
+  :\<sqsubseteq>val v\<^sub>2"} by Proposition~\ref{ordintros}\textit{(5)}
+  (as noted above @{term "v\<^sub>1"} and @{term "v\<^sub>2"} must have the
+  same underlying string).
+  In the former case we know
+  @{term "u\<^sub>1 \<in> LV r\<^sub>1 s\<^sub>1"} and @{term
+  "u\<^sub>2 \<in> LV r\<^sub>2 s\<^sub>2"}. With this we can use the
+  induction hypotheses to infer @{term "w\<^sub>1 :\<sqsubseteq>val
+  u\<^sub>1"} and @{term "w\<^sub>2 :\<sqsubseteq>val u\<^sub>2"}. By
+  Proposition~\ref{ordintros}\textit{(4,5)} we can again infer 
+  @{term "v\<^sub>1 :\<sqsubseteq>val
+  v\<^sub>2"}.
+
+  \end{itemize}
+
+  \noindent The case for \<open>P\<star>\<close> is similar to the \<open>PS\<close>-case and omitted.\qed
+  \end{proof}
+
+  \noindent This theorem shows that our \<open>POSIX\<close> value for a
+  regular expression \<open>r\<close> and string @{term s} is in fact a
+  minimal element of the values in \<open>LV r s\<close>. By
+  Proposition~\ref{ordlen}\textit{(2)} we also know that any value in
+  \<open>LV r s'\<close>, with @{term "s'"} being a strict prefix, cannot be
+  smaller than \<open>v\<^sub>1\<close>. The next theorem shows the
+  opposite---namely any minimal element in @{term "LV r s"} must be a
+  \<open>POSIX\<close> value. This can be established by induction on \<open>r\<close>, but the proof can be drastically simplified by using the fact
+  from the previous section about the existence of a \<open>POSIX\<close> value
+  whenever a string @{term "s \<in> L r"}.
+
+
+  \begin{theorem}
+  @{thm [mode=IfThen] PosOrd_Posix[where ?v1.0="v\<^sub>1"]} 
+  \end{theorem}
+
+  \begin{proof} 
+  If @{thm (prem 1) PosOrd_Posix[where ?v1.0="v\<^sub>1"]} then 
+  @{term "s \<in> L r"} by Proposition~\ref{inhabs}. Hence by Theorem~\ref{lexercorrect}(2) 
+  there exists a
+  \<open>POSIX\<close> value @{term "v\<^sub>P"} with @{term "s \<in> r \<rightarrow> v\<^sub>P"}
+  and by Lemma~\ref{LVposix} we also have \mbox{@{term "v\<^sub>P \<in> LV r s"}}.
+  By Theorem~\ref{orderone} we therefore have 
+  @{term "v\<^sub>P :\<sqsubseteq>val v\<^sub>1"}. If @{term "v\<^sub>P = v\<^sub>1"} then
+  we are done. Otherwise we have @{term "v\<^sub>P :\<sqsubset>val v\<^sub>1"}, which 
+  however contradicts the second assumption about @{term "v\<^sub>1"} being the smallest
+  element in @{term "LV r s"}. So we are done in this case too.\qed
+  \end{proof}
+
+  \noindent
+  From this we can also show 
+  that if @{term "LV r s"} is non-empty (or equivalently @{term "s \<in> L r"}) then 
+  it has a unique minimal element:
+
+  \begin{corollary}
+  @{thm [mode=IfThen] Least_existence1}
+  \end{corollary}
+
+
+
+  \noindent To sum up, we have shown that the (unique) minimal elements 
+  of the ordering by Okui and Suzuki are exactly the \<open>POSIX\<close>
+  values we defined inductively in Section~\ref{posixsec}. This provides
+  an independent confirmation that our ternary relation formalises the
+  informal POSIX rules. 
+
+\<close>
+
+section \<open>Bitcoded Lexing\<close>
+
+
+
+
+text \<open>
+
+Incremental calculation of the value. To simplify the proof we first define the function
+@{const flex} which calculates the ``iterated'' injection function. With this we can 
+rewrite the lexer as
+
+\begin{center}
+@{thm lexer_flex}
+\end{center}
+
+
+\<close>
+
+section \<open>Optimisations\<close>
+
+text \<open>
+
+  Derivatives as calculated by \Brz's method are usually more complex
+  regular expressions than the initial one; the result is that the
+  derivative-based matching and lexing algorithms are often abysmally slow.
+  However, 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}. These simplifications can speed up the
+  algorithms considerably, as noted in \cite{Sulzmann2014}. One of the
+  advantages of having a simple specification and correctness proof is that
+  the latter can be refined to prove the correctness of such simplification
+  steps. While the simplification of regular expressions according to 
+  rules like
+
+  \begin{equation}\label{Simpl}
+  \begin{array}{lcllcllcllcl}
+  @{term "ALT ZERO r"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+  @{term "ALT r ZERO"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+  @{term "SEQ ONE r"}  & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+  @{term "SEQ r ONE"}  & \<open>\<Rightarrow>\<close> & @{term r}
+  \end{array}
+  \end{equation}
+
+  \noindent is well understood, there is an obstacle with the POSIX value
+  calculation algorithm by Sulzmann and Lu: if we build a derivative regular
+  expression and then simplify it, we will calculate a POSIX value for this
+  simplified derivative regular expression, \emph{not} for the original (unsimplified)
+  derivative regular expression. Sulzmann and Lu \cite{Sulzmann2014} overcome this obstacle by
+  not just calculating a simplified regular expression, but also calculating
+  a \emph{rectification function} that ``repairs'' the incorrect value.
+  
+  The rectification functions can be (slightly clumsily) implemented  in
+  Isabelle/HOL as follows using some auxiliary functions:
+
+  \begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) F_RIGHT.simps(1)} & $\dn$ & \<open>Right (f v)\<close>\\
+  @{thm (lhs) F_LEFT.simps(1)} & $\dn$ & \<open>Left (f v)\<close>\\
+  
+  @{thm (lhs) F_ALT.simps(1)} & $\dn$ & \<open>Right (f\<^sub>2 v)\<close>\\
+  @{thm (lhs) F_ALT.simps(2)} & $\dn$ & \<open>Left (f\<^sub>1 v)\<close>\\
+  
+  @{thm (lhs) F_SEQ1.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 ()) (f\<^sub>2 v)\<close>\\
+  @{thm (lhs) F_SEQ2.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v) (f\<^sub>2 ())\<close>\\
+  @{thm (lhs) F_SEQ.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)\<close>\medskip\\
+  %\end{tabular}
+  %
+  %\begin{tabular}{lcl}
+  @{term "simp_ALT (ZERO, DUMMY) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_RIGHT f\<^sub>2)"}\\
+  @{term "simp_ALT (r\<^sub>1, f\<^sub>1) (ZERO, DUMMY)"} & $\dn$ & @{term "(r\<^sub>1, F_LEFT f\<^sub>1)"}\\
+  @{term "simp_ALT (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(ALT r\<^sub>1 r\<^sub>2, F_ALT f\<^sub>1 f\<^sub>2)"}\\
+  @{term "simp_SEQ (ONE, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_SEQ1 f\<^sub>1 f\<^sub>2)"}\\
+  @{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (ONE, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>1, F_SEQ2 f\<^sub>1 f\<^sub>2)"}\\
+  @{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(SEQ r\<^sub>1 r\<^sub>2, F_SEQ f\<^sub>1 f\<^sub>2)"}\\
+  \end{tabular}
+  \end{center}
+
+  \noindent
+  The functions \<open>simp\<^bsub>Alt\<^esub>\<close> and \<open>simp\<^bsub>Seq\<^esub>\<close> encode the simplification rules
+  in \eqref{Simpl} and compose the rectification functions (simplifications can occur
+  deep inside the regular expression). The main simplification function is then 
+
+  \begin{center}
+  \begin{tabular}{lcl}
+  @{term "simp (ALT r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_ALT (simp r\<^sub>1) (simp r\<^sub>2)"}\\
+  @{term "simp (SEQ r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_SEQ (simp r\<^sub>1) (simp r\<^sub>2)"}\\
+  @{term "simp r"} & $\dn$ & @{term "(r, id)"}\\
+  \end{tabular}
+  \end{center} 
+
+  \noindent where @{term "id"} stands for the identity function. The
+  function @{const simp} returns a simplified regular expression and a corresponding
+  rectification function. Note that we do not simplify under stars: this
+  seems to slow down the algorithm, rather than speed it up. The optimised
+  lexer is then given by the clauses:
+  
+  \begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) slexer.simps(1)} & $\dn$ & @{thm (rhs) slexer.simps(1)}\\
+  @{thm (lhs) slexer.simps(2)} & $\dn$ & 
+                         \<open>let (r\<^sub>s, f\<^sub>r) = simp (r \<close>$\backslash$\<open> c) in\<close>\\
+                     & & \<open>case\<close> @{term "slexer r\<^sub>s s"} \<open>of\<close>\\
+                     & & \phantom{$|$} @{term "None"}  \<open>\<Rightarrow>\<close> @{term None}\\
+                     & & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> \<open>Some (inj r c (f\<^sub>r v))\<close>                          
+  \end{tabular}
+  \end{center}
+
+  \noindent
+  In the second clause we first calculate the derivative @{term "der c r"}
+  and then simpli
+
+text \<open>
+
+Incremental calculation of the value. To simplify the proof we first define the function
+@{const flex} which calculates the ``iterated'' injection function. With this we can 
+rewrite the lexer as
+
+\begin{center}
+@{thm lexer_flex}
+\end{center}
+
+\begin{center}
+  \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)}\\
+  @{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)}
+\end{tabular}
+\end{center}
+
+\begin{center}
+\begin{tabular}{lcl}
+  @{term areg} & $::=$ & @{term "AZERO"}\\
+               & $\mid$ & @{term "AONE bs"}\\
+               & $\mid$ & @{term "ACHAR bs c"}\\
+               & $\mid$ & @{term "AALT bs r1 r2"}\\
+               & $\mid$ & @{term "ASEQ bs r\<^sub>1 r\<^sub>2"}\\
+               & $\mid$ & @{term "ASTAR bs r"}
+\end{tabular}
+\end{center}
+
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) intern.simps(1)} & $\dn$ & @{thm (rhs) intern.simps(1)}\\
+  @{thm (lhs) intern.simps(2)} & $\dn$ & @{thm (rhs) intern.simps(2)}\\
+  @{thm (lhs) intern.simps(3)} & $\dn$ & @{thm (rhs) intern.simps(3)}\\
+  @{thm (lhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) intern.simps(6)} & $\dn$ & @{thm (rhs) intern.simps(6)}\\
+\end{tabular}
+\end{center}
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) erase.simps(1)} & $\dn$ & @{thm (rhs) erase.simps(1)}\\
+  @{thm (lhs) erase.simps(2)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(2)[of bs]}\\
+  @{thm (lhs) erase.simps(3)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(3)[of bs]}\\
+  @{thm (lhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) erase.simps(6)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(6)[of bs]}\\
+\end{tabular}
+\end{center}
+
+Some simple facts about erase
+
+\begin{lemma}\mbox{}\\
+@{thm erase_bder}\\
+@{thm erase_intern}
+\end{lemma}
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) bnullable.simps(1)} & $\dn$ & @{thm (rhs) bnullable.simps(1)}\\
+  @{thm (lhs) bnullable.simps(2)} & $\dn$ & @{thm (rhs) bnullable.simps(2)}\\
+  @{thm (lhs) bnullable.simps(3)} & $\dn$ & @{thm (rhs) bnullable.simps(3)}\\
+  @{thm (lhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bnullable.simps(6)} & $\dn$ & @{thm (rhs) bnullable.simps(6)}\medskip\\
+
+%  \end{tabular}
+%  \end{center}
+
+%  \begin{center}
+%  \begin{tabular}{lcl}
+
+  @{thm (lhs) bder.simps(1)} & $\dn$ & @{thm (rhs) bder.simps(1)}\\
+  @{thm (lhs) bder.simps(2)} & $\dn$ & @{thm (rhs) bder.simps(2)}\\
+  @{thm (lhs) bder.simps(3)} & $\dn$ & @{thm (rhs) bder.simps(3)}\\
+  @{thm (lhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bder.simps(6)} & $\dn$ & @{thm (rhs) bder.simps(6)}
+  \end{tabular}
+  \end{center}
+
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) bmkeps.simps(1)} & $\dn$ & @{thm (rhs) bmkeps.simps(1)}\\
+  @{thm (lhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bmkeps.simps(4)} & $\dn$ & @{thm (rhs) bmkeps.simps(4)}\medskip\\
+\end{tabular}
+\end{center}
+
+
+@{thm [mode=IfThen] bder_retrieve}
+
+By induction on \<open>r\<close>
+
+\begin{theorem}[Main Lemma]\mbox{}\\
+@{thm [mode=IfThen] MAIN_decode}
+\end{theorem}
+
+\noindent
+Definition of the bitcoded lexer
+
+@{thm blexer_def}
+
+
+\begin{theorem}
+@{thm blexer_correctness}
+\end{theorem}
+
+\<close>
+
+section \<open>Optimisations\<close>
+
+text \<open>
+
+  Derivatives as calculated by \Brz's method are usually more complex
+  regular expressions than the initial one; the result is that the
+  derivative-based matching and lexing algorithms are often abysmally slow.
+  However, 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}. These simplifications can speed up the
+  algorithms considerably, as noted in \cite{Sulzmann2014}. One of the
+  advantages of having a simple specification and correctness proof is that
+  the latter can be refined to prove the correctness of such simplification
+  steps. While the simplification of regular expressions according to 
+  rules like
+
+  \begin{equation}\label{Simpl}
+  \begin{array}{lcllcllcllcl}
+  @{term "ALT ZERO r"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+  @{term "ALT r ZERO"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+  @{term "SEQ ONE r"}  & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+  @{term "SEQ r ONE"}  & \<open>\<Rightarrow>\<close> & @{term r}
+  \end{array}
+  \end{equation}
+
+  \noindent is well understood, there is an obstacle with the POSIX value
+  calculation algorithm by Sulzmann and Lu: if we build a derivative regular
+  expression and then simplify it, we will calculate a POSIX value for this
+  simplified derivative regular expression, \emph{not} for the original (unsimplified)
+  derivative regular expression. Sulzmann and Lu \cite{Sulzmann2014} overcome this obstacle by
+  not just calculating a simplified regular expression, but also calculating
+  a \emph{rectification function} that ``repairs'' the incorrect value.
+  
+  The rectification functions can be (slightly clumsily) implemented  in
+  Isabelle/HOL as follows using some auxiliary functions:
+
+  \begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) F_RIGHT.simps(1)} & $\dn$ & \<open>Right (f v)\<close>\\
+  @{thm (lhs) F_LEFT.simps(1)} & $\dn$ & \<open>Left (f v)\<close>\\
+  
+  @{thm (lhs) F_ALT.simps(1)} & $\dn$ & \<open>Right (f\<^sub>2 v)\<close>\\
+  @{thm (lhs) F_ALT.simps(2)} & $\dn$ & \<open>Left (f\<^sub>1 v)\<close>\\
+  
+  @{thm (lhs) F_SEQ1.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 ()) (f\<^sub>2 v)\<close>\\
+  @{thm (lhs) F_SEQ2.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v) (f\<^sub>2 ())\<close>\\
+  @{thm (lhs) F_SEQ.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)\<close>\medskip\\
+  %\end{tabular}
+  %
+  %\begin{tabular}{lcl}
+  @{term "simp_ALT (ZERO, DUMMY) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_RIGHT f\<^sub>2)"}\\
+  @{term "simp_ALT (r\<^sub>1, f\<^sub>1) (ZERO, DUMMY)"} & $\dn$ & @{term "(r\<^sub>1, F_LEFT f\<^sub>1)"}\\
+  @{term "simp_ALT (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(ALT r\<^sub>1 r\<^sub>2, F_ALT f\<^sub>1 f\<^sub>2)"}\\
+  @{term "simp_SEQ (ONE, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_SEQ1 f\<^sub>1 f\<^sub>2)"}\\
+  @{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (ONE, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>1, F_SEQ2 f\<^sub>1 f\<^sub>2)"}\\
+  @{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(SEQ r\<^sub>1 r\<^sub>2, F_SEQ f\<^sub>1 f\<^sub>2)"}\\
+  \end{tabular}
+  \end{center}
+
+  \noindent
+  The functions \<open>simp\<^bsub>Alt\<^esub>\<close> and \<open>simp\<^bsub>Seq\<^esub>\<close> encode the simplification rules
+  in \eqref{Simpl} and compose the rectification functions (simplifications can occur
+  deep inside the regular expression). The main simplification function is then 
+
+  \begin{center}
+  \begin{tabular}{lcl}
+  @{term "simp (ALT r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_ALT (simp r\<^sub>1) (simp r\<^sub>2)"}\\
+  @{term "simp (SEQ r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_SEQ (simp r\<^sub>1) (simp r\<^sub>2)"}\\
+  @{term "simp r"} & $\dn$ & @{term "(r, id)"}\\
+  \end{tabular}
+  \end{center} 
+
+  \noindent where @{term "id"} stands for the identity function. The
+  function @{const simp} returns a simplified regular expression and a corresponding
+  rectification function. Note that we do not simplify under stars: this
+  seems to slow down the algorithm, rather than speed it up. The optimised
+  lexer is then given by the clauses:
+  
+  \begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) slexer.simps(1)} & $\dn$ & @{thm (rhs) slexer.simps(1)}\\
+  @{thm (lhs) slexer.simps(2)} & $\dn$ & 
+                         \<open>let (r\<^sub>s, f\<^sub>r) = simp (r \<close>$\backslash$\<open> c) in\<close>\\
+                     & & \<open>case\<close> @{term "slexer r\<^sub>s s"} \<open>of\<close>\\
+                     & & \phantom{$|$} @{term "None"}  \<open>\<Rightarrow>\<close> @{term None}\\
+                     & & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> \<open>Some (inj r c (f\<^sub>r v))\<close>                          
+  \end{tabular}
+  \end{center}
+
+  \noindent
+  In the second clause we first calculate the derivative @{term "der c r"}
+  and then simplify the result. This gives us a simplified derivative
+  \<open>r\<^sub>s\<close> and a rectification function \<open>f\<^sub>r\<close>. The lexer
+  is then recursively called with the simplified derivative, but before
+  we inject the character @{term c} into the value @{term v}, we need to rectify
+  @{term v} (that is construct @{term "f\<^sub>r v"}). Before we can establish the correctness
+  of @{term "slexer"}, we need to show that simplification preserves the language
+  and simplification preserves our POSIX relation once the value is rectified
+  (recall @{const "simp"} generates a (regular expression, rectification function) pair):
+
+  \begin{lemma}\mbox{}\smallskip\\\label{slexeraux}
+  \begin{tabular}{ll}
+  (1) & @{thm L_fst_simp[symmetric]}\\
+  (2) & @{thm[mode=IfThen] Posix_simp}
+  \end{tabular}
+  \end{lemma}
+
+  \begin{proof} Both are by induction on \<open>r\<close>. There is no
+  interesting case for the first statement. For the second statement,
+  of interest are the @{term "r = ALT r\<^sub>1 r\<^sub>2"} and @{term "r = SEQ r\<^sub>1
+  r\<^sub>2"} cases. In each case we have to analyse four subcases whether
+  @{term "fst (simp r\<^sub>1)"} and @{term "fst (simp r\<^sub>2)"} equals @{const
+  ZERO} (respectively @{const ONE}). For example for @{term "r = ALT
+  r\<^sub>1 r\<^sub>2"}, consider the subcase @{term "fst (simp r\<^sub>1) = ZERO"} and
+  @{term "fst (simp r\<^sub>2) \<noteq> ZERO"}. By assumption we know @{term "s \<in>
+  fst (simp (ALT r\<^sub>1 r\<^sub>2)) \<rightarrow> v"}. From this we can infer @{term "s \<in> fst (simp r\<^sub>2) \<rightarrow> v"}
+  and by IH also (*) @{term "s \<in> r\<^sub>2 \<rightarrow> (snd (simp r\<^sub>2) v)"}. Given @{term "fst (simp r\<^sub>1) = ZERO"}
+  we know @{term "L (fst (simp r\<^sub>1)) = {}"}. By the first statement
+  @{term "L r\<^sub>1"} is the empty set, meaning (**) @{term "s \<notin> L r\<^sub>1"}.
+  Taking (*) and (**) together gives by the \mbox{\<open>P+R\<close>}-rule 
+  @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> Right (snd (simp r\<^sub>2) v)"}. In turn this
+  gives @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> snd (simp (ALT r\<^sub>1 r\<^sub>2)) v"} as we need to show.
+  The other cases are similar.\qed
+  \end{proof}
+
+  \noindent We can now prove relatively straightforwardly that the
+  optimised lexer produces the expected result:
+
+  \begin{theorem}
+  @{thm slexer_correctness}
+  \end{theorem}
+
+  \begin{proof} By induction on @{term s} generalising over @{term
+  r}. The case @{term "[]"} is trivial. For the cons-case suppose the
+  string is of the form @{term "c # s"}. By induction hypothesis we
+  know @{term "slexer r s = lexer r s"} holds for all @{term r} (in
+  particular for @{term "r"} being the derivative @{term "der c
+  r"}). Let @{term "r\<^sub>s"} be the simplified derivative regular expression, that is @{term
+  "fst (simp (der c r))"}, and @{term "f\<^sub>r"} be the rectification
+  function, that is @{term "snd (simp (der c r))"}.  We distinguish the cases
+  whether (*) @{term "s \<in> L (der c r)"} or not. In the first case we
+  have by Theorem~\ref{lexercorrect}(2) a value @{term "v"} so that @{term
+  "lexer (der c r) s = Some v"} and @{term "s \<in> der c r \<rightarrow> v"} hold.
+  By Lemma~\ref{slexeraux}(1) we can also infer from~(*) that @{term "s
+  \<in> L r\<^sub>s"} holds.  Hence we know by Theorem~\ref{lexercorrect}(2) that
+  there exists a @{term "v'"} with @{term "lexer r\<^sub>s s = Some v'"} and
+  @{term "s \<in> r\<^sub>s \<rightarrow> v'"}. From the latter we know by
+  Lemma~\ref{slexeraux}(2) that @{term "s \<in> der c r \<rightarrow> (f\<^sub>r v')"} holds.
+  By the uniqueness of the POSIX relation (Theorem~\ref{posixdeterm}) we
+  can infer that @{term v} is equal to @{term "f\<^sub>r v'"}---that is the 
+  rectification function applied to @{term "v'"}
+  produces the original @{term "v"}.  Now the case follows by the
+  definitions of @{const lexer} and @{const slexer}.
+
+  In the second case where @{term "s \<notin> L (der c r)"} we have that
+  @{term "lexer (der c r) s = None"} by Theorem~\ref{lexercorrect}(1).  We
+  also know by Lemma~\ref{slexeraux}(1) that @{term "s \<notin> L r\<^sub>s"}. Hence
+  @{term "lexer r\<^sub>s s = None"} by Theorem~\ref{lexercorrect}(1) and
+  by IH then also @{term "slexer r\<^sub>s s = None"}. With this we can
+  conclude in this case too.\qed   
+
+  \end{proof} 
+
+\<close>
+fy the result. This gives us a simplified derivative
+  \<open>r\<^sub>s\<close> and a rectification function \<open>f\<^sub>r\<close>. The lexer
+  is then recursively called with the simplified derivative, but before
+  we inject the character @{term c} into the value @{term v}, we need to rectify
+  @{term v} (that is construct @{term "f\<^sub>r v"}). Before we can establish the correctness
+  of @{term "slexer"}, we need to show that simplification preserves the language
+  and simplification preserves our POSIX relation once the value is rectified
+  (recall @{const "simp"} generates a (regular expression, rectification function) pair):
+
+  \begin{lemma}\mbox{}\smallskip\\\label{slexeraux}
+  \begin{tabular}{ll}
+  (1) & @{thm L_fst_simp[symmetric]}\\
+  (2) & @{thm[mode=IfThen] Posix_simp}
+  \end{tabular}
+  \end{lemma}
+
+  \begin{proof} Both are by induction on \<open>r\<close>. There is no
+  interesting case for the first statement. For the second statement,
+  of interest are the @{term "r = ALT r\<^sub>1 r\<^sub>2"} and @{term "r = SEQ r\<^sub>1
+  r\<^sub>2"} cases. In each case we have to analyse four subcases whether
+  @{term "fst (simp r\<^sub>1)"} and @{term "fst (simp r\<^sub>2)"} equals @{const
+  ZERO} (respectively @{const ONE}). For example for @{term "r = ALT
+  r\<^sub>1 r\<^sub>2"}, consider the subcase @{term "fst (simp r\<^sub>1) = ZERO"} and
+  @{term "fst (simp r\<^sub>2) \<noteq> ZERO"}. By assumption we know @{term "s \<in>
+  fst (simp (ALT r\<^sub>1 r\<^sub>2)) \<rightarrow> v"}. From this we can infer @{term "s \<in> fst (simp r\<^sub>2) \<rightarrow> v"}
+  and by IH also (*) @{term "s \<in> r\<^sub>2 \<rightarrow> (snd (simp r\<^sub>2) v)"}. Given @{term "fst (simp r\<^sub>1) = ZERO"}
+  we know @{term "L (fst (simp r\<^sub>1)) = {}"}. By the first statement
+  @{term "L r\<^sub>1"} is the empty set, meaning (**) @{term "s \<notin> L r\<^sub>1"}.
+  Taking (*) and (**) together gives by the \mbox{\<open>P+R\<close>}-rule 
+  @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> Right (snd (simp r\<^sub>2) v)"}. In turn this
+  gives @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> snd (simp (ALT r\<^sub>1 r\<^sub>2)) v"} as we need to show.
+  The other cases are similar.\qed
+  \end{proof}
+
+  \noindent We can now prove relatively straightforwardly that the
+  optimised lexer produces the expected result:
+
+  \begin{theorem}
+  @{thm slexer_correctness}
+  \end{theorem}
+
+  \begin{proof} By induction on @{term s} generalising over @{term
+  r}. The case @{term "[]"} is trivial. For the cons-case suppose the
+  string is of the form @{term "c # s"}. By induction hypothesis we
+  know @{term "slexer r s = lexer r s"} holds for all @{term r} (in
+  particular for @{term "r"} being the derivative @{term "der c
+  r"}). Let @{term "r\<^sub>s"} be the simplified derivative regular expression, that is @{term
+  "fst (simp (der c r))"}, and @{term "f\<^sub>r"} be the rectification
+  function, that is @{term "snd (simp (der c r))"}.  We distinguish the cases
+  whether (*) @{term "s \<in> L (der c r)"} or not. In the first case we
+  have by Theorem~\ref{lexercorrect}(2) a value @{term "v"} so that @{term
+  "lexer (der c r) s = Some v"} and @{term "s \<in> der c r \<rightarrow> v"} hold.
+  By Lemma~\ref{slexeraux}(1) we can also infer from~(*) that @{term "s
+  \<in> L r\<^sub>s"} holds.  Hence we know by Theorem~\ref{lexercorrect}(2) that
+  there exists a @{term "v'"} with @{term "lexer r\<^sub>s s = Some v'"} and
+  @{term "s \<in> r\<^sub>s \<rightarrow> v'"}. From the latter we know by
+  Lemma~\ref{slexeraux}(2) that @{term "s \<in> der c r \<rightarrow> (f\<^sub>r v')"} holds.
+  By the uniqueness of the POSIX relation (Theorem~\ref{posixdeterm}) we
+  can infer that @{term v} is equal to @{term "f\<^sub>r v'"}---that is the 
+  rectification function applied to @{term "v'"}
+  produces the original @{term "v"}.  Now the case follows by the
+  definitions of @{const lexer} and @{const slexer}.
+
+  In the second case where @{term "s \<notin> L (der c r)"} we have that
+  @{term "lexer (der c r) s = None"} by Theorem~\ref{lexercorrect}(1).  We
+  also know by Lemma~\ref{slexeraux}(1) that @{term "s \<notin> L r\<^sub>s"}. Hence
+  @{term "lexer r\<^sub>s s = None"} by Theorem~\ref{lexercorrect}(1) and
+  by IH then also @{term "slexer r\<^sub>s s = None"}. With this we can
+  conclude in this case too.\qed   
+
+  \end{proof} 
+
+\<close>
+
+
+section \<open>HERE\<close>
+
+text \<open>
+
+  \begin{lemma}
+  @{thm [mode=IfThen] bder_retrieve}
+  \end{lemma}
+
+  \begin{proof}
+  By induction on the definition of @{term "erase r"}. The cases for rule 1) and 2) are
+  straightforward as @{term "der c ZERO"} and @{term "der c ONE"} are both equal to 
+  @{term ZERO}. This means @{term "\<Turnstile> v : ZERO"} cannot hold. Similarly in case of rule 3)
+  where @{term r} is of the form @{term "ACHAR d"} with @{term "c = d"}. Then by assumption
+  we know @{term "\<Turnstile> v : ONE"}, which implies @{term "v = Void"}. The equation follows by 
+  simplification of left- and right-hand side. In  case @{term "c \<noteq> d"} we have again
+  @{term "\<Turnstile> v : ZERO"}, which cannot  hold. 
+
+  For rule 4a) we have again @{term "\<Turnstile> v : ZERO"}. The property holds by IH for rule 4b).
+  The  induction hypothesis is 
+  \[
+  @{term "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"}
+  \]
+  which is what left- and right-hand side simplify to.  The slightly more interesting case
+  is for 4c). By assumption  we have 
+  @{term "\<Turnstile> v : ALT (der c (erase r\<^sub>1)) (der c (erase (AALTs bs (r\<^sub>2 # rs))))"}. This means we 
+  have either (*) @{term "\<Turnstile> v1 : der c (erase r\<^sub>1)"} with @{term "v = Left v1"} or
+  (**) @{term "\<Turnstile> v2 : der c (erase (AALTs bs (r\<^sub>2 # rs)))"} with @{term "v = Right v2"}.
+  The former  case is straightforward by simplification. The second case is \ldots TBD.
+
+  Rule 5) TBD.
+
+  Finally for rule 6) the reasoning is as follows:   By assumption we  have
+  @{term "\<Turnstile> v : SEQ (der c (erase r)) (STAR (erase r))"}. This means we also have
+  @{term "v = Seq v1 v2"}, @{term "\<Turnstile> v1 : der c (erase r)"}  and @{term "v2 = Stars vs"}.
+  We want to prove
+  \begin{align}
+  & @{term "retrieve (ASEQ bs (fuse [Z] (bder c r)) (ASTAR [] r)) v"}\\
+  &= @{term "retrieve (ASTAR bs r) (injval (STAR (erase r)) c v)"}
+  \end{align}
+  The right-hand side @{term inj}-expression is equal to 
+  @{term "Stars (injval (erase r) c v1 # vs)"}, which means the @{term  retrieve}-expression
+  simplifies to 
+  \[
+  @{term "bs @ [Z] @ retrieve r (injval (erase r) c v1) @ retrieve (ASTAR [] r) (Stars vs)"}
+  \]
+  The left-hand side (3) above simplifies to 
+  \[
+  @{term "bs @ retrieve (fuse [Z] (bder c r)) v1 @ retrieve (ASTAR [] r) (Stars vs)"} 
+  \]
+  We can move out the @{term "fuse  [Z]"} and then use the IH to show that left-hand side
+  and right-hand side are equal. This completes the proof. 
+  \end{proof}   
+
+   
+
+  \bibliographystyle{plain}
+  \bibliography{root}
+
+\<close>
+(*<*)
+end
+(*>*)
+
+
+
+(*
+
+\begin{center}
+  \begin{tabular}
+    @{thm[mode=Rule] aggressive.intros(1)[of "bs" "bs1" "rs" "r"]}
+  end{tabular}
+\end{center}
+
+
+
+\begin{center}
+  \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)}\\
+  @{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)}
+\end{tabular}
+\end{center}
+
+\begin{center}
+\begin{tabular}{lcl}
+  @{term areg} & $::=$ & @{term "AZERO"}\\
+               & $\mid$ & @{term "AONE bs"}\\
+               & $\mid$ & @{term "ACHAR bs c"}\\
+               & $\mid$ & @{term "AALT bs r1 r2"}\\
+               & $\mid$ & @{term "ASEQ bs r\<^sub>1 r\<^sub>2"}\\
+               & $\mid$ & @{term "ASTAR bs r"}
+\end{tabular}
+\end{center}
+
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) intern.simps(1)} & $\dn$ & @{thm (rhs) intern.simps(1)}\\
+  @{thm (lhs) intern.simps(2)} & $\dn$ & @{thm (rhs) intern.simps(2)}\\
+  @{thm (lhs) intern.simps(3)} & $\dn$ & @{thm (rhs) intern.simps(3)}\\
+  @{thm (lhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) intern.simps(6)} & $\dn$ & @{thm (rhs) intern.simps(6)}\\
+\end{tabular}
+\end{center}
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) erase.simps(1)} & $\dn$ & @{thm (rhs) erase.simps(1)}\\
+  @{thm (lhs) erase.simps(2)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(2)[of bs]}\\
+  @{thm (lhs) erase.simps(3)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(3)[of bs]}\\
+  @{thm (lhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) erase.simps(6)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(6)[of bs]}\\
+\end{tabular}
+\end{center}
+
+Some simple facts about erase
+
+\begin{lemma}\mbox{}\\
+@{thm erase_bder}\\
+@{thm erase_intern}
+\end{lemma}
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) bnullable.simps(1)} & $\dn$ & @{thm (rhs) bnullable.simps(1)}\\
+  @{thm (lhs) bnullable.simps(2)} & $\dn$ & @{thm (rhs) bnullable.simps(2)}\\
+  @{thm (lhs) bnullable.simps(3)} & $\dn$ & @{thm (rhs) bnullable.simps(3)}\\
+  @{thm (lhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bnullable.simps(6)} & $\dn$ & @{thm (rhs) bnullable.simps(6)}\medskip\\
+
+%  \end{tabular}
+%  \end{center}
+
+%  \begin{center}
+%  \begin{tabular}{lcl}
+
+  @{thm (lhs) bder.simps(1)} & $\dn$ & @{thm (rhs) bder.simps(1)}\\
+  @{thm (lhs) bder.simps(2)} & $\dn$ & @{thm (rhs) bder.simps(2)}\\
+  @{thm (lhs) bder.simps(3)} & $\dn$ & @{thm (rhs) bder.simps(3)}\\
+  @{thm (lhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bder.simps(6)} & $\dn$ & @{thm (rhs) bder.simps(6)}
+  \end{tabular}
+  \end{center}
+
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) bmkeps.simps(1)} & $\dn$ & @{thm (rhs) bmkeps.simps(1)}\\
+  @{thm (lhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bmkeps.simps(4)} & $\dn$ & @{thm (rhs) bmkeps.simps(4)}\medskip\\
+\end{tabular}
+\end{center}
+
+
+@{thm [mode=IfThen] bder_retrieve}
+
+By induction on \<open>r\<close>
+
+\begin{theorem}[Main Lemma]\mbox{}\\
+@{thm [mode=IfThen] MAIN_decode}
+\end{theorem}
+
+\noindent
+Definition of the bitcoded lexer
+
+@{thm blexer_def}
+
+
+\begin{theorem}
+@{thm blexer_correctness}
+\end{theorem}
+
+
+
+
+
+\begin{center}
+  \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)}\\
+  @{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)}
+\end{tabular}
+\end{center}
+
+\begin{center}
+\begin{tabular}{lcl}
+  @{term areg} & $::=$ & @{term "AZERO"}\\
+               & $\mid$ & @{term "AONE bs"}\\
+               & $\mid$ & @{term "ACHAR bs c"}\\
+               & $\mid$ & @{term "AALT bs r1 r2"}\\
+               & $\mid$ & @{term "ASEQ bs r\<^sub>1 r\<^sub>2"}\\
+               & $\mid$ & @{term "ASTAR bs r"}
+\end{tabular}
+\end{center}
+
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) intern.simps(1)} & $\dn$ & @{thm (rhs) intern.simps(1)}\\
+  @{thm (lhs) intern.simps(2)} & $\dn$ & @{thm (rhs) intern.simps(2)}\\
+  @{thm (lhs) intern.simps(3)} & $\dn$ & @{thm (rhs) intern.simps(3)}\\
+  @{thm (lhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) intern.simps(6)} & $\dn$ & @{thm (rhs) intern.simps(6)}\\
+\end{tabular}
+\end{center}
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) erase.simps(1)} & $\dn$ & @{thm (rhs) erase.simps(1)}\\
+  @{thm (lhs) erase.simps(2)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(2)[of bs]}\\
+  @{thm (lhs) erase.simps(3)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(3)[of bs]}\\
+  @{thm (lhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) erase.simps(6)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(6)[of bs]}\\
+\end{tabular}
+\end{center}
+
+Some simple facts about erase
+
+\begin{lemma}\mbox{}\\
+@{thm erase_bder}\\
+@{thm erase_intern}
+\end{lemma}
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) bnullable.simps(1)} & $\dn$ & @{thm (rhs) bnullable.simps(1)}\\
+  @{thm (lhs) bnullable.simps(2)} & $\dn$ & @{thm (rhs) bnullable.simps(2)}\\
+  @{thm (lhs) bnullable.simps(3)} & $\dn$ & @{thm (rhs) bnullable.simps(3)}\\
+  @{thm (lhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bnullable.simps(6)} & $\dn$ & @{thm (rhs) bnullable.simps(6)}\medskip\\
+
+%  \end{tabular}
+%  \end{center}
+
+%  \begin{center}
+%  \begin{tabular}{lcl}
+
+  @{thm (lhs) bder.simps(1)} & $\dn$ & @{thm (rhs) bder.simps(1)}\\
+  @{thm (lhs) bder.simps(2)} & $\dn$ & @{thm (rhs) bder.simps(2)}\\
+  @{thm (lhs) bder.simps(3)} & $\dn$ & @{thm (rhs) bder.simps(3)}\\
+  @{thm (lhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bder.simps(6)} & $\dn$ & @{thm (rhs) bder.simps(6)}
+  \end{tabular}
+  \end{center}
+
+
+\begin{center}
+  \begin{tabular}{lcl}
+  @{thm (lhs) bmkeps.simps(1)} & $\dn$ & @{thm (rhs) bmkeps.simps(1)}\\
+  @{thm (lhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+  @{thm (lhs) bmkeps.simps(4)} & $\dn$ & @{thm (rhs) bmkeps.simps(4)}\medskip\\
+\end{tabular}
+\end{center}
+
+
+@{thm [mode=IfThen] bder_retrieve}
+
+By induction on \<open>r\<close>
+
+\begin{theorem}[Main Lemma]\mbox{}\\
+@{thm [mode=IfThen] MAIN_decode}
+\end{theorem}
+
+\noindent
+Definition of the bitcoded lexer
+
+@{thm blexer_def}
+
+
+\begin{theorem}
+@{thm blexer_correctness}
+\end{theorem}
+
+\<close>
+\<close>*)
\ No newline at end of file