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-\isamarkupsection{Core of the proof%
-}
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-\begin{isamarkuptext}%
-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}
-\isa{{\normalsize{}If\,}\ v\ {\isacharcolon}{\kern0pt}\ r{\isacharbackslash}{\kern0pt}s\ {\normalsize \,then\,}\ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardot}{\kern0pt}}
-\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}
-\isa{lexer\mbox{$_b$}\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s}
-\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}
-\isa{blexer{\isacharunderscore}{\kern0pt}simp\ r\ s\ {\isasymequiv}\ \textrm{if}\ nullable\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ \textrm{then}\ decode\ {\isacharparenleft}{\kern0pt}mkeps\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ \textrm{else}\ None}
-\end{lemma}
-
-The simplification function is given as follows:
-
-\begin{center}
- \begin{tabular}{lcl}
- \isa{bsimp\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ {\isacharparenleft}{\kern0pt}bsimp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \isa{bsimp\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ rs{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isasymemptyset}{\isacharparenright}{\kern0pt}}\\
- \isa{bsimp\ AZERO} & $\dn$ & \isa{AZERO}\\
-
-\end{tabular}
-\end{center}
-
-And the two helper functions are:
-\begin{center}
- \begin{tabular}{lcl}
- \isa{bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs\isactrlsub {\isadigit{1}}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}} & $\dn$ & \isa{bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}bsimp\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ r{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}}\\
- \isa{bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}} & $\dn$ & \isa{bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isasymemptyset}{\isacharparenright}{\kern0pt}}\\
- \isa{bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharparenleft}{\kern0pt}v\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vb\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vc{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{AZERO}\\
-
-\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}
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ AZERO\ r\isactrlsub {\isadigit{2}}\ {\isasymleadsto}\ AZERO}}}\\
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ AZERO\ {\isasymleadsto}\ AZERO}}}\\
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AONE\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymleadsto}\ fuse\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}}\\
- \isa{\mbox{}\inferrule{\mbox{r\isactrlsub {\isadigit{1}}\ {\isasymleadsto}\ r\isactrlsub {\isadigit{2}}}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ bs\ r\isactrlsub {\isadigit{2}}\ r\isactrlsub {\isadigit{3}}}}}\\
- \isa{\mbox{}\inferrule{\mbox{r\isactrlsub {\isadigit{3}}\ {\isasymleadsto}\ r\isactrlsub {\isadigit{4}}}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{4}}}}}\\
- \isa{\mbox{}\inferrule{\mbox{r\ {\isasymleadsto}\ r{\isacharprime}{\kern0pt}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}}}\\
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}AZERO{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub b{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ rs\isactrlsub b{\isacharparenright}{\kern0pt}}}}\\
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}AALTs\ bs\isactrlsub {\isadigit{1}}\ rs\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub b{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ rs\isactrlsub b{\isacharparenright}{\kern0pt}}}}\\
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}}}}\\
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ AZERO}}}\\
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ fuse\ bs\ r\isactrlsub {\isadigit{1}}}}}\\
- \isa{\mbox{}\inferrule{\mbox{a\isactrlsub {\isadigit{1}}\mbox{$^\downarrow$}\ {\isacharequal}{\kern0pt}\ a\isactrlsub {\isadigit{2}}\mbox{$^\downarrow$}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub b\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a\isactrlsub {\isadigit{2}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub c{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub b\ {\isacharat}{\kern0pt}\ rs\isactrlsub c{\isacharparenright}{\kern0pt}}}}\\
-
- \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}
-\isa{{\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp\ r\ s}
-\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}
-\isa{{\normalsize{}If\,}\ \mbox{r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}}\ {\normalsize \,and\,}\ \mbox{nullable\mbox{$_b$}\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}}\ {\normalsize \,then\,}\ mkeps\mbox{$_b$}\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ mkeps\mbox{$_b$}\ r{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isachardot}{\kern0pt}}
-\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}
-\isa{lexer\mbox{$_b$}\ r\ s\ {\isasymequiv}\ \textrm{if}\ nullable\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ \textrm{then}\ decode\ {\isacharparenleft}{\kern0pt}mkeps\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ \textrm{else}\ None}
-\end{lemma}
-
-\begin{lemma}
-\isa{blexer{\isacharunderscore}{\kern0pt}simp\ r\ s\ {\isasymequiv}\ \textrm{if}\ nullable\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ \textrm{then}\ decode\ {\isacharparenleft}{\kern0pt}mkeps\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ \textrm{else}\ None}
-\end{lemma}
-
-And that yields the correctness result:
-\begin{lemma}
-\isa{lexer\ r\ s\ {\isacharequal}{\kern0pt}\ blexer{\isacharunderscore}{\kern0pt}simp\ r\ s}
-\end{lemma}
-
-The nice thing about the above%
-\end{isamarkuptext}\isamarkuptrue%
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-\isamarkupsection{Additional Simp Rules?%
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-\begin{isamarkuptext}%
-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}
- \isa{ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isacharparenright}{\kern0pt}\ r\ {\isasymleadsto}{\isacharquery}{\kern0pt}\ AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}{\isasymlambda}r{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isacharprime}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}}\\
- \end{tabular}
-\end{center}
-
-This rule allows us to simplify \mbox{\isa{a\ {\isacharplus}{\kern0pt}\ b\ {\isasymcdot}\ c\ {\isacharplus}{\kern0pt}\ a\ {\isasymcdot}\ c}}%
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-\isamarkupsection{Introduction%
-}
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-\begin{isamarkuptext}%
-Brzozowski \cite{Brzozowski1964} introduced the notion of the {\em
-derivative} \isa{r{\isacharbackslash}{\kern0pt}c} of a regular expression \isa{r} w.r.t.\
-a character~\isa{c}, and showed that it gave a simple solution to the
-problem of matching a string \isa{s} with a regular expression \isa{r}: if the derivative of \isa{r} w.r.t.\ (in succession) all the
-characters of the string matches the empty string, then \isa{r}
-matches \isa{s} (and {\em vice versa}). The derivative has the
-property (which may almost be regarded as its specification) that, for
-every string \isa{s} and regular expression \isa{r} and character
-\isa{c}, one has \isa{cs\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}} if and only if \mbox{\isa{s\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}}}.
-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 \isa{xy} and the regular expression
-\mbox{\isa{{\isacharparenleft}{\kern0pt}x\ {\isacharplus}{\kern0pt}\ y\ {\isacharplus}{\kern0pt}\ xy{\isacharparenright}{\kern0pt}\isactrlsup {\isasymstar}}}. Either the string can be
-matched in two `iterations' by the single letter-regular expressions
-\isa{x} and \isa{y}, or directly in one iteration by \isa{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 \isa{r\isactrlbsub key\isactrlesub } and \isa{r\isactrlbsub id\isactrlesub } 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 \isa{r\isactrlbsub key\isactrlesub } for recognising keywords such as \isa{if},
-\isa{then} and so on; and \isa{r\isactrlbsub id\isactrlesub }
-recognising identifiers (say, a single character followed by
-characters or numbers). Then we can form the regular expression
-\isa{{\isacharparenleft}{\kern0pt}r\isactrlbsub key\isactrlesub \ {\isacharplus}{\kern0pt}\ r\isactrlbsub id\isactrlesub {\isacharparenright}{\kern0pt}\isactrlsup {\isasymstar}}
-and use POSIX matching to tokenise strings, say \isa{iffoo} and
-\isa{if}. For \isa{iffoo} we obtain by the Longest Match Rule
-a single identifier token, not a keyword followed by an
-identifier. For \isa{if} we obtain by the Priority Rule a keyword
-token, not an identifier token---even if \isa{r\isactrlbsub id\isactrlesub }
-matches also. By the Star Rule we know \isa{{\isacharparenleft}{\kern0pt}r\isactrlbsub key\isactrlesub \ {\isacharplus}{\kern0pt}\ r\isactrlbsub id\isactrlesub {\isacharparenright}{\kern0pt}\isactrlsup {\isasymstar}} matches \isa{iffoo},
-respectively \isa{if}, in exactly one `iteration' of the star. The
-Empty String Rule is for cases where, for example, the regular expression
-\isa{{\isacharparenleft}{\kern0pt}a\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}\isactrlsup {\isasymstar}} matches against the
-string \isa{bc}. 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 \isa{xy} and
-the regular expression \mbox{\isa{{\isacharparenleft}{\kern0pt}x\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}y\ {\isacharplus}{\kern0pt}\ xy{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isactrlsup {\isasymstar}}}
-(this time fully parenthesised). We can view this regular expression
-as tree and if the string \isa{xy} is matched by two Star
-`iterations', then the \isa{x} is matched by the left-most
-alternative in this tree and the \isa{y} by the right-left alternative. This
-suggests to record this matching as
-
-\begin{center}
-\isa{Stars\ {\isacharbrackleft}{\kern0pt}Left\ {\isacharparenleft}{\kern0pt}Char\ x{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ Right\ {\isacharparenleft}{\kern0pt}Left\ {\isacharparenleft}{\kern0pt}Char\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}}
-\end{center}
-
-\noindent where \isa{Stars}, \isa{Left}, \isa{Right} and \isa{Char} are constructors for values. \isa{Stars} records how many
-iterations were used; \isa{Left}, respectively \isa{Right}, 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 \isa{xy} in a single `iteration', i.e.~the POSIX value,
-would look as follows
-
-\begin{center}
-\isa{Stars\ {\isacharbrackleft}{\kern0pt}Seq\ {\isacharparenleft}{\kern0pt}Char\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Char\ y{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}}
-\end{center}
-
-\noindent where \isa{Stars} has only a single-element list for the
-single iteration and \isa{Seq} indicates that \isa{xy} is matched
-by a sequence regular expression.
-
-%, which we will in what follows
-%write more formally as \isa{x\ {\isasymcdot}\ 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 \isa{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 \isa{r} and
-a string \isa{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 \isa{\isactrlbold {\isadigit{0}}\ {\isacharplus}{\kern0pt}\ r},
-%\isa{r\ {\isacharplus}{\kern0pt}\ \isactrlbold {\isadigit{0}}}, \isa{\isactrlbold {\isadigit{1}}\ {\isasymcdot}\ r} and \isa{r\ {\isasymcdot}\ \isactrlbold {\isadigit{1}}} to
-%\isa{r} are applied.
-
-We extend our results to ??? Bitcoded version??%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Preliminaries%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-\noindent Strings in Isabelle/HOL are lists of characters with
-the empty string being represented by the empty list, written \isa{{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}, and list-cons being written as \isa{\underline{\hspace{2mm}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}\underline{\hspace{2mm}}}. Often
-we use the usual bracket notation for lists also for strings; for
-example a string consisting of just a single character \isa{c} is
-written \isa{{\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}}. We use the usual definitions for
-\emph{prefixes} and \emph{strict prefixes} of strings. By using the
-type \isa{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}
- \isa{r\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}}
- \isa{\isactrlbold {\isadigit{0}}} $\mid$
- \isa{\isactrlbold {\isadigit{1}}} $\mid$
- \isa{c} $\mid$
- \isa{r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}} $\mid$
- \isa{r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}} $\mid$
- \isa{r\isactrlsup {\isasymstar}}
- \end{center}
-
- \noindent where \isa{\isactrlbold {\isadigit{0}}} stands for the regular expression that does
- not match any string, \isa{\isactrlbold {\isadigit{1}}} for the regular expression that matches
- only the empty string and \isa{c} for matching a character literal. The
- language of a regular expression is also defined as usual by the
- recursive function \isa{L} with the six clauses:
-
- \begin{center}
- \begin{tabular}{l@ {\hspace{4mm}}rcl}
- \textit{(1)} & \isa{L{\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{0}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isasymemptyset}}\\
- \textit{(2)} & \isa{L{\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{1}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}}\\
- \textit{(3)} & \isa{L{\isacharparenleft}{\kern0pt}c{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}}\\
- \textit{(4)} & \isa{L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ &
- \isa{L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \textit{(5)} & \isa{L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ &
- \isa{L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymunion}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \textit{(6)} & \isa{L{\isacharparenleft}{\kern0pt}r\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isasymstar}}\\
- \end{tabular}
- \end{center}
-
- \noindent In clause \textit{(4)} we use the operation \isa{\underline{\hspace{2mm}}\ {\isacharat}{\kern0pt}\ \underline{\hspace{2mm}}} 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: \isa{{\isacharparenleft}{\kern0pt}i{\isacharparenright}{\kern0pt}} the empty string being in
- the star of a language and \isa{{\isacharparenleft}{\kern0pt}ii{\isacharparenright}{\kern0pt}} if \isa{s\isactrlsub {\isadigit{1}}} is in a
- language and \isa{s\isactrlsub {\isadigit{2}}} in the star of this language, then also \isa{s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{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}
- \isa{Der\ c\ A\ {\isasymequiv}\ {\isacharbraceleft}{\kern0pt}s\ \mbox{\boldmath$\mid$}\ c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}}\;.
- \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}
- \isa{Der\ c\ {\isasymemptyset}} & \dn & \isa{{\isasymemptyset}}\\
- \isa{Der\ c\ {\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}} & \dn & \isa{{\isasymemptyset}}\\
- \isa{Der\ c\ {\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}d{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}} & \dn & \isa{\textrm{if}\ c\ {\isacharequal}{\kern0pt}\ d\ \textrm{then}\ {\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ \textrm{else}\ {\isasymemptyset}}\\
- \isa{Der\ c\ {\isacharparenleft}{\kern0pt}A\ {\isasymunion}\ B{\isacharparenright}{\kern0pt}} & \dn & \isa{Der\ c\ A\ {\isasymunion}\ Der\ c\ B}\\
- \isa{Der\ c\ {\isacharparenleft}{\kern0pt}A\ {\isacharat}{\kern0pt}\ B{\isacharparenright}{\kern0pt}} & \dn & \isa{{\isacharparenleft}{\kern0pt}Der\ c\ A\ {\isacharat}{\kern0pt}\ B{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}\textrm{if}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymin}\ A\ \textrm{then}\ Der\ c\ B\ \textrm{else}\ {\isasymemptyset}{\isacharparenright}{\kern0pt}}\\
- \isa{Der\ c\ {\isacharparenleft}{\kern0pt}A{\isasymstar}{\isacharparenright}{\kern0pt}} & \dn & \isa{Der\ c\ A\ {\isacharat}{\kern0pt}\ A{\isasymstar}}
- \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}
- \isa{nullable\ {\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{0}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{False}\\
- \isa{nullable\ {\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{1}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{True}\\
- \isa{nullable\ {\isacharparenleft}{\kern0pt}c{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{False}\\
- \isa{nullable\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{nullable\ r\isactrlsub {\isadigit{1}}\ {\isasymor}\ nullable\ r\isactrlsub {\isadigit{2}}}\\
- \isa{nullable\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{nullable\ r\isactrlsub {\isadigit{1}}\ {\isasymand}\ nullable\ r\isactrlsub {\isadigit{2}}}\\
- \isa{nullable\ {\isacharparenleft}{\kern0pt}r\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{True}\medskip\\
-
-% \end{tabular}
-% \end{center}
-
-% \begin{center}
-% \begin{tabular}{lcl}
-
- \isa{\isactrlbold {\isadigit{0}}{\isacharbackslash}{\kern0pt}c} & $\dn$ & \isa{\isactrlbold {\isadigit{0}}}\\
- \isa{\isactrlbold {\isadigit{1}}{\isacharbackslash}{\kern0pt}c} & $\dn$ & \isa{\isactrlbold {\isadigit{0}}}\\
- \isa{d{\isacharbackslash}{\kern0pt}c} & $\dn$ & \isa{\textrm{if}\ c\ {\isacharequal}{\kern0pt}\ d\ \textrm{then}\ \isactrlbold {\isadigit{1}}\ \textrm{else}\ \isactrlbold {\isadigit{0}}}\\
- \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}}\\
- \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c} & $\dn$ & \isa{\textrm{if}\ nullable\ r\isactrlsub {\isadigit{1}}\ \textrm{then}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ \textrm{else}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}}\\
- \isa{{\isacharparenleft}{\kern0pt}r\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ r\isactrlsup {\isasymstar}}
- \end{tabular}
- \end{center}
-
- \noindent
- We may extend this definition to give derivatives w.r.t.~strings:
-
- \begin{center}
- \begin{tabular}{lcl}
- \isa{r{\isacharbackslash}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}} & $\dn$ & \isa{r}\\
- \isa{r{\isacharbackslash}{\kern0pt}{\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}s}\\
- \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)} & \isa{nullable\ r} if and only if
- \isa{{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}}, and \\
- \textit{(2)} & \isa{L{\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Der\ c\ {\isacharparenleft}{\kern0pt}L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}}.
- \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}
- \isa{match\ r\ s\ {\isasymequiv}\ nullable\ {\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}s{\isacharparenright}{\kern0pt}}
- \end{center}
-
- \noindent gives a positive answer if and only if \isa{s\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}}.
- 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.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{POSIX Regular Expression Matching\label{posixsec}%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-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}
- \isa{v\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}}
- \isa{Empty} $\mid$
- \isa{Char\ c} $\mid$
- \isa{Left\ v} $\mid$
- \isa{Right\ v} $\mid$
- \isa{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}} $\mid$
- \isa{Stars\ vs}
- \end{center}
-
- \noindent where we use \isa{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 \isa{flat} function, written
- \isa{{\isacharbar}{\kern0pt}\underline{\hspace{2mm}}{\isacharbar}{\kern0pt}} and defined as:
-
- \begin{center}
- \begin{tabular}[t]{lcl}
- \isa{{\isacharbar}{\kern0pt}Empty{\isacharbar}{\kern0pt}} & $\dn$ & \isa{{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}\\
- \isa{{\isacharbar}{\kern0pt}Char\ c{\isacharbar}{\kern0pt}} & $\dn$ & \isa{{\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}}\\
- \isa{{\isacharbar}{\kern0pt}Left\ v{\isacharbar}{\kern0pt}} & $\dn$ & \isa{{\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}}\\
- \isa{{\isacharbar}{\kern0pt}Right\ v{\isacharbar}{\kern0pt}} & $\dn$ & \isa{{\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}}
- \end{tabular}\hspace{14mm}
- \begin{tabular}[t]{lcl}
- \isa{{\isacharbar}{\kern0pt}Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}} & $\dn$ & \isa{{\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isacharat}{\kern0pt}\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}}\\
- \isa{{\isacharbar}{\kern0pt}Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbar}{\kern0pt}} & $\dn$ & \isa{{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}\\
- \isa{{\isacharbar}{\kern0pt}Stars\ {\isacharparenleft}{\kern0pt}v\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vs{\isacharparenright}{\kern0pt}{\isacharbar}{\kern0pt}} & $\dn$ & \isa{{\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ {\isacharat}{\kern0pt}\ {\isacharbar}{\kern0pt}Stars\ vs{\isacharbar}{\kern0pt}}\\
- \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 \isa{{\isacharbar}{\kern0pt}vs{\isacharbar}{\kern0pt}} 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 \isa{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 \isa{v\ {\isasymin}\ 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]
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Empty\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{1}}}}} &
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Char\ c\ {\isacharcolon}{\kern0pt}\ c}}}\\[4mm]
- \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}{\mbox{Left\ v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}}} &
- \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}{\mbox{Right\ v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{2}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}}\\[4mm]
- \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}\\\ \mbox{v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}}{\mbox{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}}}} &
- \isa{\mbox{}\inferrule{\mbox{{\isasymforall}v{\isasymin}vs{\isachardot}{\kern0pt}\ v\ {\isacharcolon}{\kern0pt}\ r\ {\isasymand}\ {\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}}{\mbox{Stars\ vs\ {\isacharcolon}{\kern0pt}\ r\isactrlsup {\isasymstar}}}}
- \end{tabular}
- \end{center}
-
- \noindent where in the clause for \isa{Stars} we use the
- notation \isa{v\ {\isasymin}\ vs} for indicating that \isa{v} is a
- member in the list \isa{vs}. We require in this rule that every
- value in \isa{vs} flattens to a non-empty string. The idea is that
- \isa{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 \isa{\isactrlbold {\isadigit{0}}}, and that the
- only value associated with the regular expression \isa{\isactrlbold {\isadigit{1}}} is
- \isa{Empty}. It is routine to establish how values ``inhabiting''
- a regular expression correspond to the language of a regular
- expression, namely
-
- \begin{proposition}\label{inhabs}
- \isa{L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ \mbox{\boldmath$\mid$}\ v\ {\isacharcolon}{\kern0pt}\ r{\isacharbraceright}{\kern0pt}}
- \end{proposition}
-
- \noindent
- Given a regular expression \isa{r} and a string \isa{s}, we define the
- set of all \emph{Lexical Values} inhabited by \isa{r} with the underlying string
- being \isa{s}:\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}
- \isa{LV\ r\ s\ {\isasymequiv}\ {\isacharbraceleft}{\kern0pt}v\ \mbox{\boldmath$\mid$}\ v\ {\isacharcolon}{\kern0pt}\ r\ {\isasymand}\ {\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ s{\isacharbraceright}{\kern0pt}}
- \end{center}
-
- \noindent The main property of \isa{LV\ r\ s} is that it is alway finite.
-
- \begin{proposition}
- \isa{finite\ {\isacharparenleft}{\kern0pt}LV\ r\ s{\isacharparenright}{\kern0pt}}
- \end{proposition}
-
- \noindent This finiteness property does not hold in general if we
- remove the side-condition about \isa{{\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}} in the
- \isa{Stars}-rule above. For example using Sulzmann and Lu's
- less restrictive definition, \isa{LV\ {\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{1}}\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}} would contain
- infinitely many values, but according to our more restricted
- definition only a single value, namely \isa{LV\ {\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{1}}\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}}.
-
- If a regular expression \isa{r} matches a string \isa{s}, then
- generally the set \isa{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 \isa{derivatives}/\isa{nullable} is the first phase of the algorithm
- (calculating successive \Brz's derivatives) and \isa{mkeps}/\isa{inj}, the path from right to left, the second
- phase. This picture shows the steps required when a regular
- expression, say \isa{r\isactrlsub {\isadigit{1}}}, matches the string \isa{{\isacharbrackleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbrackright}{\kern0pt}}. We first build the three derivatives (according to
- \isa{a}, \isa{b} and \isa{c}). We then use \isa{nullable}
- to find out whether the resulting derivative regular expression
- \isa{r\isactrlsub {\isadigit{4}}} can match the empty string. If yes, we call the
- function \isa{mkeps} that produces a value \isa{v\isactrlsub {\isadigit{4}}}
- for how \isa{r\isactrlsub {\isadigit{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) {\isa{r\isactrlsub {\isadigit{1}}}};
-\node (r2) [right=of r1]{\isa{r\isactrlsub {\isadigit{2}}}};
-\draw[->,line width=1mm](r1)--(r2) node[above,midway] {\isa{\underline{\hspace{2mm}}{\isacharbackslash}{\kern0pt}a}};
-\node (r3) [right=of r2]{\isa{r\isactrlsub {\isadigit{3}}}};
-\draw[->,line width=1mm](r2)--(r3) node[above,midway] {\isa{\underline{\hspace{2mm}}{\isacharbackslash}{\kern0pt}b}};
-\node (r4) [right=of r3]{\isa{r\isactrlsub {\isadigit{4}}}};
-\draw[->,line width=1mm](r3)--(r4) node[above,midway] {\isa{\underline{\hspace{2mm}}{\isacharbackslash}{\kern0pt}c}};
-\draw (r4) node[anchor=west] {\;\raisebox{3mm}{\isa{nullable}}};
-\node (v4) [below=of r4]{\isa{v\isactrlsub {\isadigit{4}}}};
-\draw[->,line width=1mm](r4) -- (v4);
-\node (v3) [left=of v4] {\isa{v\isactrlsub {\isadigit{3}}}};
-\draw[->,line width=1mm](v4)--(v3) node[below,midway] {\isa{inj\ r\isactrlsub {\isadigit{3}}\ c}};
-\node (v2) [left=of v3]{\isa{v\isactrlsub {\isadigit{2}}}};
-\draw[->,line width=1mm](v3)--(v2) node[below,midway] {\isa{inj\ r\isactrlsub {\isadigit{2}}\ b}};
-\node (v1) [left=of v2] {\isa{v\isactrlsub {\isadigit{1}}}};
-\draw[->,line width=1mm](v2)--(v1) node[below,midway] {\isa{inj\ r\isactrlsub {\isadigit{1}}\ a}};
-\draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{\isa{mkeps}}};
-\end{tikzpicture}
-\end{center}
-\mbox{}\\[-13mm]
-
-\caption{The two phases of the algorithm by Sulzmann \& Lu \cite{Sulzmann2014},
-matching the string \isa{{\isacharbrackleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbrackright}{\kern0pt}}. The first phase (the arrows from
-left to right) is \Brz's matcher building successive derivatives. If the
-last regular expression is \isa{nullable}, then the functions of the
-second phase are called (the top-down and right-to-left arrows): first
-\isa{mkeps} calculates a value \isa{v\isactrlsub {\isadigit{4}}} witnessing
-how the empty string has been recognised by \isa{r\isactrlsub {\isadigit{4}}}. After
-that the function \isa{inj} ``injects back'' the characters of the string into
-the values.
-\label{Sulz}}
-\end{figure}
-
- \begin{center}
- \begin{tabular}{lcl}
- \isa{mkeps\ \isactrlbold {\isadigit{1}}} & $\dn$ & \isa{Empty}\\
- \isa{mkeps\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{Seq\ {\isacharparenleft}{\kern0pt}mkeps\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mkeps\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \isa{mkeps\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{\textrm{if}\ nullable\ r\isactrlsub {\isadigit{1}}\ \textrm{then}\ Left\ {\isacharparenleft}{\kern0pt}mkeps\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ \textrm{else}\ Right\ {\isacharparenleft}{\kern0pt}mkeps\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \isa{mkeps\ {\isacharparenleft}{\kern0pt}r\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}\\
- \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 \isa{nullable} fails, the string \isa{{\isacharbrackleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbrackright}{\kern0pt}} cannot be matched by \isa{r\isactrlsub {\isadigit{1}}} and the null value \isa{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 \isa{r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}, can
- match the empty string and furthermore \isa{r\isactrlsub {\isadigit{1}}} can match the
- empty string, then we return a \isa{Left}-value. The \isa{Right}-value will only be returned if \isa{r\isactrlsub {\isadigit{1}}} cannot match the empty
- string.
-
- The most interesting idea from Sulzmann and Lu \cite{Sulzmann2014} is
- the construction of a value for how \isa{r\isactrlsub {\isadigit{1}}} can match the
- string \isa{{\isacharbrackleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbrackright}{\kern0pt}} from the value how the last derivative, \isa{r\isactrlsub {\isadigit{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 \isa{inj}, takes three
- arguments, a regular expression, a character and a value. For example in
- the first (or right-most) \isa{inj}-step in Fig.~\ref{Sulz} the regular
- expression \isa{r\isactrlsub {\isadigit{3}}}, the character \isa{c} from the last
- derivative step and \isa{v\isactrlsub {\isadigit{4}}}, which is the value corresponding
- to the derivative regular expression \isa{r\isactrlsub {\isadigit{4}}}. The result is
- the new value \isa{v\isactrlsub {\isadigit{3}}}. The final result of the algorithm is
- the value \isa{v\isactrlsub {\isadigit{1}}}. The \isa{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)} & \isa{inj\ d\ c\ {\isacharparenleft}{\kern0pt}Empty{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{Char\ d}\\
- \textit{(2)} & \isa{inj\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ c\ {\isacharparenleft}{\kern0pt}Left\ v\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}} & $\dn$ &
- \isa{Left\ {\isacharparenleft}{\kern0pt}inj\ r\isactrlsub {\isadigit{1}}\ c\ v\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}}\\
- \textit{(3)} & \isa{inj\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ c\ {\isacharparenleft}{\kern0pt}Right\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ &
- \isa{Right\ {\isacharparenleft}{\kern0pt}inj\ r\isactrlsub {\isadigit{2}}\ c\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \textit{(4)} & \isa{inj\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ c\ {\isacharparenleft}{\kern0pt}Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$
- & \isa{Seq\ {\isacharparenleft}{\kern0pt}inj\ r\isactrlsub {\isadigit{1}}\ c\ v\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ v\isactrlsub {\isadigit{2}}}\\
- \textit{(5)} & \isa{inj\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ c\ {\isacharparenleft}{\kern0pt}Left\ {\isacharparenleft}{\kern0pt}Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}} & $\dn$
- & \isa{Seq\ {\isacharparenleft}{\kern0pt}inj\ r\isactrlsub {\isadigit{1}}\ c\ v\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ v\isactrlsub {\isadigit{2}}}\\
- \textit{(6)} & \isa{inj\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ c\ {\isacharparenleft}{\kern0pt}Right\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$
- & \isa{Seq\ {\isacharparenleft}{\kern0pt}mkeps\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}inj\ r\isactrlsub {\isadigit{2}}\ c\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \textit{(7)} & \isa{inj\ {\isacharparenleft}{\kern0pt}r\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}\ c\ {\isacharparenleft}{\kern0pt}Seq\ v\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}} & $\dn$
- & \isa{Stars\ {\isacharparenleft}{\kern0pt}inj\ r\ c\ v\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vs{\isacharparenright}{\kern0pt}}\\
- \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
- \isa{r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}}. This must be a value of the form \isa{Seq\ \underline{\hspace{2mm}}\ \underline{\hspace{2mm}}}\,. Recall the clause of the \isa{derivative}-function
- for sequence regular expressions:
-
- \begin{center}
- \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c} $\dn$ \isa{\textrm{if}\ nullable\ r\isactrlsub {\isadigit{1}}\ \textrm{then}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ \textrm{else}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}}
- \end{center}
-
- \noindent Consider first the \isa{else}-branch where the derivative is \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}}. The corresponding value must therefore
- be of the form \isa{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}}, which matches the left-hand
- side in clause~\textit{(4)} of \isa{inj}. In the \isa{if}-branch the derivative is an
- alternative, namely \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}}. This means we either have to consider a \isa{Left}- or
- \isa{Right}-value. In case of the \isa{Left}-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 \isa{Left\ {\isacharparenleft}{\kern0pt}Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}},
- while in \textit{(6)} it is just \isa{Right\ v\isactrlsub {\isadigit{2}}}. One more interesting
- point is in the right-hand side of clause \textit{(6)}: since in this case the
- regular expression \isa{r\isactrlsub {\isadigit{1}}} does not ``contribute'' to
- matching the string, that means it only matches the empty string, we need to
- call \isa{mkeps} in order to construct a value for how \isa{r\isactrlsub {\isadigit{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
- \isa{Seq\ v\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}}---the derivative of a star is \isa{{\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ r\isactrlsup {\isasymstar}}. Finally, the reason for why we can ignore the second argument
- in clause \textit{(1)} of \isa{inj} is that it will only ever be called in cases
- where \isa{c\ {\isacharequal}{\kern0pt}\ 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 \isa{inj\ c\ c\ {\isacharparenleft}{\kern0pt}Empty{\isacharparenright}{\kern0pt}} $\dn$ \isa{Char\ c},
- but our deviation is harmless.}
-
- The idea of the \isa{inj}-function to ``inject'' a character, say
- \isa{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 \isa{c}; the second part shows that
- the underlying string of an \isa{mkeps}-value is always the empty
- string (given the regular expression is nullable since otherwise
- \isa{mkeps} might not be defined).
-
- \begin{lemma}\mbox{}\smallskip\\\label{Prf_injval_flat}
- \begin{tabular}{ll}
- (1) & \isa{{\normalsize{}If\,}\ v\ {\isacharcolon}{\kern0pt}\ r{\isacharbackslash}{\kern0pt}c\ {\normalsize \,then\,}\ {\isacharbar}{\kern0pt}inj\ r\ c\ v{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}{\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}{\isachardot}{\kern0pt}}\\
- (2) & \isa{{\normalsize{}If\,}\ nullable\ r\ {\normalsize \,then\,}\ {\isacharbar}{\kern0pt}mkeps\ r{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardot}{\kern0pt}}
- \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 \isa{derivatives}; the second by
- an induction on \isa{r}. There are no interesting cases.\qed
- \end{proof}
-
- Having defined the \isa{mkeps} and \isa{inj} 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}
- \isa{lexer\ r\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}} & $\dn$ & \isa{\textrm{if}\ nullable\ r\ \textrm{then}\ Some\ {\isacharparenleft}{\kern0pt}mkeps\ r{\isacharparenright}{\kern0pt}\ \textrm{else}\ None}\\
- \isa{lexer\ r\ {\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{case} \isa{lexer\ {\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ s} \isa{of}\\
- & & \phantom{$|$} \isa{None} \isa{{\isasymRightarrow}} \isa{None}\\
- & & $|$ \isa{Some\ v} \isa{{\isasymRightarrow}} \isa{Some\ {\isacharparenleft}{\kern0pt}inj\ r\ c\ v{\isacharparenright}{\kern0pt}}
- \end{tabular}
- \end{center}
-
- \noindent If the regular expression does not match the string, \isa{None} is
- returned. If the regular expression \emph{does}
- match the string, then \isa{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{\isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}}, relating
- strings, regular expressions and values; the inductive rules are given in
- Figure~\ref{POSIXrules}.
- We can prove that given a string \isa{s} and regular expression \isa{r}, the POSIX value \isa{v} is uniquely determined by \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}.
-
- %
- \begin{figure}[t]
- \begin{center}
- \begin{tabular}{c}
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{{\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ \isactrlbold {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Empty}}}\isa{P}\isa{\isactrlbold {\isadigit{1}}} \qquad
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{{\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Char\ c}}}\isa{P}\isa{c}\medskip\\
- \isa{\mbox{}\inferrule{\mbox{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}}{\mbox{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Left\ v}}}\isa{P{\isacharplus}{\kern0pt}L}\qquad
- \isa{\mbox{}\inferrule{\mbox{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}\\\ \mbox{s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}}}{\mbox{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Right\ v}}}\isa{P{\isacharplus}{\kern0pt}R}\medskip\\
- $\mprset{flushleft}
- \inferrule
- {\isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\isactrlsub {\isadigit{1}}} \qquad
- \isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\isactrlsub {\isadigit{2}}} \\\\
- \isa{{\isasymnexists}s\isactrlsub {\isadigit{3}}\ s\isactrlsub {\isadigit{4}}{\isachardot}{\kern0pt}a{\isachardot}{\kern0pt}\ s\isactrlsub {\isadigit{3}}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{3}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{4}}\ {\isacharequal}{\kern0pt}\ s\isactrlsub {\isadigit{2}}\ {\isasymand}\ s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{3}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{4}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}}
- {\isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}}}$\isa{PS}\\
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{{\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ r\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}}}\isa{P{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}\medskip\\
- $\mprset{flushleft}
- \inferrule
- {\isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v} \qquad
- \isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ r\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Stars\ vs} \qquad
- \isa{{\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}} \\\\
- \isa{{\isasymnexists}s\isactrlsub {\isadigit{3}}\ s\isactrlsub {\isadigit{4}}{\isachardot}{\kern0pt}a{\isachardot}{\kern0pt}\ s\isactrlsub {\isadigit{3}}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{3}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{4}}\ {\isacharequal}{\kern0pt}\ s\isactrlsub {\isadigit{2}}\ {\isasymand}\ s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{3}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{4}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}}}
- {\isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ r\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Stars\ {\isacharparenleft}{\kern0pt}v\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vs{\isacharparenright}{\kern0pt}}}$\isa{P{\isasymstar}}
- \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 \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v} then \isa{s\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}} and \isa{{\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ s}.\\
- (2) & \isa{{\normalsize{}If\,}\ \mbox{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}\ {\normalsize \,and\,}\ \mbox{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v{\isacharprime}{\kern0pt}}\ {\normalsize \,then\,}\ v\ {\isacharequal}{\kern0pt}\ v{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}}
- \end{tabular}
- \end{theorem}
-
- \begin{proof} Both by induction on the definition of \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}.
- The second parts follows by a case analysis of \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v{\isacharprime}{\kern0pt}} and
- the first part.\qed
- \end{proof}
-
- \noindent
- We claim that our \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v} relation captures the idea behind the four
- informal POSIX rules shown in the Introduction: Consider for example the
- rules \isa{P{\isacharplus}{\kern0pt}L} and \isa{P{\isacharplus}{\kern0pt}R} where the POSIX value for a string
- and an alternative regular expression, that is \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}},
- is specified---it is always a \isa{Left}-value, \emph{except} when the
- string to be matched is not in the language of \isa{r\isactrlsub {\isadigit{1}}}; only then it
- is a \isa{Right}-value (see the side-condition in \isa{P{\isacharplus}{\kern0pt}R}).
- Interesting is also the rule for sequence regular expressions (\isa{PS}). The first two premises state that \isa{v\isactrlsub {\isadigit{1}}} and \isa{v\isactrlsub {\isadigit{2}}}
- are the POSIX values for \isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}} and \isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}
- respectively. Consider now the third premise and note that the POSIX value
- of this rule should match the string \mbox{\isa{s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}}}. According to the
- Longest Match Rule, we want that the \isa{s\isactrlsub {\isadigit{1}}} is the longest initial
- split of \mbox{\isa{s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}}} such that \isa{s\isactrlsub {\isadigit{2}}} is still recognised
- by \isa{r\isactrlsub {\isadigit{2}}}. Let us assume, contrary to the third premise, that there
- \emph{exist} an \isa{s\isactrlsub {\isadigit{3}}} and \isa{s\isactrlsub {\isadigit{4}}} such that \isa{s\isactrlsub {\isadigit{2}}}
- can be split up into a non-empty string \isa{s\isactrlsub {\isadigit{3}}} and a possibly empty
- string \isa{s\isactrlsub {\isadigit{4}}}. Moreover the longer string \isa{s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{3}}} can be
- matched by \isa{r\isactrlsub {\isadigit{1}}} and the shorter \isa{s\isactrlsub {\isadigit{4}}} can still be
- matched by \isa{r\isactrlsub {\isadigit{2}}}. In this case \isa{s\isactrlsub {\isadigit{1}}} would \emph{not} be the
- longest initial split of \mbox{\isa{s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}}} and therefore \isa{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}} cannot be a POSIX value for \isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}.
- 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 \isa{P{\isasymstar}}-rule. Also there we want that \isa{s\isactrlsub {\isadigit{1}}} is the longest initial
- split of \isa{s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}} and furthermore the corresponding value
- \isa{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 \isa{Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}} as the POSIX value. Indeed we can show that our POSIX values
- are lexical values which exclude those \isa{Stars} that contain subvalues
- that flatten to the empty string.
-
- \begin{lemma}\label{LVposix}
- \isa{{\normalsize{}If\,}\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\ {\normalsize \,then\,}\ v\ {\isasymin}\ LV\ r\ s{\isachardot}{\kern0pt}}
- \end{lemma}
-
- \begin{proof}
- By routine induction on \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}.\qed
- \end{proof}
-
- \noindent
- Next is the lemma that shows the function \isa{mkeps} calculates
- the POSIX value for the empty string and a nullable regular expression.
-
- \begin{lemma}\label{lemmkeps}
- \isa{{\normalsize{}If\,}\ nullable\ r\ {\normalsize \,then\,}\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ mkeps\ r{\isachardot}{\kern0pt}}
- \end{lemma}
-
- \begin{proof}
- By routine induction on \isa{r}.\qed
- \end{proof}
-
- \noindent
- The central lemma for our POSIX relation is that the \isa{inj}-function
- preserves POSIX values.
-
- \begin{lemma}\label{Posix2}
- \isa{{\normalsize{}If\,}\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\ {\normalsize \,then\,}\ {\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ inj\ r\ c\ v{\isachardot}{\kern0pt}}
- \end{lemma}
-
- \begin{proof}
- By induction on \isa{r}. We explain two cases.
-
- \begin{itemize}
- \item[$\bullet$] Case \isa{r\ {\isacharequal}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}. There are
- two subcases, namely \isa{{\isacharparenleft}{\kern0pt}a{\isacharparenright}{\kern0pt}} \mbox{\isa{v\ {\isacharequal}{\kern0pt}\ Left\ v{\isacharprime}{\kern0pt}}} and \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v{\isacharprime}{\kern0pt}}; and \isa{{\isacharparenleft}{\kern0pt}b{\isacharparenright}{\kern0pt}} \isa{v\ {\isacharequal}{\kern0pt}\ Right\ v{\isacharprime}{\kern0pt}}, \isa{s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}} and \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v{\isacharprime}{\kern0pt}}. In \isa{{\isacharparenleft}{\kern0pt}a{\isacharparenright}{\kern0pt}} we
- know \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v{\isacharprime}{\kern0pt}}, from which we can infer \isa{{\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ inj\ r\isactrlsub {\isadigit{1}}\ c\ v{\isacharprime}{\kern0pt}} by induction hypothesis and hence \isa{{\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ inj\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ c\ {\isacharparenleft}{\kern0pt}Left\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}} as needed. Similarly
- in subcase \isa{{\isacharparenleft}{\kern0pt}b{\isacharparenright}{\kern0pt}} where, however, in addition we have to use
- Proposition~\ref{derprop}(2) in order to infer \isa{c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}} from \isa{s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}}.\smallskip
-
- \item[$\bullet$] Case \isa{r\ {\isacharequal}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}}. There are three subcases:
-
- \begin{quote}
- \begin{description}
- \item[\isa{{\isacharparenleft}{\kern0pt}a{\isacharparenright}{\kern0pt}}] \isa{v\ {\isacharequal}{\kern0pt}\ Left\ {\isacharparenleft}{\kern0pt}Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} and \isa{nullable\ r\isactrlsub {\isadigit{1}}}
- \item[\isa{{\isacharparenleft}{\kern0pt}b{\isacharparenright}{\kern0pt}}] \isa{v\ {\isacharequal}{\kern0pt}\ Right\ v\isactrlsub {\isadigit{1}}} and \isa{nullable\ r\isactrlsub {\isadigit{1}}}
- \item[\isa{{\isacharparenleft}{\kern0pt}c{\isacharparenright}{\kern0pt}}] \isa{v\ {\isacharequal}{\kern0pt}\ Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}} and \isa{{\isasymnot}\ nullable\ r\isactrlsub {\isadigit{1}}}
- \end{description}
- \end{quote}
-
- \noindent For \isa{{\isacharparenleft}{\kern0pt}a{\isacharparenright}{\kern0pt}} we know \isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\isactrlsub {\isadigit{1}}} and
- \isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\isactrlsub {\isadigit{2}}} as well as
- %
- \[\isa{{\isasymnexists}s\isactrlsub {\isadigit{3}}\ s\isactrlsub {\isadigit{4}}{\isachardot}{\kern0pt}a{\isachardot}{\kern0pt}\ s\isactrlsub {\isadigit{3}}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{3}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{4}}\ {\isacharequal}{\kern0pt}\ s\isactrlsub {\isadigit{2}}\ {\isasymand}\ s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{3}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{4}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\]
-
- \noindent From the latter we can infer by Proposition~\ref{derprop}(2):
- %
- \[\isa{{\isasymnexists}s\isactrlsub {\isadigit{3}}\ s\isactrlsub {\isadigit{4}}{\isachardot}{\kern0pt}a{\isachardot}{\kern0pt}\ s\isactrlsub {\isadigit{3}}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{3}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{4}}\ {\isacharequal}{\kern0pt}\ s\isactrlsub {\isadigit{2}}\ {\isasymand}\ c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{3}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{4}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\]
-
- \noindent We can use the induction hypothesis for \isa{r\isactrlsub {\isadigit{1}}} to obtain
- \isa{{\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ inj\ r\isactrlsub {\isadigit{1}}\ c\ v\isactrlsub {\isadigit{1}}}. Putting this all together allows us to infer
- \isa{{\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Seq\ {\isacharparenleft}{\kern0pt}inj\ r\isactrlsub {\isadigit{1}}\ c\ v\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ v\isactrlsub {\isadigit{2}}}. The case \isa{{\isacharparenleft}{\kern0pt}c{\isacharparenright}{\kern0pt}}
- is similar.
-
- For \isa{{\isacharparenleft}{\kern0pt}b{\isacharparenright}{\kern0pt}} we know \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\isactrlsub {\isadigit{1}}} and
- \isa{s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}. From the former
- we have \isa{{\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ inj\ r\isactrlsub {\isadigit{2}}\ c\ v\isactrlsub {\isadigit{1}}} by induction hypothesis
- for \isa{r\isactrlsub {\isadigit{2}}}. From the latter we can infer
- %
- \[\isa{{\isasymnexists}s\isactrlsub {\isadigit{3}}\ s\isactrlsub {\isadigit{4}}{\isachardot}{\kern0pt}a{\isachardot}{\kern0pt}\ s\isactrlsub {\isadigit{3}}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{3}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{4}}\ {\isacharequal}{\kern0pt}\ c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s\ {\isasymand}\ s\isactrlsub {\isadigit{3}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ s\isactrlsub {\isadigit{4}}\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\]
-
- \noindent By Lemma~\ref{lemmkeps} we know \isa{{\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ mkeps\ r\isactrlsub {\isadigit{1}}}
- holds. Putting this all together, we can conclude with \isa{{\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Seq\ {\isacharparenleft}{\kern0pt}mkeps\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}inj\ r\isactrlsub {\isadigit{2}}\ c\ v\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}}, as required.
-
- Finally suppose \isa{r\ {\isacharequal}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\isactrlsup {\isasymstar}}. This case is very similar to the
- sequence case, except that we need to also ensure that \isa{{\isacharbar}{\kern0pt}inj\ r\isactrlsub {\isadigit{1}}\ c\ v\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}. This follows from \isa{{\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ inj\ r\isactrlsub {\isadigit{1}}\ c\ v\isactrlsub {\isadigit{1}}} (which in turn follows from \isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\isactrlsub {\isadigit{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 \isa{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) & \isa{s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}} if and only if \isa{lexer\ r\ s\ {\isacharequal}{\kern0pt}\ None}\\
- (2) & \isa{s\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}} if and only if \isa{{\isasymexists}v{\isachardot}{\kern0pt}\ lexer\ r\ s\ {\isacharequal}{\kern0pt}\ Some\ v\ {\isasymand}\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}\\
- \end{tabular}
- \end{theorem}
-
- \begin{proof}
- By induction on \isa{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) & \isa{lexer\ r\ s\ {\isacharequal}{\kern0pt}\ None} if and only if \isa{{\isasymnexists}v{\isachardot}{\kern0pt}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}\\
- (2) & \isa{lexer\ r\ s\ {\isacharequal}{\kern0pt}\ Some\ v} if and only if \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}\\
- \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.%
-\end{isamarkuptext}\isamarkuptrue%
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-\isamarkupsection{Ordering of Values according to Okui and Suzuki%
-}
-\isamarkuptrue%
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-\endisatagdocument
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-\isadelimdocument
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-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-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 \isa{v}
-
- \begin{center}
- \isa{v\ {\isasymequiv}\ Stars\ {\isacharbrackleft}{\kern0pt}Seq\ {\isacharparenleft}{\kern0pt}Char\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Char\ y{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ Char\ z{\isacharbrackright}{\kern0pt}}
- \end{center}
-
- \noindent
- At position \isa{{\isacharbrackleft}{\kern0pt}{\isadigit{0}}{\isacharcomma}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}} of this value is the
- subvalue \isa{Char\ y} and at position \isa{{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}} the
- subvalue \isa{Char\ z}. At the `root' position, or empty list
- \isa{{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}, is the whole value \isa{v}. Positions such as \isa{{\isacharbrackleft}{\kern0pt}{\isadigit{0}}{\isacharcomma}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}{\isadigit{0}}{\isacharbrackright}{\kern0pt}} or \isa{{\isacharbrackleft}{\kern0pt}{\isadigit{2}}{\isacharbrackright}{\kern0pt}} are outside of \isa{v}. If it exists, the subvalue of \isa{v} at a position \isa{p}, written \isa{v\mbox{$\downharpoonleft$}\isactrlbsub p\isactrlesub }, can be recursively defined by
-
- \begin{center}
- \begin{tabular}{r@ {\hspace{0mm}}lcl}
- \isa{v} & \isa{{\isasymdownharpoonleft}\isactrlbsub {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\isactrlesub } & \isa{{\isasymequiv}}& \isa{v}\\
- \isa{Left\ v} & \isa{{\isasymdownharpoonleft}\isactrlbsub {\isadigit{0}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}ps\isactrlesub } & \isa{{\isasymequiv}}& \isa{v\mbox{$\downharpoonleft$}\isactrlbsub ps\isactrlesub }\\
- \isa{Right\ v} & \isa{{\isasymdownharpoonleft}\isactrlbsub {\isadigit{1}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}ps\isactrlesub } & \isa{{\isasymequiv}} &
- \isa{v\mbox{$\downharpoonleft$}\isactrlbsub ps\isactrlesub }\\
- \isa{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}} & \isa{{\isasymdownharpoonleft}\isactrlbsub {\isadigit{0}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}ps\isactrlesub } & \isa{{\isasymequiv}} &
- \isa{v\isactrlsub {\isadigit{1}}\mbox{$\downharpoonleft$}\isactrlbsub ps\isactrlesub } \\
- \isa{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}} & \isa{{\isasymdownharpoonleft}\isactrlbsub {\isadigit{1}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}ps\isactrlesub }
- & \isa{{\isasymequiv}} &
- \isa{v\isactrlsub {\isadigit{2}}\mbox{$\downharpoonleft$}\isactrlbsub ps\isactrlesub } \\
- \isa{Stars\ vs} & \isa{{\isasymdownharpoonleft}\isactrlbsub n{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}ps\isactrlesub } & \isa{{\isasymequiv}}& \isa{vs\ensuremath{_{[\mathit{n}]}}\mbox{$\downharpoonleft$}\isactrlbsub ps\isactrlesub }\\
- \end{tabular}
- \end{center}
-
- \noindent In the last clause we use Isabelle's notation \isa{vs\ensuremath{_{[\mathit{n}]}}} for the
- \isa{n}th element in a list. The set of positions inside a value \isa{v},
- written \isa{Pos\ v}, is given by
-
- \begin{center}
- \begin{tabular}{lcl}
- \isa{Pos\ {\isacharparenleft}{\kern0pt}Empty{\isacharparenright}{\kern0pt}} & \isa{{\isasymequiv}} & \isa{{\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}}\\
- \isa{Pos\ {\isacharparenleft}{\kern0pt}Char\ c{\isacharparenright}{\kern0pt}} & \isa{{\isasymequiv}} & \isa{{\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}}\\
- \isa{Pos\ {\isacharparenleft}{\kern0pt}Left\ v{\isacharparenright}{\kern0pt}} & \isa{{\isasymequiv}} & \isa{{\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps\ \mbox{\boldmath$\mid$}\ ps\ {\isasymin}\ Pos\ v{\isacharbraceright}{\kern0pt}}\\
- \isa{Pos\ {\isacharparenleft}{\kern0pt}Right\ v{\isacharparenright}{\kern0pt}} & \isa{{\isasymequiv}} & \isa{{\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}{\isadigit{1}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps\ \mbox{\boldmath$\mid$}\ ps\ {\isasymin}\ Pos\ v{\isacharbraceright}{\kern0pt}}\\
- \isa{Pos\ {\isacharparenleft}{\kern0pt}Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}
- & \isa{{\isasymequiv}}
- & \isa{{\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps\ \mbox{\boldmath$\mid$}\ ps\ {\isasymin}\ Pos\ v\isactrlsub {\isadigit{1}}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}{\isadigit{1}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps\ \mbox{\boldmath$\mid$}\ ps\ {\isasymin}\ Pos\ v\isactrlsub {\isadigit{2}}{\isacharbraceright}{\kern0pt}}\\
- \isa{Pos\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}} & \isa{{\isasymequiv}} & \isa{{\isacharbraceleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}{\isasymUnion}n\ {\isacharless}{\kern0pt}\ len\ vs\ {\isacharbraceleft}{\kern0pt}n\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps\ \mbox{\boldmath$\mid$}\ ps\ {\isasymin}\ Pos\ vs\ensuremath{_{[\mathit{n}]}}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}}\\
- \end{tabular}
- \end{center}
-
- \noindent
- whereby \isa{len} 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
- \isa{v} and compare it with the following \isa{w}:
-
- \begin{center}
- \begin{tabular}{l}
- \isa{v\ {\isasymequiv}\ Stars\ {\isacharbrackleft}{\kern0pt}Seq\ {\isacharparenleft}{\kern0pt}Char\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Char\ y{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ Char\ z{\isacharbrackright}{\kern0pt}}\\
- \isa{w\ {\isasymequiv}\ Stars\ {\isacharbrackleft}{\kern0pt}Char\ x{\isacharcomma}{\kern0pt}\ Char\ y{\isacharcomma}{\kern0pt}\ Char\ z{\isacharbrackright}{\kern0pt}}
- \end{tabular}
- \end{center}
-
- \noindent Both values match the string \isa{xyz}, that means if
- we flatten these values at their respective root position, we obtain
- \isa{xyz}. However, at position \isa{{\isacharbrackleft}{\kern0pt}{\isadigit{0}}{\isacharbrackright}{\kern0pt}}, \isa{v} matches
- \isa{xy} whereas \isa{w} matches only the shorter \isa{x}. So
- according to the Longest Match Rule, we should prefer \isa{v},
- rather than \isa{w} as POSIX value for string \isa{xyz} (and
- corresponding regular expression). In order to
- formalise this idea, Okui and Suzuki introduce a measure for
- subvalues at position \isa{p}, called the \emph{norm} of \isa{v}
- at position \isa{p}. We can define this measure in Isabelle as an
- integer as follows
-
- \begin{center}
- \isa{{\isasymparallel}v{\isasymparallel}\isactrlbsub p\isactrlesub \ {\isasymequiv}\ \textrm{if}\ p\ {\isasymin}\ Pos\ v\ \textrm{then}\ len\ {\isacharbar}{\kern0pt}v\mbox{$\downharpoonleft$}\isactrlbsub p\isactrlesub {\isacharbar}{\kern0pt}\ \textrm{else}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}}
- \end{center}
-
- \noindent where we take the length of the flattened value at
- position \isa{p}, provided the position is inside \isa{v}; if
- not, then the norm is \isa{{\isacharminus}{\kern0pt}{\isadigit{1}}}. The default for outside
- positions is crucial for the POSIX requirement of preferring a
- \isa{Left}-value over a \isa{Right}-value (if they can match the
- same string---see the Priority Rule from the Introduction). For this
- consider
-
- \begin{center}
- \isa{v\ {\isasymequiv}\ Left\ {\isacharparenleft}{\kern0pt}Char\ x{\isacharparenright}{\kern0pt}} \qquad and \qquad \isa{w\ {\isasymequiv}\ Right\ {\isacharparenleft}{\kern0pt}Char\ x{\isacharparenright}{\kern0pt}}
- \end{center}
-
- \noindent Both values match \isa{x}. At position \isa{{\isacharbrackleft}{\kern0pt}{\isadigit{0}}{\isacharbrackright}{\kern0pt}}
- the norm of \isa{v} is \isa{{\isadigit{1}}} (the subvalue matches \isa{x}),
- but the norm of \isa{w} is \isa{{\isacharminus}{\kern0pt}{\isadigit{1}}} (the position is outside
- \isa{w} according to how we defined the `inside' positions of
- \isa{Left}- and \isa{Right}-values). Of course at position
- \isa{{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}}, the norms \isa{{\isasymparallel}v{\isasymparallel}\isactrlbsub {\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isactrlesub } and \isa{{\isasymparallel}w{\isasymparallel}\isactrlbsub {\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isactrlesub } are reversed, but the point is that subvalues
- will be analysed according to lexicographically ordered
- positions. According to this ordering, the position \isa{{\isacharbrackleft}{\kern0pt}{\isadigit{0}}{\isacharbrackright}{\kern0pt}}
- takes precedence over \isa{{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}} and thus also \isa{v} will be
- preferred over \isa{w}. The lexicographic ordering of positions, written
- \isa{\underline{\hspace{2mm}}\ {\isasymprec}\isactrlbsub lex\isactrlesub \ \underline{\hspace{2mm}}}, can be conveniently formalised
- by three inference rules
-
- \begin{center}
- \begin{tabular}{ccc}
- \isa{\mbox{}\inferrule{\mbox{}}{\mbox{{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymprec}\isactrlbsub lex\isactrlesub \ p\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps}}}\hspace{1cm} &
- \isa{\mbox{}\inferrule{\mbox{p\isactrlsub {\isadigit{1}}\ {\isacharless}{\kern0pt}\ p\isactrlsub {\isadigit{2}}}}{\mbox{p\isactrlsub {\isadigit{1}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps\isactrlsub {\isadigit{1}}\ {\isasymprec}\isactrlbsub lex\isactrlesub \ p\isactrlsub {\isadigit{2}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps\isactrlsub {\isadigit{2}}}}}\hspace{1cm} &
- \isa{\mbox{}\inferrule{\mbox{ps\isactrlsub {\isadigit{1}}\ {\isasymprec}\isactrlbsub lex\isactrlesub \ ps\isactrlsub {\isadigit{2}}}}{\mbox{p\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps\isactrlsub {\isadigit{1}}\ {\isasymprec}\isactrlbsub lex\isactrlesub \ p\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}ps\isactrlsub {\isadigit{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 \isa{v\isactrlsub {\isadigit{1}}} is \emph{smaller at position \isa{p}} than
- \isa{v\isactrlsub {\isadigit{2}}}, written \isa{v\isactrlsub {\isadigit{1}}\ {\isasymprec}\isactrlbsub p\isactrlesub \ v\isactrlsub {\isadigit{2}}},
- if and only if $(i)$ the norm at position \isa{p} is
- greater in \isa{v\isactrlsub {\isadigit{1}}} (that is the string \isa{{\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{1}}\mbox{$\downharpoonleft$}\isactrlbsub p\isactrlesub {\isacharbar}{\kern0pt}} is longer
- than \isa{{\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}\mbox{$\downharpoonleft$}\isactrlbsub p\isactrlesub {\isacharbar}{\kern0pt}}) and $(ii)$ all subvalues at
- positions that are inside \isa{v\isactrlsub {\isadigit{1}}} or \isa{v\isactrlsub {\isadigit{2}}} and that are
- lexicographically smaller than \isa{p}, we have the same norm, namely
-
- \begin{center}
- \begin{tabular}{c}
- \isa{v\isactrlsub {\isadigit{1}}\ {\isasymprec}\isactrlbsub p\isactrlesub \ v\isactrlsub {\isadigit{2}}}
- \isa{{\isasymequiv}}
- $\begin{cases}
- (i) & \isa{{\isasymparallel}v\isactrlsub {\isadigit{2}}{\isasymparallel}\isactrlbsub p\isactrlesub \ {\isacharless}{\kern0pt}\ {\isasymparallel}v\isactrlsub {\isadigit{1}}{\isasymparallel}\isactrlbsub p\isactrlesub } \quad\text{and}\smallskip \\
- (ii) & \isa{{\isasymforall}q{\isasymin}Pos\ v\isactrlsub {\isadigit{1}}\ {\isasymunion}\ Pos\ v\isactrlsub {\isadigit{2}}{\isachardot}{\kern0pt}\ q\ {\isasymprec}\isactrlbsub lex\isactrlesub \ p\ {\isasymlongrightarrow}\ {\isasymparallel}v\isactrlsub {\isadigit{1}}{\isasymparallel}\isactrlbsub q\isactrlesub \ {\isacharequal}{\kern0pt}\ {\isasymparallel}v\isactrlsub {\isadigit{2}}{\isasymparallel}\isactrlbsub q\isactrlesub }
- \end{cases}$
- \end{tabular}
- \end{center}
-
- \noindent The position \isa{p} in this definition acts as the
- \emph{first distinct position} of \isa{v\isactrlsub {\isadigit{1}}} and \isa{v\isactrlsub {\isadigit{2}}}, where both values match strings of different length
- \cite{OkuiSuzuki2010}. Since at \isa{p} the values \isa{v\isactrlsub {\isadigit{1}}} and \isa{v\isactrlsub {\isadigit{2}}} match different strings, the
- ordering is irreflexive. Derived from the definition above
- are the following two orderings:
-
- \begin{center}
- \begin{tabular}{l}
- \isa{v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{2}}\ {\isasymequiv}\ {\isasymexists}p{\isachardot}{\kern0pt}\ v\isactrlsub {\isadigit{1}}\ {\isasymprec}\isactrlbsub p\isactrlesub \ v\isactrlsub {\isadigit{2}}}\\
- \isa{v\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ v\isactrlsub {\isadigit{2}}\ {\isasymequiv}\ v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{2}}\ {\isasymor}\ v\isactrlsub {\isadigit{1}}\ {\isacharequal}{\kern0pt}\ v\isactrlsub {\isadigit{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
- \isa{\underline{\hspace{2mm}}\ {\isasymprec}\ \underline{\hspace{2mm}}} and \isa{\underline{\hspace{2mm}}\ \mbox{$\preccurlyeq$}\ \underline{\hspace{2mm}}}
- by Okui and Suzuki.
-
- \begin{lemma}[Transitivity]\label{transitivity}
- \isa{{\normalsize{}If\,}\ \mbox{v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{2}}}\ {\normalsize \,and\,}\ \mbox{v\isactrlsub {\isadigit{2}}\ {\isasymprec}\ v\isactrlsub {\isadigit{3}}}\ {\normalsize \,then\,}\ v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{3}}{\isachardot}{\kern0pt}}
- \end{lemma}
-
- \begin{proof} From the assumption we obtain two positions \isa{p}
- and \isa{q}, where the values \isa{v\isactrlsub {\isadigit{1}}} and \isa{v\isactrlsub {\isadigit{2}}} (respectively \isa{v\isactrlsub {\isadigit{2}}} and \isa{v\isactrlsub {\isadigit{3}}}) are `distinct'. Since \isa{{\isasymprec}\isactrlbsub lex\isactrlesub } is trichotomous, we need to consider
- three cases, namely \isa{p\ {\isacharequal}{\kern0pt}\ q}, \isa{p\ {\isasymprec}\isactrlbsub lex\isactrlesub \ q} and
- \isa{q\ {\isasymprec}\isactrlbsub lex\isactrlesub \ p}. Let us look at the first case. Clearly
- \isa{{\isasymparallel}v\isactrlsub {\isadigit{2}}{\isasymparallel}\isactrlbsub p\isactrlesub \ {\isacharless}{\kern0pt}\ {\isasymparallel}v\isactrlsub {\isadigit{1}}{\isasymparallel}\isactrlbsub p\isactrlesub } and \isa{{\isasymparallel}v\isactrlsub {\isadigit{3}}{\isasymparallel}\isactrlbsub p\isactrlesub \ {\isacharless}{\kern0pt}\ {\isasymparallel}v\isactrlsub {\isadigit{2}}{\isasymparallel}\isactrlbsub p\isactrlesub } imply \isa{{\isasymparallel}v\isactrlsub {\isadigit{3}}{\isasymparallel}\isactrlbsub p\isactrlesub \ {\isacharless}{\kern0pt}\ {\isasymparallel}v\isactrlsub {\isadigit{1}}{\isasymparallel}\isactrlbsub p\isactrlesub }. It remains to show
- that for a \isa{p{\isacharprime}{\kern0pt}\ {\isasymin}\ Pos\ v\isactrlsub {\isadigit{1}}\ {\isasymunion}\ Pos\ v\isactrlsub {\isadigit{3}}}
- with \isa{p{\isacharprime}{\kern0pt}\ {\isasymprec}\isactrlbsub lex\isactrlesub \ p} that \isa{{\isasymparallel}v\isactrlsub {\isadigit{1}}{\isasymparallel}\isactrlbsub p{\isacharprime}{\kern0pt}\isactrlesub \ {\isacharequal}{\kern0pt}\ {\isasymparallel}v\isactrlsub {\isadigit{3}}{\isasymparallel}\isactrlbsub p{\isacharprime}{\kern0pt}\isactrlesub } holds. Suppose \isa{p{\isacharprime}{\kern0pt}\ {\isasymin}\ Pos\ v\isactrlsub {\isadigit{1}}}, then we can infer from the first assumption that \isa{{\isasymparallel}v\isactrlsub {\isadigit{1}}{\isasymparallel}\isactrlbsub p{\isacharprime}{\kern0pt}\isactrlesub \ {\isacharequal}{\kern0pt}\ {\isasymparallel}v\isactrlsub {\isadigit{2}}{\isasymparallel}\isactrlbsub p{\isacharprime}{\kern0pt}\isactrlesub }. But this means
- that \isa{p{\isacharprime}{\kern0pt}} must be in \isa{Pos\ v\isactrlsub {\isadigit{2}}} too (the norm
- cannot be \isa{{\isacharminus}{\kern0pt}{\isadigit{1}}} given \isa{p{\isacharprime}{\kern0pt}\ {\isasymin}\ Pos\ v\isactrlsub {\isadigit{1}}}).
- Hence we can use the second assumption and
- infer \isa{{\isasymparallel}v\isactrlsub {\isadigit{2}}{\isasymparallel}\isactrlbsub p{\isacharprime}{\kern0pt}\isactrlesub \ {\isacharequal}{\kern0pt}\ {\isasymparallel}v\isactrlsub {\isadigit{3}}{\isasymparallel}\isactrlbsub p{\isacharprime}{\kern0pt}\isactrlesub },
- which concludes this case with \isa{v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{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 \isa{{\isasymprec}} 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 \isa{r} and \isa{s}, the orderings have a unique
- minimal element on the set \isa{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 \isa{{\isasymprec}}-ordering.
-
- \begin{proposition}\mbox{}\smallskip\\\label{ordlen}
- \begin{tabular}{@ {}ll}
- (1) &
- \isa{{\normalsize{}If\,}\ v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{2}}\ {\normalsize \,then\,}\ len\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}\ {\isasymle}\ len\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}{\isachardot}{\kern0pt}}\\
- (2) &
- \isa{{\normalsize{}If\,}\ len\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}\ {\isacharless}{\kern0pt}\ len\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\normalsize \,then\,}\ v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{2}}{\isachardot}{\kern0pt}}
- \end{tabular}
- \end{proposition}
-
- \noindent Both properties follow from the definition of the ordering. Note that
- \textit{(2)} entails that a value, say \isa{v\isactrlsub {\isadigit{2}}}, whose underlying
- string is a strict prefix of another flattened value, say \isa{v\isactrlsub {\isadigit{1}}}, then
- \isa{v\isactrlsub {\isadigit{1}}} must be smaller than \isa{v\isactrlsub {\isadigit{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)} &
- \isa{{\normalsize{}If\,}\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}\ {\normalsize \,then\,}\ Left\ v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ Right\ v\isactrlsub {\isadigit{2}}{\isachardot}{\kern0pt}}\\
- \textit{(2)} & If
- \isa{{\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}} \;then\;
- \isa{Left\ v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ Left\ v\isactrlsub {\isadigit{2}}} \;iff\;
- \isa{v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{2}}}\\
- \textit{(3)} & If
- \isa{{\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}} \;then\;
- \isa{Right\ v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ Right\ v\isactrlsub {\isadigit{2}}} \;iff\;
- \isa{v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{2}}}\\
- \textit{(4)} & If
- \isa{{\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}w\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}} \;then\;
- \isa{Seq\ v\ v\isactrlsub {\isadigit{2}}\ {\isasymprec}\ Seq\ v\ w\isactrlsub {\isadigit{2}}} \;iff\;
- \isa{v\isactrlsub {\isadigit{2}}\ {\isasymprec}\ w\isactrlsub {\isadigit{2}}}\\
- \textit{(5)} & If
- \isa{{\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isacharat}{\kern0pt}\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}w\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isacharat}{\kern0pt}\ {\isacharbar}{\kern0pt}w\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}} \;and\;
- \isa{v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ w\isactrlsub {\isadigit{1}}} \;then\;
- \isa{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}\ {\isasymprec}\ Seq\ w\isactrlsub {\isadigit{1}}\ w\isactrlsub {\isadigit{2}}}\\
- \textit{(6)} & If
- \isa{{\isacharbar}{\kern0pt}vs\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}vs\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}} \;then\;
- \isa{Stars\ {\isacharparenleft}{\kern0pt}vs\ {\isacharat}{\kern0pt}\ vs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymprec}\ Stars\ {\isacharparenleft}{\kern0pt}vs\ {\isacharat}{\kern0pt}\ vs\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} \;iff\;
- \isa{Stars\ vs\isactrlsub {\isadigit{1}}\ {\isasymprec}\ Stars\ vs\isactrlsub {\isadigit{2}}}\\
-
- \textit{(7)} & If
- \isa{{\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{1}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vs\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vs\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}} \;and\;
- \isa{v\isactrlsub {\isadigit{1}}\ {\isasymprec}\ v\isactrlsub {\isadigit{2}}} \;then\;
- \isa{Stars\ {\isacharparenleft}{\kern0pt}v\isactrlsub {\isadigit{1}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymprec}\ Stars\ {\isacharparenleft}{\kern0pt}v\isactrlsub {\isadigit{2}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vs\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \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 \isa{Left} and \isa{Right}). Unfortunately this cannot be done easily: such
- a single statement would require an additional assumption about the
- two values \isa{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}} and \isa{Seq\ w\isactrlsub {\isadigit{1}}\ w\isactrlsub {\isadigit{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 \isa{POSIX} value \isa{v\isactrlsub {\isadigit{1}}}
- for \isa{r} and \isa{s}, then any other lexical value \isa{v\isactrlsub {\isadigit{2}}} in \isa{LV\ r\ s} is greater or equal than \isa{v\isactrlsub {\isadigit{1}}}, namely:
-
-
- \begin{theorem}\label{orderone}
- \isa{{\normalsize{}If\,}\ \mbox{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\isactrlsub {\isadigit{1}}}\ {\normalsize \,and\,}\ \mbox{v\isactrlsub {\isadigit{2}}\ {\isasymin}\ LV\ r\ s}\ {\normalsize \,then\,}\ v\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ v\isactrlsub {\isadigit{2}}{\isachardot}{\kern0pt}}
- \end{theorem}
-
- \begin{proof} By induction on our POSIX rules. By
- Theorem~\ref{posixdeterm} and the definition of \isa{LV}, it is clear
- that \isa{v\isactrlsub {\isadigit{1}}} and \isa{v\isactrlsub {\isadigit{2}}} have the same
- underlying string \isa{s}. The three base cases are
- straightforward: for example for \isa{v\isactrlsub {\isadigit{1}}\ {\isacharequal}{\kern0pt}\ Empty}, we have
- that \isa{v\isactrlsub {\isadigit{2}}\ {\isasymin}\ LV\ \isactrlbold {\isadigit{1}}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}} must also be of the form
- \mbox{\isa{v\isactrlsub {\isadigit{2}}\ {\isacharequal}{\kern0pt}\ Empty}}. Therefore we have \isa{v\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ v\isactrlsub {\isadigit{2}}}. The inductive cases for
- \isa{r} being of the form \isa{r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}} and
- \isa{r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}} are as follows:
-
-
- \begin{itemize}
-
- \item[$\bullet$] Case \isa{P{\isacharplus}{\kern0pt}L} with \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Left\ w\isactrlsub {\isadigit{1}}}: In this case the value
- \isa{v\isactrlsub {\isadigit{2}}} is either of the
- form \isa{Left\ w\isactrlsub {\isadigit{2}}} or \isa{Right\ w\isactrlsub {\isadigit{2}}}. In the
- latter case we can immediately conclude with \mbox{\isa{v\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ v\isactrlsub {\isadigit{2}}}} since a \isa{Left}-value with the
- same underlying string \isa{s} is always smaller than a
- \isa{Right}-value by Proposition~\ref{ordintros}\textit{(1)}.
- In the former case we have \isa{w\isactrlsub {\isadigit{2}}\ {\isasymin}\ LV\ r\isactrlsub {\isadigit{1}}\ s} and can use the induction hypothesis to infer
- \isa{w\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ w\isactrlsub {\isadigit{2}}}. Because \isa{w\isactrlsub {\isadigit{1}}} and \isa{w\isactrlsub {\isadigit{2}}} have the same underlying string
- \isa{s}, we can conclude with \isa{Left\ w\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ Left\ w\isactrlsub {\isadigit{2}}} using
- Proposition~\ref{ordintros}\textit{(2)}.\smallskip
-
- \item[$\bullet$] Case \isa{P{\isacharplus}{\kern0pt}R} with \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Right\ w\isactrlsub {\isadigit{1}}}: This case similar to the previous
- case, except that we additionally know \isa{s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}}. This is needed when \isa{v\isactrlsub {\isadigit{2}}} is of the form
- \mbox{\isa{Left\ w\isactrlsub {\isadigit{2}}}}. Since \mbox{\isa{{\isacharbar}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}w\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}} \isa{{\isacharequal}{\kern0pt}\ s}} and \isa{w\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}, we can derive a contradiction for \mbox{\isa{s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}}} using
- Proposition~\ref{inhabs}. So also in this case \mbox{\isa{v\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ v\isactrlsub {\isadigit{2}}}}.\smallskip
-
- \item[$\bullet$] Case \isa{PS} with \isa{{\isacharparenleft}{\kern0pt}s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Seq\ w\isactrlsub {\isadigit{1}}\ w\isactrlsub {\isadigit{2}}}: We can assume \isa{v\isactrlsub {\isadigit{2}}\ {\isacharequal}{\kern0pt}\ Seq\ u\isactrlsub {\isadigit{1}}\ u\isactrlsub {\isadigit{2}}} with \isa{u\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}} and \mbox{\isa{u\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}}. We have \isa{s\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ s\isactrlsub {\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}u\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}\ {\isacharat}{\kern0pt}\ {\isacharbar}{\kern0pt}u\isactrlsub {\isadigit{2}}{\isacharbar}{\kern0pt}}. By the side-condition of the
- \isa{PS}-rule we know that either \isa{s\isactrlsub {\isadigit{1}}\ {\isacharequal}{\kern0pt}\ {\isacharbar}{\kern0pt}u\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}} or that \isa{{\isacharbar}{\kern0pt}u\isactrlsub {\isadigit{1}}{\isacharbar}{\kern0pt}} is a strict prefix of
- \isa{s\isactrlsub {\isadigit{1}}}. In the latter case we can infer \isa{w\isactrlsub {\isadigit{1}}\ {\isasymprec}\ u\isactrlsub {\isadigit{1}}} by
- Proposition~\ref{ordlen}\textit{(2)} and from this \isa{v\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ v\isactrlsub {\isadigit{2}}} by Proposition~\ref{ordintros}\textit{(5)}
- (as noted above \isa{v\isactrlsub {\isadigit{1}}} and \isa{v\isactrlsub {\isadigit{2}}} must have the
- same underlying string).
- In the former case we know
- \isa{u\isactrlsub {\isadigit{1}}\ {\isasymin}\ LV\ r\isactrlsub {\isadigit{1}}\ s\isactrlsub {\isadigit{1}}} and \isa{u\isactrlsub {\isadigit{2}}\ {\isasymin}\ LV\ r\isactrlsub {\isadigit{2}}\ s\isactrlsub {\isadigit{2}}}. With this we can use the
- induction hypotheses to infer \isa{w\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ u\isactrlsub {\isadigit{1}}} and \isa{w\isactrlsub {\isadigit{2}}\ \mbox{$\preccurlyeq$}\ u\isactrlsub {\isadigit{2}}}. By
- Proposition~\ref{ordintros}\textit{(4,5)} we can again infer
- \isa{v\isactrlsub {\isadigit{1}}\ \mbox{$\preccurlyeq$}\ v\isactrlsub {\isadigit{2}}}.
-
- \end{itemize}
-
- \noindent The case for \isa{P{\isasymstar}} is similar to the \isa{PS}-case and omitted.\qed
- \end{proof}
-
- \noindent This theorem shows that our \isa{POSIX} value for a
- regular expression \isa{r} and string \isa{s} is in fact a
- minimal element of the values in \isa{LV\ r\ s}. By
- Proposition~\ref{ordlen}\textit{(2)} we also know that any value in
- \isa{LV\ r\ s{\isacharprime}{\kern0pt}}, with \isa{s{\isacharprime}{\kern0pt}} being a strict prefix, cannot be
- smaller than \isa{v\isactrlsub {\isadigit{1}}}. The next theorem shows the
- opposite---namely any minimal element in \isa{LV\ r\ s} must be a
- \isa{POSIX} value. This can be established by induction on \isa{r}, but the proof can be drastically simplified by using the fact
- from the previous section about the existence of a \isa{POSIX} value
- whenever a string \isa{s\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}}.
-
-
- \begin{theorem}
- \isa{{\normalsize{}If\,}\ \mbox{v\isactrlsub {\isadigit{1}}\ {\isasymin}\ LV\ r\ s}\ {\normalsize \,and\,}\ \mbox{{\isasymforall}v\isactrlsub {\isadigit{2}}{\isasymin}LV\ r\ s{\isachardot}{\kern0pt}\ v\isactrlsub {\isadigit{2}}\ \mbox{$\not\prec$}\ v\isactrlsub {\isadigit{1}}}\ {\normalsize \,then\,}\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\isactrlsub {\isadigit{1}}{\isachardot}{\kern0pt}}
- \end{theorem}
-
- \begin{proof}
- If \isa{v\isactrlsub {\isadigit{1}}\ {\isasymin}\ LV\ r\ s} then
- \isa{s\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}} by Proposition~\ref{inhabs}. Hence by Theorem~\ref{lexercorrect}(2)
- there exists a
- \isa{POSIX} value \isa{v\isactrlsub P} with \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\isactrlsub P}
- and by Lemma~\ref{LVposix} we also have \mbox{\isa{v\isactrlsub P\ {\isasymin}\ LV\ r\ s}}.
- By Theorem~\ref{orderone} we therefore have
- \isa{v\isactrlsub P\ \mbox{$\preccurlyeq$}\ v\isactrlsub {\isadigit{1}}}. If \isa{v\isactrlsub P\ {\isacharequal}{\kern0pt}\ v\isactrlsub {\isadigit{1}}} then
- we are done. Otherwise we have \isa{v\isactrlsub P\ {\isasymprec}\ v\isactrlsub {\isadigit{1}}}, which
- however contradicts the second assumption about \isa{v\isactrlsub {\isadigit{1}}} being the smallest
- element in \isa{LV\ r\ s}. So we are done in this case too.\qed
- \end{proof}
-
- \noindent
- From this we can also show
- that if \isa{LV\ r\ s} is non-empty (or equivalently \isa{s\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}}) then
- it has a unique minimal element:
-
- \begin{corollary}
- \isa{{\normalsize{}If\,}\ LV\ r\ s\ {\isasymnoteq}\ {\isasymemptyset}\ {\normalsize \,then\,}\ {\isasymexists}{\isacharbang}{\kern0pt}vmin{\isachardot}{\kern0pt}\ vmin\ {\isasymin}\ LV\ r\ s\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}v{\isasymin}LV\ r\ s{\isachardot}{\kern0pt}\ vmin\ \mbox{$\preccurlyeq$}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}}
- \end{corollary}
-
-
-
- \noindent To sum up, we have shown that the (unique) minimal elements
- of the ordering by Okui and Suzuki are exactly the \isa{POSIX}
- values we defined inductively in Section~\ref{posixsec}. This provides
- an independent confirmation that our ternary relation formalises the
- informal POSIX rules.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
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-\isatagdocument
-%
-\isamarkupsection{Bitcoded Lexing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
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-%
-\isadelimdocument
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-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-Incremental calculation of the value. To simplify the proof we first define the function
-\isa{flex} which calculates the ``iterated'' injection function. With this we can
-rewrite the lexer as
-
-\begin{center}
-\isa{lexer\ r\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\textrm{if}\ nullable\ {\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}s{\isacharparenright}{\kern0pt}\ \textrm{then}\ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \textrm{else}\ None{\isacharparenright}{\kern0pt}}
-\end{center}%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Optimisations%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-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 \isa{\isactrlbold {\isadigit{0}}\ {\isacharplus}{\kern0pt}\ r}, \isa{r\ {\isacharplus}{\kern0pt}\ \isactrlbold {\isadigit{0}}}, \isa{\isactrlbold {\isadigit{1}}\ {\isasymcdot}\ r} and
- \isa{r\ {\isasymcdot}\ \isactrlbold {\isadigit{1}}} to \isa{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}
- \isa{\isactrlbold {\isadigit{0}}\ {\isacharplus}{\kern0pt}\ r} & \isa{{\isasymRightarrow}} & \isa{r} \hspace{8mm}%\\
- \isa{r\ {\isacharplus}{\kern0pt}\ \isactrlbold {\isadigit{0}}} & \isa{{\isasymRightarrow}} & \isa{r} \hspace{8mm}%\\
- \isa{\isactrlbold {\isadigit{1}}\ {\isasymcdot}\ r} & \isa{{\isasymRightarrow}} & \isa{r} \hspace{8mm}%\\
- \isa{r\ {\isasymcdot}\ \isactrlbold {\isadigit{1}}} & \isa{{\isasymRightarrow}} & \isa{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}
- \isa{F\isactrlbsub Right\isactrlesub \ f\ v} & $\dn$ & \isa{Right\ {\isacharparenleft}{\kern0pt}f\ v{\isacharparenright}{\kern0pt}}\\
- \isa{F\isactrlbsub Left\isactrlesub \ f\ v} & $\dn$ & \isa{Left\ {\isacharparenleft}{\kern0pt}f\ v{\isacharparenright}{\kern0pt}}\\
-
- \isa{F\isactrlbsub Alt\isactrlesub \ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}Right\ v{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{Right\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{2}}\ v{\isacharparenright}{\kern0pt}}\\
- \isa{F\isactrlbsub Alt\isactrlesub \ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}Left\ v{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{Left\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{1}}\ v{\isacharparenright}{\kern0pt}}\\
-
- \isa{F\isactrlbsub Seq{\isadigit{1}}\isactrlesub \ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}\ v} & $\dn$ & \isa{Seq\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{2}}\ v{\isacharparenright}{\kern0pt}}\\
- \isa{F\isactrlbsub Seq{\isadigit{2}}\isactrlesub \ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}\ v} & $\dn$ & \isa{Seq\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{1}}\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}}\\
- \isa{F\isactrlbsub Seq\isactrlesub \ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{Seq\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{2}}\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\medskip\\
- %\end{tabular}
- %
- %\begin{tabular}{lcl}
- \isa{simp\isactrlbsub Alt\isactrlesub \ {\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{0}}{\isacharcomma}{\kern0pt}\ \underline{\hspace{2mm}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ F\isactrlbsub Right\isactrlesub \ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \isa{simp\isactrlbsub Alt\isactrlesub \ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{0}}{\isacharcomma}{\kern0pt}\ \underline{\hspace{2mm}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ F\isactrlbsub Left\isactrlesub \ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}}\\
- \isa{simp\isactrlbsub Alt\isactrlesub \ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ F\isactrlbsub Alt\isactrlesub \ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \isa{simp\isactrlbsub Seq\isactrlesub \ {\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ F\isactrlbsub Seq{\isadigit{1}}\isactrlesub \ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \isa{simp\isactrlbsub Seq\isactrlesub \ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isactrlbold {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ F\isactrlbsub Seq{\isadigit{2}}\isactrlesub \ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \isa{simp\isactrlbsub Seq\isactrlesub \ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ F\isactrlbsub Seq\isactrlesub \ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \end{tabular}
- \end{center}
-
- \noindent
- The functions \isa{simp\isactrlbsub Alt\isactrlesub } and \isa{simp\isactrlbsub Seq\isactrlesub } 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}
- \isa{simp\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{simp\isactrlbsub Alt\isactrlesub \ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \isa{simp\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{simp\isactrlbsub Seq\isactrlesub \ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
- \isa{simp\ r} & $\dn$ & \isa{{\isacharparenleft}{\kern0pt}r{\isacharcomma}{\kern0pt}\ id{\isacharparenright}{\kern0pt}}\\
- \end{tabular}
- \end{center}
-
- \noindent where \isa{id} stands for the identity function. The
- function \isa{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}
- \isa{lexer\isactrlsup {\isacharplus}{\kern0pt}\ r\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}} & $\dn$ & \isa{\textrm{if}\ nullable\ r\ \textrm{then}\ Some\ {\isacharparenleft}{\kern0pt}mkeps\ r{\isacharparenright}{\kern0pt}\ \textrm{else}\ None}\\
- \isa{lexer\isactrlsup {\isacharplus}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s{\isacharparenright}{\kern0pt}} & $\dn$ &
- \isa{let\ {\isacharparenleft}{\kern0pt}r\isactrlsub s{\isacharcomma}{\kern0pt}\ f\isactrlsub r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ simp\ {\isacharparenleft}{\kern0pt}r}$\backslash$\isa{c{\isacharparenright}{\kern0pt}\ in}\\
- & & \isa{case} \isa{lexer\isactrlsup {\isacharplus}{\kern0pt}\ r\isactrlsub s\ s} \isa{of}\\
- & & \phantom{$|$} \isa{None} \isa{{\isasymRightarrow}} \isa{None}\\
- & & $|$ \isa{Some\ v} \isa{{\isasymRightarrow}} \isa{Some\ {\isacharparenleft}{\kern0pt}inj\ r\ c\ {\isacharparenleft}{\kern0pt}f\isactrlsub r\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}}
- \end{tabular}
- \end{center}
-
- \noindent
- In the second clause we first calculate the derivative \isa{r{\isacharbackslash}{\kern0pt}c}
- and then simpli
-
-text \isa{\ \ Incremental\ calculation\ of\ the\ value{\isachardot}{\kern0pt}\ To\ simplify\ the\ proof\ we\ first\ define\ the\ function\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}const\ flex{\isacharbraceright}{\kern0pt}\ which\ calculates\ the\ {\isacharbackquote}{\kern0pt}{\isacharbackquote}{\kern0pt}iterated{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ injection\ function{\isachardot}{\kern0pt}\ With\ this\ we\ can\ rewrite\ the\ lexer\ as\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ lexer{\isacharunderscore}{\kern0pt}flex{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequote}{\kern0pt}v\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequote}{\kern0pt}v\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}v\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ code{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ areg{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}AZERO{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}mid{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}AONE\ bs{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}mid{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}ACHAR\ bs\ c{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}mid{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}AALT\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}mid{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}mid{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}ASTAR\ bs\ r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ intern{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ Some\ simple\ facts\ about\ erase\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}lemma{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}mbox{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ erase{\isacharunderscore}{\kern0pt}bder{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ erase{\isacharunderscore}{\kern0pt}intern{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}lemma{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}medskip{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ \ {\isacharpercent}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharpercent}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ {\isacharpercent}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharpercent}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ \ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ bs\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}\ {\isachardoublequote}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}medskip{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ \ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharbrackleft}{\kern0pt}mode{\isacharequal}{\kern0pt}IfThen{\isacharbrackright}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}retrieve{\isacharbraceright}{\kern0pt}\ \ By\ induction\ on\ {\isasymopen}r{\isasymclose}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}theorem{\isacharbraceright}{\kern0pt}{\isacharbrackleft}{\kern0pt}Main\ Lemma{\isacharbrackright}{\kern0pt}{\isacharbackslash}{\kern0pt}mbox{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharbrackleft}{\kern0pt}mode{\isacharequal}{\kern0pt}IfThen{\isacharbrackright}{\kern0pt}\ MAIN{\isacharunderscore}{\kern0pt}decode{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}theorem{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}noindent\ Definition\ of\ the\ bitcoded\ lexer\ \ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ blexer{\isacharunderscore}{\kern0pt}def{\isacharbraceright}{\kern0pt}\ \ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}theorem{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ blexer{\isacharunderscore}{\kern0pt}correctness{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}theorem{\isacharbraceright}{\kern0pt}\ \ }
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-section \isa{Optimisations}
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-text \isa{\ \ Derivatives\ as\ calculated\ by\ {\isacharbackslash}{\kern0pt}Brz{\isacharprime}{\kern0pt}s\ method\ are\ usually\ more\ complex\ regular\ expressions\ than\ the\ initial\ one{\isacharsemicolon}{\kern0pt}\ the\ result\ is\ that\ the\ derivative{\isacharminus}{\kern0pt}based\ matching\ and\ lexing\ algorithms\ are\ often\ abysmally\ slow{\isachardot}{\kern0pt}\ However{\isacharcomma}{\kern0pt}\ various\ optimisations\ are\ possible{\isacharcomma}{\kern0pt}\ such\ as\ the\ simplifications\ of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}ALT\ ZERO\ r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}ALT\ r\ ZERO{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}SEQ\ ONE\ r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ and\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}SEQ\ r\ ONE{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ to\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ r{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ These\ simplifications\ can\ speed\ up\ the\ algorithms\ considerably{\isacharcomma}{\kern0pt}\ as\ noted\ in\ {\isacharbackslash}{\kern0pt}cite{\isacharbraceleft}{\kern0pt}Sulzmann{\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ 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{\isachardot}{\kern0pt}\ While\ the\ simplification\ of\ regular\ expressions\ according\ to\ rules\ like\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}equation{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}label{\isacharbraceleft}{\kern0pt}Simpl{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}array{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcllcllcllcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}ALT\ ZERO\ r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}{\isasymRightarrow}{\isasymclose}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ r{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}hspace{\isacharbraceleft}{\kern0pt}{\isadigit{8}}mm{\isacharbraceright}{\kern0pt}{\isacharpercent}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}ALT\ r\ ZERO{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}{\isasymRightarrow}{\isasymclose}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ r{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}hspace{\isacharbraceleft}{\kern0pt}{\isadigit{8}}mm{\isacharbraceright}{\kern0pt}{\isacharpercent}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}SEQ\ ONE\ r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ \ {\isacharampersand}{\kern0pt}\ {\isasymopen}{\isasymRightarrow}{\isasymclose}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ r{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}hspace{\isacharbraceleft}{\kern0pt}{\isadigit{8}}mm{\isacharbraceright}{\kern0pt}{\isacharpercent}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}SEQ\ r\ ONE{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ \ {\isacharampersand}{\kern0pt}\ {\isasymopen}{\isasymRightarrow}{\isasymclose}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ r{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}array{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}equation{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}noindent\ is\ well\ understood{\isacharcomma}{\kern0pt}\ there\ is\ an\ obstacle\ with\ the\ POSIX\ value\ calculation\ algorithm\ by\ Sulzmann\ and\ Lu{\isacharcolon}{\kern0pt}\ if\ we\ build\ a\ derivative\ regular\ expression\ and\ then\ simplify\ it{\isacharcomma}{\kern0pt}\ we\ will\ calculate\ a\ POSIX\ value\ for\ this\ simplified\ derivative\ regular\ expression{\isacharcomma}{\kern0pt}\ {\isacharbackslash}{\kern0pt}emph{\isacharbraceleft}{\kern0pt}not{\isacharbraceright}{\kern0pt}\ for\ the\ original\ {\isacharparenleft}{\kern0pt}unsimplified{\isacharparenright}{\kern0pt}\ derivative\ regular\ expression{\isachardot}{\kern0pt}\ Sulzmann\ and\ Lu\ {\isacharbackslash}{\kern0pt}cite{\isacharbraceleft}{\kern0pt}Sulzmann{\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}{\isacharbraceright}{\kern0pt}\ overcome\ this\ obstacle\ by\ not\ just\ calculating\ a\ simplified\ regular\ expression{\isacharcomma}{\kern0pt}\ but\ also\ calculating\ a\ {\isacharbackslash}{\kern0pt}emph{\isacharbraceleft}{\kern0pt}rectification\ function{\isacharbraceright}{\kern0pt}\ that\ {\isacharbackquote}{\kern0pt}{\isacharbackquote}{\kern0pt}repairs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ the\ incorrect\ value{\isachardot}{\kern0pt}\ \ The\ rectification\ functions\ can\ be\ {\isacharparenleft}{\kern0pt}slightly\ clumsily{\isacharparenright}{\kern0pt}\ implemented\ \ in\ Isabelle{\isacharslash}{\kern0pt}HOL\ as\ follows\ using\ some\ auxiliary\ functions{\isacharcolon}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}RIGHT{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}Right\ {\isacharparenleft}{\kern0pt}f\ v{\isacharparenright}{\kern0pt}{\isasymclose}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}LEFT{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}Left\ {\isacharparenleft}{\kern0pt}f\ v{\isacharparenright}{\kern0pt}{\isasymclose}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ \ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}ALT{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}Right\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{2}}\ v{\isacharparenright}{\kern0pt}{\isasymclose}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}ALT{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}Left\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{1}}\ v{\isacharparenright}{\kern0pt}{\isasymclose}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ \ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}SEQ{\isadigit{1}}{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}Seq\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{2}}\ v{\isacharparenright}{\kern0pt}{\isasymclose}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}SEQ{\isadigit{2}}{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}Seq\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{1}}\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isasymclose}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}SEQ{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}Seq\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\isactrlsub {\isadigit{2}}\ v\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isasymclose}{\isacharbackslash}{\kern0pt}medskip{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharpercent}{\kern0pt}{\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharpercent}{\kern0pt}\ {\isacharpercent}{\kern0pt}{\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp{\isacharunderscore}{\kern0pt}ALT\ {\isacharparenleft}{\kern0pt}ZERO{\isacharcomma}{\kern0pt}\ DUMMY{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}RIGHT\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp{\isacharunderscore}{\kern0pt}ALT\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ZERO{\isacharcomma}{\kern0pt}\ DUMMY{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}LEFT\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp{\isacharunderscore}{\kern0pt}ALT\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}ALT\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}ALT\ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp{\isacharunderscore}{\kern0pt}SEQ\ {\isacharparenleft}{\kern0pt}ONE{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}SEQ{\isadigit{1}}\ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp{\isacharunderscore}{\kern0pt}SEQ\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ONE{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}SEQ{\isadigit{2}}\ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp{\isacharunderscore}{\kern0pt}SEQ\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}SEQ\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharcomma}{\kern0pt}\ F{\isacharunderscore}{\kern0pt}SEQ\ f\isactrlsub {\isadigit{1}}\ f\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}noindent\ The\ functions\ {\isasymopen}simp\isactrlbsub Alt\isactrlesub {\isasymclose}\ and\ {\isasymopen}simp\isactrlbsub Seq\isactrlesub {\isasymclose}\ encode\ the\ simplification\ rules\ in\ {\isacharbackslash}{\kern0pt}eqref{\isacharbraceleft}{\kern0pt}Simpl{\isacharbraceright}{\kern0pt}\ and\ compose\ the\ rectification\ functions\ {\isacharparenleft}{\kern0pt}simplifications\ can\ occur\ deep\ inside\ the\ regular\ expression{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ The\ main\ simplification\ function\ is\ then\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}ALT\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp{\isacharunderscore}{\kern0pt}ALT\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}SEQ\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp{\isacharunderscore}{\kern0pt}SEQ\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}simp\ r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}r{\isacharcomma}{\kern0pt}\ id{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}noindent\ where\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}id{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ stands\ for\ the\ identity\ function{\isachardot}{\kern0pt}\ The\ function\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}const\ simp{\isacharbraceright}{\kern0pt}\ returns\ a\ simplified\ regular\ expression\ and\ a\ corresponding\ rectification\ function{\isachardot}{\kern0pt}\ Note\ that\ we\ do\ not\ simplify\ under\ stars{\isacharcolon}{\kern0pt}\ this\ seems\ to\ slow\ down\ the\ algorithm{\isacharcomma}{\kern0pt}\ rather\ than\ speed\ it\ up{\isachardot}{\kern0pt}\ The\ optimised\ lexer\ is\ then\ given\ by\ the\ clauses{\isacharcolon}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}lcl{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ slexer{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\ slexer{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ {\isacharparenleft}{\kern0pt}lhs{\isacharparenright}{\kern0pt}\ slexer{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}dn{\isachardollar}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}let\ {\isacharparenleft}{\kern0pt}r\isactrlsub s{\isacharcomma}{\kern0pt}\ f\isactrlsub r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ simp\ {\isacharparenleft}{\kern0pt}r\ {\isasymclose}{\isachardollar}{\kern0pt}{\isacharbackslash}{\kern0pt}backslash{\isachardollar}{\kern0pt}{\isasymopen}\ c{\isacharparenright}{\kern0pt}\ in{\isasymclose}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isasymopen}case{\isasymclose}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}slexer\ r\isactrlsub s\ s{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymopen}of{\isasymclose}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharbackslash}{\kern0pt}phantom{\isacharbraceleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharbar}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}None{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ \ {\isasymopen}{\isasymRightarrow}{\isasymclose}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ None{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isachardollar}{\kern0pt}{\isacharbar}{\kern0pt}{\isachardollar}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}Some\ v{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymopen}{\isasymRightarrow}{\isasymclose}\ {\isasymopen}Some\ {\isacharparenleft}{\kern0pt}inj\ r\ c\ {\isacharparenleft}{\kern0pt}f\isactrlsub r\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isasymclose}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}center{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}noindent\ In\ the\ second\ clause\ we\ first\ calculate\ the\ derivative\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}der\ c\ r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ and\ then\ simplify\ the\ result{\isachardot}{\kern0pt}\ This\ gives\ us\ a\ simplified\ derivative\ {\isasymopen}r\isactrlsub s{\isasymclose}\ and\ a\ rectification\ function\ {\isasymopen}f\isactrlsub r{\isasymclose}{\isachardot}{\kern0pt}\ The\ lexer\ is\ then\ recursively\ called\ with\ the\ simplified\ derivative{\isacharcomma}{\kern0pt}\ but\ before\ we\ inject\ the\ character\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ c{\isacharbraceright}{\kern0pt}\ into\ the\ value\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ v{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ we\ need\ to\ rectify\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ v{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}that\ is\ construct\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}f\isactrlsub r\ v{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ Before\ we\ can\ establish\ the\ correctness\ of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}slexer{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ we\ need\ to\ show\ that\ simplification\ preserves\ the\ language\ and\ simplification\ preserves\ our\ POSIX\ relation\ once\ the\ value\ is\ rectified\ {\isacharparenleft}{\kern0pt}recall\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}const\ {\isachardoublequote}{\kern0pt}simp{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ generates\ a\ {\isacharparenleft}{\kern0pt}regular\ expression{\isacharcomma}{\kern0pt}\ rectification\ function{\isacharparenright}{\kern0pt}\ pair{\isacharparenright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}lemma{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}mbox{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}smallskip{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}label{\isacharbraceleft}{\kern0pt}slexeraux{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}{\isacharbraceleft}{\kern0pt}ll{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ L{\isacharunderscore}{\kern0pt}fst{\isacharunderscore}{\kern0pt}simp{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbackslash}{\kern0pt}{\isacharbackslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharampersand}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm{\isacharbrackleft}{\kern0pt}mode{\isacharequal}{\kern0pt}IfThen{\isacharbrackright}{\kern0pt}\ Posix{\isacharunderscore}{\kern0pt}simp{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}tabular{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}lemma{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}proof{\isacharbraceright}{\kern0pt}\ Both\ are\ by\ induction\ on\ {\isasymopen}r{\isasymclose}{\isachardot}{\kern0pt}\ There\ is\ no\ interesting\ case\ for\ the\ first\ statement{\isachardot}{\kern0pt}\ For\ the\ second\ statement{\isacharcomma}{\kern0pt}\ of\ interest\ are\ the\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}r\ {\isacharequal}{\kern0pt}\ ALT\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ and\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}r\ {\isacharequal}{\kern0pt}\ SEQ\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ cases{\isachardot}{\kern0pt}\ In\ each\ case\ we\ have\ to\ analyse\ four\ subcases\ whether\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ and\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ equals\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}const\ ZERO{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}respectively\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}const\ ONE{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ For\ example\ for\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}r\ {\isacharequal}{\kern0pt}\ ALT\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ consider\ the\ subcase\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ZERO{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ and\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ ZERO{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ By\ assumption\ we\ know\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}ALT\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ From\ this\ we\ can\ infer\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ and\ by\ IH\ also\ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ r\isactrlsub {\isadigit{2}}\ {\isasymrightarrow}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ Given\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ZERO{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ we\ know\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}L\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ By\ the\ first\ statement\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}L\ r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ is\ the\ empty\ set{\isacharcomma}{\kern0pt}\ meaning\ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymnotin}\ L\ r\isactrlsub {\isadigit{1}}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ Taking\ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\ and\ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\ together\ gives\ by\ the\ {\isacharbackslash}{\kern0pt}mbox{\isacharbraceleft}{\kern0pt}{\isasymopen}P{\isacharplus}{\kern0pt}R{\isasymclose}{\isacharbraceright}{\kern0pt}{\isacharminus}{\kern0pt}rule\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ ALT\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}\ {\isasymrightarrow}\ Right\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ In\ turn\ this\ gives\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ ALT\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}\ {\isasymrightarrow}\ snd\ {\isacharparenleft}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}ALT\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ as\ we\ need\ to\ show{\isachardot}{\kern0pt}\ The\ other\ cases\ are\ similar{\isachardot}{\kern0pt}{\isacharbackslash}{\kern0pt}qed\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}proof{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}noindent\ We\ can\ now\ prove\ relatively\ straightforwardly\ that\ the\ optimised\ lexer\ produces\ the\ expected\ result{\isacharcolon}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}theorem{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}thm\ slexer{\isacharunderscore}{\kern0pt}correctness{\isacharbraceright}{\kern0pt}\ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}theorem{\isacharbraceright}{\kern0pt}\ \ {\isacharbackslash}{\kern0pt}begin{\isacharbraceleft}{\kern0pt}proof{\isacharbraceright}{\kern0pt}\ By\ induction\ on\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ s{\isacharbraceright}{\kern0pt}\ generalising\ over\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ r{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ The\ case\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ is\ trivial{\isachardot}{\kern0pt}\ For\ the\ cons{\isacharminus}{\kern0pt}case\ suppose\ the\ string\ is\ of\ the\ form\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}c\ {\isacharhash}{\kern0pt}\ s{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ By\ induction\ hypothesis\ we\ know\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}slexer\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ holds\ for\ all\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ r{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}in\ particular\ for\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ being\ the\ derivative\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}der\ c\ r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ Let\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}r\isactrlsub s{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ be\ the\ simplified\ derivative\ regular\ expression{\isacharcomma}{\kern0pt}\ that\ is\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ and\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}f\isactrlsub r{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ be\ the\ rectification\ function{\isacharcomma}{\kern0pt}\ that\ is\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ \ We\ distinguish\ the\ cases\ whether\ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ L\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ or\ not{\isachardot}{\kern0pt}\ In\ the\ first\ case\ we\ have\ by\ Theorem{\isachartilde}{\kern0pt}{\isacharbackslash}{\kern0pt}ref{\isacharbraceleft}{\kern0pt}lexercorrect{\isacharbraceright}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ a\ value\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}v{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ so\ that\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}lexer\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ Some\ v{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ and\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ der\ c\ r\ {\isasymrightarrow}\ v{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ hold{\isachardot}{\kern0pt}\ By\ Lemma{\isachartilde}{\kern0pt}{\isacharbackslash}{\kern0pt}ref{\isacharbraceleft}{\kern0pt}slexeraux{\isacharbraceright}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ we\ can\ also\ infer\ from{\isachartilde}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\ that\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ L\ r\isactrlsub s{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ holds{\isachardot}{\kern0pt}\ \ Hence\ we\ know\ by\ Theorem{\isachartilde}{\kern0pt}{\isacharbackslash}{\kern0pt}ref{\isacharbraceleft}{\kern0pt}lexercorrect{\isacharbraceright}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ that\ there\ exists\ a\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}v{\isacharprime}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ with\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}lexer\ r\isactrlsub s\ s\ {\isacharequal}{\kern0pt}\ Some\ v{\isacharprime}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ and\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ r\isactrlsub s\ {\isasymrightarrow}\ v{\isacharprime}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ From\ the\ latter\ we\ know\ by\ Lemma{\isachartilde}{\kern0pt}{\isacharbackslash}{\kern0pt}ref{\isacharbraceleft}{\kern0pt}slexeraux{\isacharbraceright}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ that\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymin}\ der\ c\ r\ {\isasymrightarrow}\ {\isacharparenleft}{\kern0pt}f\isactrlsub r\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ holds{\isachardot}{\kern0pt}\ By\ the\ uniqueness\ of\ the\ POSIX\ relation\ {\isacharparenleft}{\kern0pt}Theorem{\isachartilde}{\kern0pt}{\isacharbackslash}{\kern0pt}ref{\isacharbraceleft}{\kern0pt}posixdeterm{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ we\ can\ infer\ that\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ v{\isacharbraceright}{\kern0pt}\ is\ equal\ to\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}f\isactrlsub r\ v{\isacharprime}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharminus}{\kern0pt}{\isacharminus}{\kern0pt}{\isacharminus}{\kern0pt}that\ is\ the\ rectification\ function\ applied\ to\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}v{\isacharprime}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ produces\ the\ original\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}v{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ \ Now\ the\ case\ follows\ by\ the\ definitions\ of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}const\ lexer{\isacharbraceright}{\kern0pt}\ and\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}const\ slexer{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ \ In\ the\ second\ case\ where\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymnotin}\ L\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ we\ have\ that\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}lexer\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ None{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ by\ Theorem{\isachartilde}{\kern0pt}{\isacharbackslash}{\kern0pt}ref{\isacharbraceleft}{\kern0pt}lexercorrect{\isacharbraceright}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ \ We\ also\ know\ by\ Lemma{\isachartilde}{\kern0pt}{\isacharbackslash}{\kern0pt}ref{\isacharbraceleft}{\kern0pt}slexeraux{\isacharbraceright}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ that\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}s\ {\isasymnotin}\ L\ r\isactrlsub s{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ Hence\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}lexer\ r\isactrlsub s\ s\ {\isacharequal}{\kern0pt}\ None{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}\ by\ Theorem{\isachartilde}{\kern0pt}{\isacharbackslash}{\kern0pt}ref{\isacharbraceleft}{\kern0pt}lexercorrect{\isacharbraceright}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ and\ by\ IH\ then\ also\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}term\ {\isachardoublequote}{\kern0pt}slexer\ r\isactrlsub s\ s\ {\isacharequal}{\kern0pt}\ None{\isachardoublequote}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ With\ this\ we\ can\ conclude\ in\ this\ case\ too{\isachardot}{\kern0pt}{\isacharbackslash}{\kern0pt}qed\ \ {\isacharbackslash}{\kern0pt}end{\isacharbraceleft}{\kern0pt}proof{\isacharbraceright}{\kern0pt}\ \ }
-fy the result. This gives us a simplified derivative
- \isa{r\isactrlsub s} and a rectification function \isa{f\isactrlsub r}. The lexer
- is then recursively called with the simplified derivative, but before
- we inject the character \isa{c} into the value \isa{v}, we need to rectify
- \isa{v} (that is construct \isa{f\isactrlsub r\ v}). Before we can establish the correctness
- of \isa{lexer\isactrlsup {\isacharplus}{\kern0pt}}, we need to show that simplification preserves the language
- and simplification preserves our POSIX relation once the value is rectified
- (recall \isa{simp} generates a (regular expression, rectification function) pair):
-
- \begin{lemma}\mbox{}\smallskip\\\label{slexeraux}
- \begin{tabular}{ll}
- (1) & \isa{L{\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}simp\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}}\\
- (2) & \isa{{\normalsize{}If\,}\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}simp\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v\ {\normalsize \,then\,}\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ snd\ {\isacharparenleft}{\kern0pt}simp\ r{\isacharparenright}{\kern0pt}\ v{\isachardot}{\kern0pt}}
- \end{tabular}
- \end{lemma}
-
- \begin{proof} Both are by induction on \isa{r}. There is no
- interesting case for the first statement. For the second statement,
- of interest are the \isa{r\ {\isacharequal}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}} and \isa{r\ {\isacharequal}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}} cases. In each case we have to analyse four subcases whether
- \isa{fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}} and \isa{fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} equals \isa{\isactrlbold {\isadigit{0}}} (respectively \isa{\isactrlbold {\isadigit{1}}}). For example for \isa{r\ {\isacharequal}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}, consider the subcase \isa{fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isactrlbold {\isadigit{0}}} and
- \isa{fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ \isactrlbold {\isadigit{0}}}. By assumption we know \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}. From this we can infer \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v}
- and by IH also (*) \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ snd\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ v}. Given \isa{fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isactrlbold {\isadigit{0}}}
- we know \isa{L{\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymemptyset}}. By the first statement
- \isa{L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}} is the empty set, meaning (**) \isa{s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}}.
- Taking (*) and (**) together gives by the \mbox{\isa{P{\isacharplus}{\kern0pt}R}}-rule
- \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ Right\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}simp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}}. In turn this
- gives \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ snd\ {\isacharparenleft}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ 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}
- \isa{lexer\isactrlsup {\isacharplus}{\kern0pt}\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s}
- \end{theorem}
-
- \begin{proof} By induction on \isa{s} generalising over \isa{r}. The case \isa{{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}} is trivial. For the cons-case suppose the
- string is of the form \isa{c\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}s}. By induction hypothesis we
- know \isa{lexer\isactrlsup {\isacharplus}{\kern0pt}\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s} holds for all \isa{r} (in
- particular for \isa{r} being the derivative \isa{r{\isacharbackslash}{\kern0pt}c}). Let \isa{r\isactrlsub s} be the simplified derivative regular expression, that is \isa{fst\ {\isacharparenleft}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}}, and \isa{f\isactrlsub r} be the rectification
- function, that is \isa{snd\ {\isacharparenleft}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}}. We distinguish the cases
- whether (*) \isa{s\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}} or not. In the first case we
- have by Theorem~\ref{lexercorrect}(2) a value \isa{v} so that \isa{lexer\ {\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ Some\ v} and \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v} hold.
- By Lemma~\ref{slexeraux}(1) we can also infer from~(*) that \isa{s\ {\isasymin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub s{\isacharparenright}{\kern0pt}} holds. Hence we know by Theorem~\ref{lexercorrect}(2) that
- there exists a \isa{v{\isacharprime}{\kern0pt}} with \isa{lexer\ r\isactrlsub s\ s\ {\isacharequal}{\kern0pt}\ Some\ v{\isacharprime}{\kern0pt}} and
- \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r\isactrlsub s{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ v{\isacharprime}{\kern0pt}}. From the latter we know by
- Lemma~\ref{slexeraux}(2) that \isa{{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymrightarrow}\ f\isactrlsub r\ v{\isacharprime}{\kern0pt}} holds.
- By the uniqueness of the POSIX relation (Theorem~\ref{posixdeterm}) we
- can infer that \isa{v} is equal to \isa{f\isactrlsub r\ v{\isacharprime}{\kern0pt}}---that is the
- rectification function applied to \isa{v{\isacharprime}{\kern0pt}}
- produces the original \isa{v}. Now the case follows by the
- definitions of \isa{lexer} and \isa{lexer\isactrlsup {\isacharplus}{\kern0pt}}.
-
- In the second case where \isa{s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}} we have that
- \isa{lexer\ {\isacharparenleft}{\kern0pt}r{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ None} by Theorem~\ref{lexercorrect}(1). We
- also know by Lemma~\ref{slexeraux}(1) that \isa{s\ {\isasymnotin}\ L{\isacharparenleft}{\kern0pt}r\isactrlsub s{\isacharparenright}{\kern0pt}}. Hence
- \isa{lexer\ r\isactrlsub s\ s\ {\isacharequal}{\kern0pt}\ None} by Theorem~\ref{lexercorrect}(1) and
- by IH then also \isa{lexer\isactrlsup {\isacharplus}{\kern0pt}\ r\isactrlsub s\ s\ {\isacharequal}{\kern0pt}\ None}. With this we can
- conclude in this case too.\qed
-
- \end{proof}%
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-\begin{isamarkuptext}%
-\begin{lemma}
- \isa{{\normalsize{}If\,}\ v\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c\ {\normalsize \,then\,}\ retrieve\ {\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}c{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}inj\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}}
- \end{lemma}
-
- \begin{proof}
- By induction on the definition of \isa{r\mbox{$^\downarrow$}}. The cases for rule 1) and 2) are
- straightforward as \isa{\isactrlbold {\isadigit{0}}{\isacharbackslash}{\kern0pt}c} and \isa{\isactrlbold {\isadigit{1}}{\isacharbackslash}{\kern0pt}c} are both equal to
- \isa{\isactrlbold {\isadigit{0}}}. This means \isa{v\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{0}}} cannot hold. Similarly in case of rule 3)
- where \isa{r} is of the form \isa{ACHAR\ d} with \isa{c\ {\isacharequal}{\kern0pt}\ d}. Then by assumption
- we know \isa{v\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{1}}}, which implies \isa{v\ {\isacharequal}{\kern0pt}\ Empty}. The equation follows by
- simplification of left- and right-hand side. In case \isa{c\ {\isasymnoteq}\ d} we have again
- \isa{v\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{0}}}, which cannot hold.
-
- For rule 4a) we have again \isa{v\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{0}}}. The property holds by IH for rule 4b).
- The induction hypothesis is
- \[
- \isa{retrieve\ {\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}c{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}inj\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}}
- \]
- which is what left- and right-hand side simplify to. The slightly more interesting case
- is for 4c). By assumption we have
- \isa{v\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}}. This means we
- have either (*) \isa{v{\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c} with \isa{v\ {\isacharequal}{\kern0pt}\ Left\ v{\isadigit{1}}} or
- (**) \isa{v{\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c} with \isa{v\ {\isacharequal}{\kern0pt}\ Right\ v{\isadigit{2}}}.
- 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
- \isa{v\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymcdot}\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}\isactrlsup {\isasymstar}}. This means we also have
- \isa{v\ {\isacharequal}{\kern0pt}\ Seq\ v{\isadigit{1}}\ v{\isadigit{2}}}, \isa{v{\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c} and \isa{v{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ Stars\ vs}.
- We want to prove
- \begin{align}
- & \isa{retrieve\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}c{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v}\\
- &= \isa{retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}inj\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}\isactrlsup {\isasymstar}{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}}
- \end{align}
- The right-hand side \isa{inj}-expression is equal to
- \isa{Stars\ {\isacharparenleft}{\kern0pt}inj\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}\ c\ v{\isadigit{1}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}vs{\isacharparenright}{\kern0pt}}, which means the \isa{retrieve}-expression
- simplifies to
- \[
- \isa{bs\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}inj\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}\ c\ v{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}}
- \]
- The left-hand side (3) above simplifies to
- \[
- \isa{bs\ {\isacharat}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}c{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isadigit{1}}\ {\isacharat}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}}
- \]
- We can move out the \isa{fuse\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}} and then use the IH to show that left-hand side
- and right-hand side are equal. This completes the proof.
- \end{proof}
-
-
-
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- \bibliography{root}%
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\ No newline at end of file