218
+ − 1
(*<*)
+ − 2
theory Paper
+ − 3
imports
+ − 4
"../Lexer"
+ − 5
"../Simplifying"
280
+ − 6
"../Positions"
287
+ − 7
"../Sulzmann"
218
+ − 8
"~~/src/HOL/Library/LaTeXsugar"
+ − 9
begin
+ − 10
265
+ − 11
lemma Suc_0_fold:
+ − 12
"Suc 0 = 1"
+ − 13
by simp
+ − 14
+ − 15
+ − 16
218
+ − 17
declare [[show_question_marks = false]]
+ − 18
267
+ − 19
syntax (latex output)
274
+ − 20
"_Collect" :: "pttrn => bool => 'a set" ("(1{_ \<^latex>\<open>\\mbox{\\boldmath$\\mid$}\<close> _})")
267
+ − 21
"_CollectIn" :: "pttrn => 'a set => bool => 'a set" ("(1{_ \<in> _ |e _})")
+ − 22
273
+ − 23
syntax
+ − 24
"_Not_Ex" :: "idts \<Rightarrow> bool \<Rightarrow> bool" ("(3\<nexists>_.a./ _)" [0, 10] 10)
+ − 25
"_Not_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool" ("(3\<nexists>!_.a./ _)" [0, 10] 10)
+ − 26
267
+ − 27
218
+ − 28
abbreviation
265
+ − 29
"der_syn r c \<equiv> der c r"
218
+ − 30
+ − 31
abbreviation
265
+ − 32
"ders_syn r s \<equiv> ders s r"
+ − 33
289
+ − 34
abbreviation
+ − 35
"bder_syn r c \<equiv> bder c r"
+ − 36
+ − 37
abbreviation
+ − 38
"bders_syn r s \<equiv> bders r s"
+ − 39
265
+ − 40
+ − 41
abbreviation
+ − 42
"nprec v1 v2 \<equiv> \<not>(v1 :\<sqsubset>val v2)"
+ − 43
218
+ − 44
267
+ − 45
+ − 46
218
+ − 47
notation (latex output)
274
+ − 48
If ("(\<^latex>\<open>\\textrm{\<close>if\<^latex>\<open>}\<close> (_)/ \<^latex>\<open>\\textrm{\<close>then\<^latex>\<open>}\<close> (_)/ \<^latex>\<open>\\textrm{\<close>else\<^latex>\<open>}\<close> (_))" 10) and
+ − 49
Cons ("_\<^latex>\<open>\\mbox{$\\,$}\<close>::\<^latex>\<open>\\mbox{$\\,$}\<close>_" [75,73] 73) and
218
+ − 50
273
+ − 51
ZERO ("\<^bold>0" 81) and
+ − 52
ONE ("\<^bold>1" 81) and
218
+ − 53
CHAR ("_" [1000] 80) and
+ − 54
ALT ("_ + _" [77,77] 78) and
+ − 55
SEQ ("_ \<cdot> _" [77,77] 78) and
287
+ − 56
STAR ("_\<^sup>\<star>" [79] 78) and
218
+ − 57
265
+ − 58
val.Void ("Empty" 78) and
218
+ − 59
val.Char ("Char _" [1000] 78) and
+ − 60
val.Left ("Left _" [79] 78) and
+ − 61
val.Right ("Right _" [1000] 78) and
+ − 62
val.Seq ("Seq _ _" [79,79] 78) and
+ − 63
val.Stars ("Stars _" [79] 78) and
+ − 64
+ − 65
L ("L'(_')" [10] 78) and
272
+ − 66
LV ("LV _ _" [80,73] 78) and
218
+ − 67
der_syn ("_\\_" [79, 1000] 76) and
+ − 68
ders_syn ("_\\_" [79, 1000] 76) and
+ − 69
flat ("|_|" [75] 74) and
273
+ − 70
flats ("|_|" [72] 74) and
218
+ − 71
Sequ ("_ @ _" [78,77] 63) and
+ − 72
injval ("inj _ _ _" [79,77,79] 76) and
+ − 73
mkeps ("mkeps _" [79] 76) and
+ − 74
length ("len _" [73] 73) and
266
+ − 75
intlen ("len _" [73] 73) and
267
+ − 76
set ("_" [73] 73) and
218
+ − 77
267
+ − 78
Prf ("_ : _" [75,75] 75) and
218
+ − 79
Posix ("'(_, _') \<rightarrow> _" [63,75,75] 75) and
+ − 80
+ − 81
lexer ("lexer _ _" [78,78] 77) and
+ − 82
F_RIGHT ("F\<^bsub>Right\<^esub> _") and
+ − 83
F_LEFT ("F\<^bsub>Left\<^esub> _") and
+ − 84
F_ALT ("F\<^bsub>Alt\<^esub> _ _") and
+ − 85
F_SEQ1 ("F\<^bsub>Seq1\<^esub> _ _") and
+ − 86
F_SEQ2 ("F\<^bsub>Seq2\<^esub> _ _") and
+ − 87
F_SEQ ("F\<^bsub>Seq\<^esub> _ _") and
+ − 88
simp_SEQ ("simp\<^bsub>Seq\<^esub> _ _" [1000, 1000] 1) and
+ − 89
simp_ALT ("simp\<^bsub>Alt\<^esub> _ _" [1000, 1000] 1) and
+ − 90
slexer ("lexer\<^sup>+" 1000) and
+ − 91
274
+ − 92
at ("_\<^latex>\<open>\\mbox{$\\downharpoonleft$}\<close>\<^bsub>_\<^esub>") and
265
+ − 93
lex_list ("_ \<prec>\<^bsub>lex\<^esub> _") and
+ − 94
PosOrd ("_ \<prec>\<^bsub>_\<^esub> _" [77,77,77] 77) and
+ − 95
PosOrd_ex ("_ \<prec> _" [77,77] 77) and
274
+ − 96
PosOrd_ex_eq ("_ \<^latex>\<open>\\mbox{$\\preccurlyeq$}\<close> _" [77,77] 77) and
265
+ − 97
pflat_len ("\<parallel>_\<parallel>\<^bsub>_\<^esub>") and
274
+ − 98
nprec ("_ \<^latex>\<open>\\mbox{$\\not\\prec$}\<close> _" [77,77] 77) and
265
+ − 99
289
+ − 100
bder_syn ("_\<^latex>\<open>\\mbox{$\\bbslash$}\<close>_" [79, 1000] 76) and
+ − 101
bders_syn ("_\<^latex>\<open>\\mbox{$\\bbslash$}\<close>_" [79, 1000] 76) and
+ − 102
intern ("_\<^latex>\<open>\\mbox{$^\\uparrow$}\<close>" [900] 80) and
+ − 103
erase ("_\<^latex>\<open>\\mbox{$^\\downarrow$}\<close>" [1000] 74) and
+ − 104
bnullable ("nullable\<^latex>\<open>\\mbox{$_b$}\<close> _" [1000] 80) and
+ − 105
bmkeps ("mkeps\<^latex>\<open>\\mbox{$_b$}\<close> _" [1000] 80) and
+ − 106
blexer ("lexer\<^latex>\<open>\\mbox{$_b$}\<close> _ _" [77, 77] 80) and
+ − 107
code ("code _" [79] 74) and
+ − 108
274
+ − 109
DUMMY ("\<^latex>\<open>\\underline{\\hspace{2mm}}\<close>")
268
+ − 110
218
+ − 111
+ − 112
definition
+ − 113
"match r s \<equiv> nullable (ders s r)"
+ − 114
267
+ − 115
268
+ − 116
lemma LV_STAR_ONE_empty:
+ − 117
shows "LV (STAR ONE) [] = {Stars []}"
+ − 118
by(auto simp add: LV_def elim: Prf.cases intro: Prf.intros)
267
+ − 119
+ − 120
+ − 121
218
+ − 122
(*
+ − 123
comments not implemented
+ − 124
272
+ − 125
p9. The condition "not exists s3 s4..." appears often enough (in particular in
218
+ − 126
the proof of Lemma 3) to warrant a definition.
+ − 127
+ − 128
*)
+ − 129
273
+ − 130
218
+ − 131
(*>*)
+ − 132
267
+ − 133
+ − 134
330
+ − 135
section \<open>Introduction\<close>
218
+ − 136
+ − 137
330
+ − 138
text \<open>
218
+ − 139
+ − 140
Brzozowski \cite{Brzozowski1964} introduced the notion of the {\em
330
+ − 141
derivative} @{term "der c r"} of a regular expression \<open>r\<close> w.r.t.\
+ − 142
a character~\<open>c\<close>, and showed that it gave a simple solution to the
287
+ − 143
problem of matching a string @{term s} with a regular expression @{term
+ − 144
r}: if the derivative of @{term r} w.r.t.\ (in succession) all the
+ − 145
characters of the string matches the empty string, then @{term r}
+ − 146
matches @{term s} (and {\em vice versa}). The derivative has the
+ − 147
property (which may almost be regarded as its specification) that, for
+ − 148
every string @{term s} and regular expression @{term r} and character
+ − 149
@{term c}, one has @{term "cs \<in> L(r)"} if and only if \mbox{@{term "s \<in> L(der c r)"}}.
+ − 150
The beauty of Brzozowski's derivatives is that
+ − 151
they are neatly expressible in any functional language, and easily
+ − 152
definable and reasoned about in theorem provers---the definitions just
+ − 153
consist of inductive datatypes and simple recursive functions. A
218
+ − 154
mechanised correctness proof of Brzozowski's matcher in for example HOL4
287
+ − 155
has been mentioned by Owens and Slind~\cite{Owens2008}. Another one in
+ − 156
Isabelle/HOL is part of the work by Krauss and Nipkow \cite{Krauss2011}.
+ − 157
And another one in Coq is given by Coquand and Siles \cite{Coquand2012}.
218
+ − 158
287
+ − 159
If a regular expression matches a string, then in general there is more
+ − 160
than one way of how the string is matched. There are two commonly used
+ − 161
disambiguation strategies to generate a unique answer: one is called
+ − 162
GREEDY matching \cite{Frisch2004} and the other is POSIX
+ − 163
matching~\cite{POSIX,Kuklewicz,OkuiSuzuki2010,Sulzmann2014,Vansummeren2006}.
+ − 164
For example consider the string @{term xy} and the regular expression
+ − 165
\mbox{@{term "STAR (ALT (ALT x y) xy)"}}. Either the string can be
+ − 166
matched in two `iterations' by the single letter-regular expressions
+ − 167
@{term x} and @{term y}, or directly in one iteration by @{term xy}. The
+ − 168
first case corresponds to GREEDY matching, which first matches with the
+ − 169
left-most symbol and only matches the next symbol in case of a mismatch
+ − 170
(this is greedy in the sense of preferring instant gratification to
+ − 171
delayed repletion). The second case is POSIX matching, which prefers the
+ − 172
longest match.
218
+ − 173
268
+ − 174
In the context of lexing, where an input string needs to be split up
+ − 175
into a sequence of tokens, POSIX is the more natural disambiguation
+ − 176
strategy for what programmers consider basic syntactic building blocks
+ − 177
in their programs. These building blocks are often specified by some
330
+ − 178
regular expressions, say \<open>r\<^bsub>key\<^esub>\<close> and \<open>r\<^bsub>id\<^esub>\<close> for recognising keywords and identifiers,
268
+ − 179
respectively. There are a few underlying (informal) rules behind
287
+ − 180
tokenising a string in a POSIX \cite{POSIX} fashion:
218
+ − 181
+ − 182
\begin{itemize}
265
+ − 183
\item[$\bullet$] \emph{The Longest Match Rule} (or \emph{``{M}aximal {M}unch {R}ule''}):
218
+ − 184
The longest initial substring matched by any regular expression is taken as
+ − 185
next token.\smallskip
+ − 186
+ − 187
\item[$\bullet$] \emph{Priority Rule:}
265
+ − 188
For a particular longest initial substring, the first (leftmost) regular expression
+ − 189
that can match determines the token.\smallskip
+ − 190
+ − 191
\item[$\bullet$] \emph{Star Rule:} A subexpression repeated by ${}^\star$ shall
+ − 192
not match an empty string unless this is the only match for the repetition.\smallskip
+ − 193
+ − 194
\item[$\bullet$] \emph{Empty String Rule:} An empty string shall be considered to
268
+ − 195
be longer than no match at all.
218
+ − 196
\end{itemize}
+ − 197
330
+ − 198
\noindent Consider for example a regular expression \<open>r\<^bsub>key\<^esub>\<close> for recognising keywords such as \<open>if\<close>,
+ − 199
\<open>then\<close> and so on; and \<open>r\<^bsub>id\<^esub>\<close>
218
+ − 200
recognising identifiers (say, a single character followed by
+ − 201
characters or numbers). Then we can form the regular expression
330
+ − 202
\<open>(r\<^bsub>key\<^esub> + r\<^bsub>id\<^esub>)\<^sup>\<star>\<close>
+ − 203
and use POSIX matching to tokenise strings, say \<open>iffoo\<close> and
+ − 204
\<open>if\<close>. For \<open>iffoo\<close> we obtain by the Longest Match Rule
268
+ − 205
a single identifier token, not a keyword followed by an
330
+ − 206
identifier. For \<open>if\<close> we obtain by the Priority Rule a keyword
+ − 207
token, not an identifier token---even if \<open>r\<^bsub>id\<^esub>\<close>
+ − 208
matches also. By the Star Rule we know \<open>(r\<^bsub>key\<^esub> +
+ − 209
r\<^bsub>id\<^esub>)\<^sup>\<star>\<close> matches \<open>iffoo\<close>,
+ − 210
respectively \<open>if\<close>, in exactly one `iteration' of the star. The
273
+ − 211
Empty String Rule is for cases where, for example, the regular expression
330
+ − 212
\<open>(a\<^sup>\<star>)\<^sup>\<star>\<close> matches against the
+ − 213
string \<open>bc\<close>. Then the longest initial matched substring is the
268
+ − 214
empty string, which is matched by both the whole regular expression
272
+ − 215
and the parenthesised subexpression.
267
+ − 216
218
+ − 217
+ − 218
One limitation of Brzozowski's matcher is that it only generates a
+ − 219
YES/NO answer for whether a string is being matched by a regular
+ − 220
expression. Sulzmann and Lu~\cite{Sulzmann2014} extended this matcher
+ − 221
to allow generation not just of a YES/NO answer but of an actual
272
+ − 222
matching, called a [lexical] {\em value}. Assuming a regular
+ − 223
expression matches a string, values encode the information of
+ − 224
\emph{how} the string is matched by the regular expression---that is,
+ − 225
which part of the string is matched by which part of the regular
330
+ − 226
expression. For this consider again the string \<open>xy\<close> and
+ − 227
the regular expression \mbox{\<open>(x + (y + xy))\<^sup>\<star>\<close>}
273
+ − 228
(this time fully parenthesised). We can view this regular expression
330
+ − 229
as tree and if the string \<open>xy\<close> is matched by two Star
+ − 230
`iterations', then the \<open>x\<close> is matched by the left-most
+ − 231
alternative in this tree and the \<open>y\<close> by the right-left alternative. This
273
+ − 232
suggests to record this matching as
+ − 233
+ − 234
\begin{center}
+ − 235
@{term "Stars [Left(Char x), Right(Left(Char y))]"}
+ − 236
\end{center}
272
+ − 237
330
+ − 238
\noindent where @{const Stars}, \<open>Left\<close>, \<open>Right\<close> and \<open>Char\<close> are constructors for values. \<open>Stars\<close> records how many
+ − 239
iterations were used; \<open>Left\<close>, respectively \<open>Right\<close>, which
275
+ − 240
alternative is used. This `tree view' leads naturally to the idea that
+ − 241
regular expressions act as types and values as inhabiting those types
+ − 242
(see, for example, \cite{HosoyaVouillonPierce2005}). The value for
330
+ − 243
matching \<open>xy\<close> in a single `iteration', i.e.~the POSIX value,
287
+ − 244
would look as follows
272
+ − 245
273
+ − 246
\begin{center}
+ − 247
@{term "Stars [Seq (Char x) (Char y)]"}
+ − 248
\end{center}
+ − 249
+ − 250
\noindent where @{const Stars} has only a single-element list for the
+ − 251
single iteration and @{const Seq} indicates that @{term xy} is matched
287
+ − 252
by a sequence regular expression.
+ − 253
+ − 254
%, which we will in what follows
+ − 255
%write more formally as @{term "SEQ x y"}.
272
+ − 256
218
+ − 257
272
+ − 258
Sulzmann and Lu give a simple algorithm to calculate a value that
+ − 259
appears to be the value associated with POSIX matching. The challenge
+ − 260
then is to specify that value, in an algorithm-independent fashion,
+ − 261
and to show that Sulzmann and Lu's derivative-based algorithm does
+ − 262
indeed calculate a value that is correct according to the
+ − 263
specification. The answer given by Sulzmann and Lu
+ − 264
\cite{Sulzmann2014} is to define a relation (called an ``order
+ − 265
relation'') on the set of values of @{term r}, and to show that (once
+ − 266
a string to be matched is chosen) there is a maximum element and that
+ − 267
it is computed by their derivative-based algorithm. This proof idea is
+ − 268
inspired by work of Frisch and Cardelli \cite{Frisch2004} on a GREEDY
+ − 269
regular expression matching algorithm. However, we were not able to
+ − 270
establish transitivity and totality for the ``order relation'' by
+ − 271
Sulzmann and Lu. There are some inherent problems with their approach
+ − 272
(of which some of the proofs are not published in
+ − 273
\cite{Sulzmann2014}); perhaps more importantly, we give in this paper
+ − 274
a simple inductive (and algorithm-independent) definition of what we
+ − 275
call being a {\em POSIX value} for a regular expression @{term r} and
+ − 276
a string @{term s}; we show that the algorithm by Sulzmann and Lu
+ − 277
computes such a value and that such a value is unique. Our proofs are
+ − 278
both done by hand and checked in Isabelle/HOL. The experience of
+ − 279
doing our proofs has been that this mechanical checking was absolutely
+ − 280
essential: this subject area has hidden snares. This was also noted by
+ − 281
Kuklewicz \cite{Kuklewicz} who found that nearly all POSIX matching
+ − 282
implementations are ``buggy'' \cite[Page 203]{Sulzmann2014} and by
+ − 283
Grathwohl et al \cite[Page 36]{CrashCourse2014} who wrote:
218
+ − 284
+ − 285
\begin{quote}
+ − 286
\it{}``The POSIX strategy is more complicated than the greedy because of
+ − 287
the dependence on information about the length of matched strings in the
+ − 288
various subexpressions.''
+ − 289
\end{quote}
+ − 290
+ − 291
+ − 292
+ − 293
\noindent {\bf Contributions:} We have implemented in Isabelle/HOL the
+ − 294
derivative-based regular expression matching algorithm of
+ − 295
Sulzmann and Lu \cite{Sulzmann2014}. We have proved the correctness of this
+ − 296
algorithm according to our specification of what a POSIX value is (inspired
+ − 297
by work of Vansummeren \cite{Vansummeren2006}). Sulzmann
+ − 298
and Lu sketch in \cite{Sulzmann2014} an informal correctness proof: but to
+ − 299
us it contains unfillable gaps.\footnote{An extended version of
+ − 300
\cite{Sulzmann2014} is available at the website of its first author; this
+ − 301
extended version already includes remarks in the appendix that their
+ − 302
informal proof contains gaps, and possible fixes are not fully worked out.}
+ − 303
Our specification of a POSIX value consists of a simple inductive definition
+ − 304
that given a string and a regular expression uniquely determines this value.
267
+ − 305
We also show that our definition is equivalent to an ordering
268
+ − 306
of values based on positions by Okui and Suzuki \cite{OkuiSuzuki2010}.
287
+ − 307
+ − 308
%Derivatives as calculated by Brzozowski's method are usually more complex
+ − 309
%regular expressions than the initial one; various optimisations are
+ − 310
%possible. We prove the correctness when simplifications of @{term "ALT ZERO r"},
+ − 311
%@{term "ALT r ZERO"}, @{term "SEQ ONE r"} and @{term "SEQ r ONE"} to
+ − 312
%@{term r} are applied.
+ − 313
288
+ − 314
We extend our results to ??? Bitcoded version??
218
+ − 315
330
+ − 316
\<close>
218
+ − 317
330
+ − 318
section \<open>Preliminaries\<close>
218
+ − 319
330
+ − 320
text \<open>\noindent Strings in Isabelle/HOL are lists of characters with
273
+ − 321
the empty string being represented by the empty list, written @{term
+ − 322
"[]"}, and list-cons being written as @{term "DUMMY # DUMMY"}. Often
+ − 323
we use the usual bracket notation for lists also for strings; for
+ − 324
example a string consisting of just a single character @{term c} is
+ − 325
written @{term "[c]"}. We use the usual definitions for
+ − 326
\emph{prefixes} and \emph{strict prefixes} of strings. By using the
218
+ − 327
type @{type char} for characters we have a supply of finitely many
+ − 328
characters roughly corresponding to the ASCII character set. Regular
273
+ − 329
expressions are defined as usual as the elements of the following
+ − 330
inductive datatype:
218
+ − 331
+ − 332
\begin{center}
330
+ − 333
\<open>r :=\<close>
218
+ − 334
@{const "ZERO"} $\mid$
+ − 335
@{const "ONE"} $\mid$
+ − 336
@{term "CHAR c"} $\mid$
+ − 337
@{term "ALT r\<^sub>1 r\<^sub>2"} $\mid$
+ − 338
@{term "SEQ r\<^sub>1 r\<^sub>2"} $\mid$
+ − 339
@{term "STAR r"}
+ − 340
\end{center}
+ − 341
+ − 342
\noindent where @{const ZERO} stands for the regular expression that does
+ − 343
not match any string, @{const ONE} for the regular expression that matches
+ − 344
only the empty string and @{term c} for matching a character literal. The
+ − 345
language of a regular expression is also defined as usual by the
+ − 346
recursive function @{term L} with the six clauses:
+ − 347
+ − 348
\begin{center}
267
+ − 349
\begin{tabular}{l@ {\hspace{4mm}}rcl}
273
+ − 350
\textit{(1)} & @{thm (lhs) L.simps(1)} & $\dn$ & @{thm (rhs) L.simps(1)}\\
+ − 351
\textit{(2)} & @{thm (lhs) L.simps(2)} & $\dn$ & @{thm (rhs) L.simps(2)}\\
+ − 352
\textit{(3)} & @{thm (lhs) L.simps(3)} & $\dn$ & @{thm (rhs) L.simps(3)}\\
+ − 353
\textit{(4)} & @{thm (lhs) L.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ &
+ − 354
@{thm (rhs) L.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
+ − 355
\textit{(5)} & @{thm (lhs) L.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ &
+ − 356
@{thm (rhs) L.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
+ − 357
\textit{(6)} & @{thm (lhs) L.simps(6)} & $\dn$ & @{thm (rhs) L.simps(6)}\\
218
+ − 358
\end{tabular}
+ − 359
\end{center}
+ − 360
273
+ − 361
\noindent In clause \textit{(4)} we use the operation @{term "DUMMY ;;
218
+ − 362
DUMMY"} for the concatenation of two languages (it is also list-append for
+ − 363
strings). We use the star-notation for regular expressions and for
+ − 364
languages (in the last clause above). The star for languages is defined
330
+ − 365
inductively by two clauses: \<open>(i)\<close> the empty string being in
+ − 366
the star of a language and \<open>(ii)\<close> if @{term "s\<^sub>1"} is in a
218
+ − 367
language and @{term "s\<^sub>2"} in the star of this language, then also @{term
+ − 368
"s\<^sub>1 @ s\<^sub>2"} is in the star of this language. It will also be convenient
+ − 369
to use the following notion of a \emph{semantic derivative} (or \emph{left
+ − 370
quotient}) of a language defined as
+ − 371
%
+ − 372
\begin{center}
+ − 373
@{thm Der_def}\;.
+ − 374
\end{center}
+ − 375
+ − 376
\noindent
+ − 377
For semantic derivatives we have the following equations (for example
+ − 378
mechanically proved in \cite{Krauss2011}):
+ − 379
%
+ − 380
\begin{equation}\label{SemDer}
+ − 381
\begin{array}{lcl}
+ − 382
@{thm (lhs) Der_null} & \dn & @{thm (rhs) Der_null}\\
+ − 383
@{thm (lhs) Der_empty} & \dn & @{thm (rhs) Der_empty}\\
+ − 384
@{thm (lhs) Der_char} & \dn & @{thm (rhs) Der_char}\\
+ − 385
@{thm (lhs) Der_union} & \dn & @{thm (rhs) Der_union}\\
+ − 386
@{thm (lhs) Der_Sequ} & \dn & @{thm (rhs) Der_Sequ}\\
+ − 387
@{thm (lhs) Der_star} & \dn & @{thm (rhs) Der_star}
+ − 388
\end{array}
+ − 389
\end{equation}
+ − 390
+ − 391
+ − 392
\noindent \emph{\Brz's derivatives} of regular expressions
+ − 393
\cite{Brzozowski1964} can be easily defined by two recursive functions:
+ − 394
the first is from regular expressions to booleans (implementing a test
+ − 395
when a regular expression can match the empty string), and the second
+ − 396
takes a regular expression and a character to a (derivative) regular
+ − 397
expression:
+ − 398
+ − 399
\begin{center}
+ − 400
\begin{tabular}{lcl}
+ − 401
@{thm (lhs) nullable.simps(1)} & $\dn$ & @{thm (rhs) nullable.simps(1)}\\
+ − 402
@{thm (lhs) nullable.simps(2)} & $\dn$ & @{thm (rhs) nullable.simps(2)}\\
+ − 403
@{thm (lhs) nullable.simps(3)} & $\dn$ & @{thm (rhs) nullable.simps(3)}\\
+ − 404
@{thm (lhs) nullable.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) nullable.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
+ − 405
@{thm (lhs) nullable.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) nullable.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
273
+ − 406
@{thm (lhs) nullable.simps(6)} & $\dn$ & @{thm (rhs) nullable.simps(6)}\medskip\\
218
+ − 407
273
+ − 408
% \end{tabular}
+ − 409
% \end{center}
+ − 410
+ − 411
% \begin{center}
+ − 412
% \begin{tabular}{lcl}
+ − 413
218
+ − 414
@{thm (lhs) der.simps(1)} & $\dn$ & @{thm (rhs) der.simps(1)}\\
+ − 415
@{thm (lhs) der.simps(2)} & $\dn$ & @{thm (rhs) der.simps(2)}\\
+ − 416
@{thm (lhs) der.simps(3)} & $\dn$ & @{thm (rhs) der.simps(3)}\\
+ − 417
@{thm (lhs) der.simps(4)[of c "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) der.simps(4)[of c "r\<^sub>1" "r\<^sub>2"]}\\
+ − 418
@{thm (lhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]}\\
+ − 419
@{thm (lhs) der.simps(6)} & $\dn$ & @{thm (rhs) der.simps(6)}
+ − 420
\end{tabular}
+ − 421
\end{center}
+ − 422
+ − 423
\noindent
+ − 424
We may extend this definition to give derivatives w.r.t.~strings:
+ − 425
+ − 426
\begin{center}
+ − 427
\begin{tabular}{lcl}
+ − 428
@{thm (lhs) ders.simps(1)} & $\dn$ & @{thm (rhs) ders.simps(1)}\\
+ − 429
@{thm (lhs) ders.simps(2)} & $\dn$ & @{thm (rhs) ders.simps(2)}\\
+ − 430
\end{tabular}
+ − 431
\end{center}
+ − 432
+ − 433
\noindent Given the equations in \eqref{SemDer}, it is a relatively easy
+ − 434
exercise in mechanical reasoning to establish that
+ − 435
+ − 436
\begin{proposition}\label{derprop}\mbox{}\\
+ − 437
\begin{tabular}{ll}
273
+ − 438
\textit{(1)} & @{thm (lhs) nullable_correctness} if and only if
218
+ − 439
@{thm (rhs) nullable_correctness}, and \\
273
+ − 440
\textit{(2)} & @{thm[mode=IfThen] der_correctness}.
218
+ − 441
\end{tabular}
+ − 442
\end{proposition}
+ − 443
+ − 444
\noindent With this in place it is also very routine to prove that the
+ − 445
regular expression matcher defined as
+ − 446
%
+ − 447
\begin{center}
+ − 448
@{thm match_def}
+ − 449
\end{center}
+ − 450
+ − 451
\noindent gives a positive answer if and only if @{term "s \<in> L r"}.
+ − 452
Consequently, this regular expression matching algorithm satisfies the
+ − 453
usual specification for regular expression matching. While the matcher
+ − 454
above calculates a provably correct YES/NO answer for whether a regular
+ − 455
expression matches a string or not, the novel idea of Sulzmann and Lu
+ − 456
\cite{Sulzmann2014} is to append another phase to this algorithm in order
+ − 457
to calculate a [lexical] value. We will explain the details next.
+ − 458
330
+ − 459
\<close>
218
+ − 460
330
+ − 461
section \<open>POSIX Regular Expression Matching\label{posixsec}\<close>
218
+ − 462
330
+ − 463
text \<open>
218
+ − 464
268
+ − 465
There have been many previous works that use values for encoding
+ − 466
\emph{how} a regular expression matches a string.
+ − 467
The clever idea by Sulzmann and Lu \cite{Sulzmann2014} is to
+ − 468
define a function on values that mirrors (but inverts) the
218
+ − 469
construction of the derivative on regular expressions. \emph{Values}
+ − 470
are defined as the inductive datatype
+ − 471
+ − 472
\begin{center}
330
+ − 473
\<open>v :=\<close>
218
+ − 474
@{const "Void"} $\mid$
+ − 475
@{term "val.Char c"} $\mid$
+ − 476
@{term "Left v"} $\mid$
+ − 477
@{term "Right v"} $\mid$
+ − 478
@{term "Seq v\<^sub>1 v\<^sub>2"} $\mid$
+ − 479
@{term "Stars vs"}
+ − 480
\end{center}
+ − 481
+ − 482
\noindent where we use @{term vs} to stand for a list of
+ − 483
values. (This is similar to the approach taken by Frisch and
+ − 484
Cardelli for GREEDY matching \cite{Frisch2004}, and Sulzmann and Lu
+ − 485
for POSIX matching \cite{Sulzmann2014}). The string underlying a
+ − 486
value can be calculated by the @{const flat} function, written
+ − 487
@{term "flat DUMMY"} and defined as:
+ − 488
+ − 489
\begin{center}
+ − 490
\begin{tabular}[t]{lcl}
+ − 491
@{thm (lhs) flat.simps(1)} & $\dn$ & @{thm (rhs) flat.simps(1)}\\
+ − 492
@{thm (lhs) flat.simps(2)} & $\dn$ & @{thm (rhs) flat.simps(2)}\\
+ − 493
@{thm (lhs) flat.simps(3)} & $\dn$ & @{thm (rhs) flat.simps(3)}\\
+ − 494
@{thm (lhs) flat.simps(4)} & $\dn$ & @{thm (rhs) flat.simps(4)}
+ − 495
\end{tabular}\hspace{14mm}
+ − 496
\begin{tabular}[t]{lcl}
+ − 497
@{thm (lhs) flat.simps(5)[of "v\<^sub>1" "v\<^sub>2"]} & $\dn$ & @{thm (rhs) flat.simps(5)[of "v\<^sub>1" "v\<^sub>2"]}\\
+ − 498
@{thm (lhs) flat.simps(6)} & $\dn$ & @{thm (rhs) flat.simps(6)}\\
+ − 499
@{thm (lhs) flat.simps(7)} & $\dn$ & @{thm (rhs) flat.simps(7)}\\
+ − 500
\end{tabular}
+ − 501
\end{center}
+ − 502
273
+ − 503
\noindent We will sometimes refer to the underlying string of a
+ − 504
value as \emph{flattened value}. We will also overload our notation and
+ − 505
use @{term "flats vs"} for flattening a list of values and concatenating
+ − 506
the resulting strings.
+ − 507
+ − 508
Sulzmann and Lu define
+ − 509
inductively an \emph{inhabitation relation} that associates values to
+ − 510
regular expressions. We define this relation as
+ − 511
follows:\footnote{Note that the rule for @{term Stars} differs from
+ − 512
our earlier paper \cite{AusafDyckhoffUrban2016}. There we used the
+ − 513
original definition by Sulzmann and Lu which does not require that
+ − 514
the values @{term "v \<in> set vs"} flatten to a non-empty
+ − 515
string. The reason for introducing the more restricted version of
+ − 516
lexical values is convenience later on when reasoning about an
+ − 517
ordering relation for values.}
218
+ − 518
+ − 519
\begin{center}
280
+ − 520
\begin{tabular}{c@ {\hspace{12mm}}c}\label{prfintros}
218
+ − 521
\\[-8mm]
268
+ − 522
@{thm[mode=Axiom] Prf.intros(4)} &
218
+ − 523
@{thm[mode=Axiom] Prf.intros(5)[of "c"]}\\[4mm]
268
+ − 524
@{thm[mode=Rule] Prf.intros(2)[of "v\<^sub>1" "r\<^sub>1" "r\<^sub>2"]} &
218
+ − 525
@{thm[mode=Rule] Prf.intros(3)[of "v\<^sub>2" "r\<^sub>1" "r\<^sub>2"]}\\[4mm]
268
+ − 526
@{thm[mode=Rule] Prf.intros(1)[of "v\<^sub>1" "r\<^sub>1" "v\<^sub>2" "r\<^sub>2"]} &
266
+ − 527
@{thm[mode=Rule] Prf.intros(6)[of "vs"]}
218
+ − 528
\end{tabular}
+ − 529
\end{center}
+ − 530
268
+ − 531
\noindent where in the clause for @{const "Stars"} we use the
330
+ − 532
notation @{term "v \<in> set vs"} for indicating that \<open>v\<close> is a
+ − 533
member in the list \<open>vs\<close>. We require in this rule that every
268
+ − 534
value in @{term vs} flattens to a non-empty string. The idea is that
+ − 535
@{term "Stars"}-values satisfy the informal Star Rule (see Introduction)
+ − 536
where the $^\star$ does not match the empty string unless this is
+ − 537
the only match for the repetition. Note also that no values are
+ − 538
associated with the regular expression @{term ZERO}, and that the
+ − 539
only value associated with the regular expression @{term ONE} is
+ − 540
@{term Void}. It is routine to establish how values ``inhabiting''
+ − 541
a regular expression correspond to the language of a regular
+ − 542
expression, namely
218
+ − 543
269
+ − 544
\begin{proposition}\label{inhabs}
218
+ − 545
@{thm L_flat_Prf}
+ − 546
\end{proposition}
+ − 547
267
+ − 548
\noindent
330
+ − 549
Given a regular expression \<open>r\<close> and a string \<open>s\<close>, we define the
+ − 550
set of all \emph{Lexical Values} inhabited by \<open>r\<close> with the underlying string
+ − 551
being \<open>s\<close>:\footnote{Okui and Suzuki refer to our lexical values
268
+ − 552
as \emph{canonical values} in \cite{OkuiSuzuki2010}. The notion of \emph{non-problematic
273
+ − 553
values} by Cardelli and Frisch \cite{Frisch2004} is related, but not identical
268
+ − 554
to our lexical values.}
267
+ − 555
+ − 556
\begin{center}
+ − 557
@{thm LV_def}
+ − 558
\end{center}
+ − 559
268
+ − 560
\noindent The main property of @{term "LV r s"} is that it is alway finite.
+ − 561
+ − 562
\begin{proposition}
+ − 563
@{thm LV_finite}
+ − 564
\end{proposition}
267
+ − 565
268
+ − 566
\noindent This finiteness property does not hold in general if we
+ − 567
remove the side-condition about @{term "flat v \<noteq> []"} in the
+ − 568
@{term Stars}-rule above. For example using Sulzmann and Lu's
+ − 569
less restrictive definition, @{term "LV (STAR ONE) []"} would contain
+ − 570
infinitely many values, but according to our more restricted
273
+ − 571
definition only a single value, namely @{thm LV_STAR_ONE_empty}.
267
+ − 572
330
+ − 573
If a regular expression \<open>r\<close> matches a string \<open>s\<close>, then
268
+ − 574
generally the set @{term "LV r s"} is not just a singleton set. In
+ − 575
case of POSIX matching the problem is to calculate the unique lexical value
+ − 576
that satisfies the (informal) POSIX rules from the Introduction.
+ − 577
Graphically the POSIX value calculation algorithm by Sulzmann and Lu
+ − 578
can be illustrated by the picture in Figure~\ref{Sulz} where the
+ − 579
path from the left to the right involving @{term
+ − 580
derivatives}/@{const nullable} is the first phase of the algorithm
+ − 581
(calculating successive \Brz's derivatives) and @{const
330
+ − 582
mkeps}/\<open>inj\<close>, the path from right to left, the second
268
+ − 583
phase. This picture shows the steps required when a regular
330
+ − 584
expression, say \<open>r\<^sub>1\<close>, matches the string @{term
268
+ − 585
"[a,b,c]"}. We first build the three derivatives (according to
+ − 586
@{term a}, @{term b} and @{term c}). We then use @{const nullable}
+ − 587
to find out whether the resulting derivative regular expression
+ − 588
@{term "r\<^sub>4"} can match the empty string. If yes, we call the
+ − 589
function @{const mkeps} that produces a value @{term "v\<^sub>4"}
+ − 590
for how @{term "r\<^sub>4"} can match the empty string (taking into
+ − 591
account the POSIX constraints in case there are several ways). This
+ − 592
function is defined by the clauses:
218
+ − 593
+ − 594
\begin{figure}[t]
+ − 595
\begin{center}
+ − 596
\begin{tikzpicture}[scale=2,node distance=1.3cm,
+ − 597
every node/.style={minimum size=6mm}]
+ − 598
\node (r1) {@{term "r\<^sub>1"}};
+ − 599
\node (r2) [right=of r1]{@{term "r\<^sub>2"}};
+ − 600
\draw[->,line width=1mm](r1)--(r2) node[above,midway] {@{term "der a DUMMY"}};
+ − 601
\node (r3) [right=of r2]{@{term "r\<^sub>3"}};
+ − 602
\draw[->,line width=1mm](r2)--(r3) node[above,midway] {@{term "der b DUMMY"}};
+ − 603
\node (r4) [right=of r3]{@{term "r\<^sub>4"}};
+ − 604
\draw[->,line width=1mm](r3)--(r4) node[above,midway] {@{term "der c DUMMY"}};
+ − 605
\draw (r4) node[anchor=west] {\;\raisebox{3mm}{@{term nullable}}};
+ − 606
\node (v4) [below=of r4]{@{term "v\<^sub>4"}};
+ − 607
\draw[->,line width=1mm](r4) -- (v4);
+ − 608
\node (v3) [left=of v4] {@{term "v\<^sub>3"}};
330
+ − 609
\draw[->,line width=1mm](v4)--(v3) node[below,midway] {\<open>inj r\<^sub>3 c\<close>};
218
+ − 610
\node (v2) [left=of v3]{@{term "v\<^sub>2"}};
330
+ − 611
\draw[->,line width=1mm](v3)--(v2) node[below,midway] {\<open>inj r\<^sub>2 b\<close>};
218
+ − 612
\node (v1) [left=of v2] {@{term "v\<^sub>1"}};
330
+ − 613
\draw[->,line width=1mm](v2)--(v1) node[below,midway] {\<open>inj r\<^sub>1 a\<close>};
218
+ − 614
\draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{@{term "mkeps"}}};
+ − 615
\end{tikzpicture}
+ − 616
\end{center}
+ − 617
\mbox{}\\[-13mm]
+ − 618
+ − 619
\caption{The two phases of the algorithm by Sulzmann \& Lu \cite{Sulzmann2014},
+ − 620
matching the string @{term "[a,b,c]"}. The first phase (the arrows from
+ − 621
left to right) is \Brz's matcher building successive derivatives. If the
+ − 622
last regular expression is @{term nullable}, then the functions of the
+ − 623
second phase are called (the top-down and right-to-left arrows): first
+ − 624
@{term mkeps} calculates a value @{term "v\<^sub>4"} witnessing
+ − 625
how the empty string has been recognised by @{term "r\<^sub>4"}. After
+ − 626
that the function @{term inj} ``injects back'' the characters of the string into
+ − 627
the values.
+ − 628
\label{Sulz}}
+ − 629
\end{figure}
+ − 630
+ − 631
\begin{center}
+ − 632
\begin{tabular}{lcl}
+ − 633
@{thm (lhs) mkeps.simps(1)} & $\dn$ & @{thm (rhs) mkeps.simps(1)}\\
+ − 634
@{thm (lhs) mkeps.simps(2)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) mkeps.simps(2)[of "r\<^sub>1" "r\<^sub>2"]}\\
+ − 635
@{thm (lhs) mkeps.simps(3)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) mkeps.simps(3)[of "r\<^sub>1" "r\<^sub>2"]}\\
+ − 636
@{thm (lhs) mkeps.simps(4)} & $\dn$ & @{thm (rhs) mkeps.simps(4)}\\
+ − 637
\end{tabular}
+ − 638
\end{center}
+ − 639
+ − 640
\noindent Note that this function needs only to be partially defined,
+ − 641
namely only for regular expressions that are nullable. In case @{const
+ − 642
nullable} fails, the string @{term "[a,b,c]"} cannot be matched by @{term
+ − 643
"r\<^sub>1"} and the null value @{term "None"} is returned. Note also how this function
+ − 644
makes some subtle choices leading to a POSIX value: for example if an
+ − 645
alternative regular expression, say @{term "ALT r\<^sub>1 r\<^sub>2"}, can
+ − 646
match the empty string and furthermore @{term "r\<^sub>1"} can match the
330
+ − 647
empty string, then we return a \<open>Left\<close>-value. The \<open>Right\<close>-value will only be returned if @{term "r\<^sub>1"} cannot match the empty
218
+ − 648
string.
+ − 649
+ − 650
The most interesting idea from Sulzmann and Lu \cite{Sulzmann2014} is
+ − 651
the construction of a value for how @{term "r\<^sub>1"} can match the
+ − 652
string @{term "[a,b,c]"} from the value how the last derivative, @{term
+ − 653
"r\<^sub>4"} in Fig.~\ref{Sulz}, can match the empty string. Sulzmann and
+ − 654
Lu achieve this by stepwise ``injecting back'' the characters into the
+ − 655
values thus inverting the operation of building derivatives, but on the level
+ − 656
of values. The corresponding function, called @{term inj}, takes three
+ − 657
arguments, a regular expression, a character and a value. For example in
+ − 658
the first (or right-most) @{term inj}-step in Fig.~\ref{Sulz} the regular
+ − 659
expression @{term "r\<^sub>3"}, the character @{term c} from the last
+ − 660
derivative step and @{term "v\<^sub>4"}, which is the value corresponding
+ − 661
to the derivative regular expression @{term "r\<^sub>4"}. The result is
+ − 662
the new value @{term "v\<^sub>3"}. The final result of the algorithm is
+ − 663
the value @{term "v\<^sub>1"}. The @{term inj} function is defined by recursion on regular
+ − 664
expressions and by analysing the shape of values (corresponding to
+ − 665
the derivative regular expressions).
+ − 666
%
+ − 667
\begin{center}
+ − 668
\begin{tabular}{l@ {\hspace{5mm}}lcl}
273
+ − 669
\textit{(1)} & @{thm (lhs) injval.simps(1)} & $\dn$ & @{thm (rhs) injval.simps(1)}\\
+ − 670
\textit{(2)} & @{thm (lhs) injval.simps(2)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1"]} & $\dn$ &
218
+ − 671
@{thm (rhs) injval.simps(2)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1"]}\\
273
+ − 672
\textit{(3)} & @{thm (lhs) injval.simps(3)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]} & $\dn$ &
218
+ − 673
@{thm (rhs) injval.simps(3)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]}\\
273
+ − 674
\textit{(4)} & @{thm (lhs) injval.simps(4)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]} & $\dn$
218
+ − 675
& @{thm (rhs) injval.simps(4)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]}\\
273
+ − 676
\textit{(5)} & @{thm (lhs) injval.simps(5)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]} & $\dn$
218
+ − 677
& @{thm (rhs) injval.simps(5)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]}\\
273
+ − 678
\textit{(6)} & @{thm (lhs) injval.simps(6)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]} & $\dn$
218
+ − 679
& @{thm (rhs) injval.simps(6)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]}\\
273
+ − 680
\textit{(7)} & @{thm (lhs) injval.simps(7)[of "r" "c" "v" "vs"]} & $\dn$
218
+ − 681
& @{thm (rhs) injval.simps(7)[of "r" "c" "v" "vs"]}\\
+ − 682
\end{tabular}
+ − 683
\end{center}
+ − 684
+ − 685
\noindent To better understand what is going on in this definition it
+ − 686
might be instructive to look first at the three sequence cases (clauses
273
+ − 687
\textit{(4)} -- \textit{(6)}). In each case we need to construct an ``injected value'' for
218
+ − 688
@{term "SEQ r\<^sub>1 r\<^sub>2"}. This must be a value of the form @{term
330
+ − 689
"Seq DUMMY DUMMY"}\,. Recall the clause of the \<open>derivative\<close>-function
218
+ − 690
for sequence regular expressions:
+ − 691
+ − 692
\begin{center}
+ − 693
@{thm (lhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]} $\dn$ @{thm (rhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]}
+ − 694
\end{center}
+ − 695
330
+ − 696
\noindent Consider first the \<open>else\<close>-branch where the derivative is @{term
218
+ − 697
"SEQ (der c r\<^sub>1) r\<^sub>2"}. The corresponding value must therefore
+ − 698
be of the form @{term "Seq v\<^sub>1 v\<^sub>2"}, which matches the left-hand
330
+ − 699
side in clause~\textit{(4)} of @{term inj}. In the \<open>if\<close>-branch the derivative is an
218
+ − 700
alternative, namely @{term "ALT (SEQ (der c r\<^sub>1) r\<^sub>2) (der c
330
+ − 701
r\<^sub>2)"}. This means we either have to consider a \<open>Left\<close>- or
+ − 702
\<open>Right\<close>-value. In case of the \<open>Left\<close>-value we know further it
218
+ − 703
must be a value for a sequence regular expression. Therefore the pattern
273
+ − 704
we match in the clause \textit{(5)} is @{term "Left (Seq v\<^sub>1 v\<^sub>2)"},
+ − 705
while in \textit{(6)} it is just @{term "Right v\<^sub>2"}. One more interesting
+ − 706
point is in the right-hand side of clause \textit{(6)}: since in this case the
330
+ − 707
regular expression \<open>r\<^sub>1\<close> does not ``contribute'' to
218
+ − 708
matching the string, that means it only matches the empty string, we need to
+ − 709
call @{const mkeps} in order to construct a value for how @{term "r\<^sub>1"}
+ − 710
can match this empty string. A similar argument applies for why we can
273
+ − 711
expect in the left-hand side of clause \textit{(7)} that the value is of the form
218
+ − 712
@{term "Seq v (Stars vs)"}---the derivative of a star is @{term "SEQ (der c r)
+ − 713
(STAR r)"}. Finally, the reason for why we can ignore the second argument
273
+ − 714
in clause \textit{(1)} of @{term inj} is that it will only ever be called in cases
218
+ − 715
where @{term "c=d"}, but the usual linearity restrictions in patterns do
+ − 716
not allow us to build this constraint explicitly into our function
+ − 717
definition.\footnote{Sulzmann and Lu state this clause as @{thm (lhs)
+ − 718
injval.simps(1)[of "c" "c"]} $\dn$ @{thm (rhs) injval.simps(1)[of "c"]},
+ − 719
but our deviation is harmless.}
+ − 720
+ − 721
The idea of the @{term inj}-function to ``inject'' a character, say
+ − 722
@{term c}, into a value can be made precise by the first part of the
+ − 723
following lemma, which shows that the underlying string of an injected
289
+ − 724
value has a prepended character @{term c}; the second part shows that
+ − 725
the underlying string of an @{const mkeps}-value is always the empty
+ − 726
string (given the regular expression is nullable since otherwise
330
+ − 727
\<open>mkeps\<close> might not be defined).
218
+ − 728
+ − 729
\begin{lemma}\mbox{}\smallskip\\\label{Prf_injval_flat}
+ − 730
\begin{tabular}{ll}
+ − 731
(1) & @{thm[mode=IfThen] Prf_injval_flat}\\
+ − 732
(2) & @{thm[mode=IfThen] mkeps_flat}
+ − 733
\end{tabular}
+ − 734
\end{lemma}
+ − 735
+ − 736
\begin{proof}
+ − 737
Both properties are by routine inductions: the first one can, for example,
+ − 738
be proved by induction over the definition of @{term derivatives}; the second by
+ − 739
an induction on @{term r}. There are no interesting cases.\qed
+ − 740
\end{proof}
+ − 741
330
+ − 742
Having defined the @{const mkeps} and \<open>inj\<close> function we can extend
267
+ − 743
\Brz's matcher so that a value is constructed (assuming the
218
+ − 744
regular expression matches the string). The clauses of the Sulzmann and Lu lexer are
+ − 745
+ − 746
\begin{center}
+ − 747
\begin{tabular}{lcl}
+ − 748
@{thm (lhs) lexer.simps(1)} & $\dn$ & @{thm (rhs) lexer.simps(1)}\\
330
+ − 749
@{thm (lhs) lexer.simps(2)} & $\dn$ & \<open>case\<close> @{term "lexer (der c r) s"} \<open>of\<close>\\
+ − 750
& & \phantom{$|$} @{term "None"} \<open>\<Rightarrow>\<close> @{term None}\\
+ − 751
& & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> @{term "Some (injval r c v)"}
218
+ − 752
\end{tabular}
+ − 753
\end{center}
+ − 754
+ − 755
\noindent If the regular expression does not match the string, @{const None} is
+ − 756
returned. If the regular expression \emph{does}
+ − 757
match the string, then @{const Some} value is returned. One important
+ − 758
virtue of this algorithm is that it can be implemented with ease in any
+ − 759
functional programming language and also in Isabelle/HOL. In the remaining
+ − 760
part of this section we prove that this algorithm is correct.
+ − 761
267
+ − 762
The well-known idea of POSIX matching is informally defined by some
273
+ − 763
rules such as the Longest Match and Priority Rules (see
267
+ − 764
Introduction); as correctly argued in \cite{Sulzmann2014}, this
218
+ − 765
needs formal specification. Sulzmann and Lu define an ``ordering
267
+ − 766
relation'' between values and argue that there is a maximum value,
+ − 767
as given by the derivative-based algorithm. In contrast, we shall
+ − 768
introduce a simple inductive definition that specifies directly what
+ − 769
a \emph{POSIX value} is, incorporating the POSIX-specific choices
+ − 770
into the side-conditions of our rules. Our definition is inspired by
273
+ − 771
the matching relation given by Vansummeren~\cite{Vansummeren2006}.
+ − 772
The relation we define is ternary and
267
+ − 773
written as \mbox{@{term "s \<in> r \<rightarrow> v"}}, relating
+ − 774
strings, regular expressions and values; the inductive rules are given in
+ − 775
Figure~\ref{POSIXrules}.
+ − 776
We can prove that given a string @{term s} and regular expression @{term
+ − 777
r}, the POSIX value @{term v} is uniquely determined by @{term "s \<in> r \<rightarrow> v"}.
+ − 778
218
+ − 779
%
267
+ − 780
\begin{figure}[t]
218
+ − 781
\begin{center}
+ − 782
\begin{tabular}{c}
330
+ − 783
@{thm[mode=Axiom] Posix.intros(1)}\<open>P\<close>@{term "ONE"} \qquad
+ − 784
@{thm[mode=Axiom] Posix.intros(2)}\<open>P\<close>@{term "c"}\medskip\\
+ − 785
@{thm[mode=Rule] Posix.intros(3)[of "s" "r\<^sub>1" "v" "r\<^sub>2"]}\<open>P+L\<close>\qquad
+ − 786
@{thm[mode=Rule] Posix.intros(4)[of "s" "r\<^sub>2" "v" "r\<^sub>1"]}\<open>P+R\<close>\medskip\\
218
+ − 787
$\mprset{flushleft}
+ − 788
\inferrule
+ − 789
{@{thm (prem 1) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \qquad
+ − 790
@{thm (prem 2) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \\\\
+ − 791
@{thm (prem 3) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}
330
+ − 792
{@{thm (concl) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}$\<open>PS\<close>\\
+ − 793
@{thm[mode=Axiom] Posix.intros(7)}\<open>P[]\<close>\medskip\\
218
+ − 794
$\mprset{flushleft}
+ − 795
\inferrule
+ − 796
{@{thm (prem 1) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
+ − 797
@{thm (prem 2) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
+ − 798
@{thm (prem 3) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \\\\
+ − 799
@{thm (prem 4) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}
330
+ − 800
{@{thm (concl) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}$\<open>P\<star>\<close>
218
+ − 801
\end{tabular}
+ − 802
\end{center}
267
+ − 803
\caption{Our inductive definition of POSIX values.}\label{POSIXrules}
+ − 804
\end{figure}
218
+ − 805
267
+ − 806
218
+ − 807
+ − 808
\begin{theorem}\mbox{}\smallskip\\\label{posixdeterm}
+ − 809
\begin{tabular}{ll}
272
+ − 810
(1) & If @{thm (prem 1) Posix1(1)} then @{thm (concl)
218
+ − 811
Posix1(1)} and @{thm (concl) Posix1(2)}.\\
272
+ − 812
(2) & @{thm[mode=IfThen] Posix_determ(1)[of _ _ "v" "v'"]}
218
+ − 813
\end{tabular}
+ − 814
\end{theorem}
+ − 815
+ − 816
\begin{proof} Both by induction on the definition of @{term "s \<in> r \<rightarrow> v"}.
+ − 817
The second parts follows by a case analysis of @{term "s \<in> r \<rightarrow> v'"} and
+ − 818
the first part.\qed
+ − 819
\end{proof}
+ − 820
+ − 821
\noindent
267
+ − 822
We claim that our @{term "s \<in> r \<rightarrow> v"} relation captures the idea behind the four
218
+ − 823
informal POSIX rules shown in the Introduction: Consider for example the
330
+ − 824
rules \<open>P+L\<close> and \<open>P+R\<close> where the POSIX value for a string
218
+ − 825
and an alternative regular expression, that is @{term "(s, ALT r\<^sub>1 r\<^sub>2)"},
330
+ − 826
is specified---it is always a \<open>Left\<close>-value, \emph{except} when the
218
+ − 827
string to be matched is not in the language of @{term "r\<^sub>1"}; only then it
330
+ − 828
is a \<open>Right\<close>-value (see the side-condition in \<open>P+R\<close>).
+ − 829
Interesting is also the rule for sequence regular expressions (\<open>PS\<close>). The first two premises state that @{term "v\<^sub>1"} and @{term "v\<^sub>2"}
218
+ − 830
are the POSIX values for @{term "(s\<^sub>1, r\<^sub>1)"} and @{term "(s\<^sub>2, r\<^sub>2)"}
+ − 831
respectively. Consider now the third premise and note that the POSIX value
+ − 832
of this rule should match the string \mbox{@{term "s\<^sub>1 @ s\<^sub>2"}}. According to the
272
+ − 833
Longest Match Rule, we want that the @{term "s\<^sub>1"} is the longest initial
218
+ − 834
split of \mbox{@{term "s\<^sub>1 @ s\<^sub>2"}} such that @{term "s\<^sub>2"} is still recognised
+ − 835
by @{term "r\<^sub>2"}. Let us assume, contrary to the third premise, that there
+ − 836
\emph{exist} an @{term "s\<^sub>3"} and @{term "s\<^sub>4"} such that @{term "s\<^sub>2"}
+ − 837
can be split up into a non-empty string @{term "s\<^sub>3"} and a possibly empty
+ − 838
string @{term "s\<^sub>4"}. Moreover the longer string @{term "s\<^sub>1 @ s\<^sub>3"} can be
330
+ − 839
matched by \<open>r\<^sub>1\<close> and the shorter @{term "s\<^sub>4"} can still be
218
+ − 840
matched by @{term "r\<^sub>2"}. In this case @{term "s\<^sub>1"} would \emph{not} be the
+ − 841
longest initial split of \mbox{@{term "s\<^sub>1 @ s\<^sub>2"}} and therefore @{term "Seq v\<^sub>1
+ − 842
v\<^sub>2"} cannot be a POSIX value for @{term "(s\<^sub>1 @ s\<^sub>2, SEQ r\<^sub>1 r\<^sub>2)"}.
272
+ − 843
The main point is that our side-condition ensures the Longest
+ − 844
Match Rule is satisfied.
218
+ − 845
330
+ − 846
A similar condition is imposed on the POSIX value in the \<open>P\<star>\<close>-rule. Also there we want that @{term "s\<^sub>1"} is the longest initial
218
+ − 847
split of @{term "s\<^sub>1 @ s\<^sub>2"} and furthermore the corresponding value
+ − 848
@{term v} cannot be flattened to the empty string. In effect, we require
+ − 849
that in each ``iteration'' of the star, some non-empty substring needs to
+ − 850
be ``chipped'' away; only in case of the empty string we accept @{term
273
+ − 851
"Stars []"} as the POSIX value. Indeed we can show that our POSIX values
330
+ − 852
are lexical values which exclude those \<open>Stars\<close> that contain subvalues
267
+ − 853
that flatten to the empty string.
218
+ − 854
272
+ − 855
\begin{lemma}\label{LVposix}
268
+ − 856
@{thm [mode=IfThen] Posix_LV}
267
+ − 857
\end{lemma}
+ − 858
+ − 859
\begin{proof}
268
+ − 860
By routine induction on @{thm (prem 1) Posix_LV}.\qed
267
+ − 861
\end{proof}
+ − 862
+ − 863
\noindent
218
+ − 864
Next is the lemma that shows the function @{term "mkeps"} calculates
+ − 865
the POSIX value for the empty string and a nullable regular expression.
+ − 866
+ − 867
\begin{lemma}\label{lemmkeps}
+ − 868
@{thm[mode=IfThen] Posix_mkeps}
+ − 869
\end{lemma}
+ − 870
+ − 871
\begin{proof}
+ − 872
By routine induction on @{term r}.\qed
+ − 873
\end{proof}
+ − 874
+ − 875
\noindent
330
+ − 876
The central lemma for our POSIX relation is that the \<open>inj\<close>-function
218
+ − 877
preserves POSIX values.
+ − 878
+ − 879
\begin{lemma}\label{Posix2}
+ − 880
@{thm[mode=IfThen] Posix_injval}
+ − 881
\end{lemma}
+ − 882
+ − 883
\begin{proof}
330
+ − 884
By induction on \<open>r\<close>. We explain two cases.
218
+ − 885
+ − 886
\begin{itemize}
+ − 887
\item[$\bullet$] Case @{term "r = ALT r\<^sub>1 r\<^sub>2"}. There are
330
+ − 888
two subcases, namely \<open>(a)\<close> \mbox{@{term "v = Left v'"}} and @{term
+ − 889
"s \<in> der c r\<^sub>1 \<rightarrow> v'"}; and \<open>(b)\<close> @{term "v = Right v'"}, @{term
+ − 890
"s \<notin> L (der c r\<^sub>1)"} and @{term "s \<in> der c r\<^sub>2 \<rightarrow> v'"}. In \<open>(a)\<close> we
218
+ − 891
know @{term "s \<in> der c r\<^sub>1 \<rightarrow> v'"}, from which we can infer @{term "(c # s)
+ − 892
\<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v'"} by induction hypothesis and hence @{term "(c #
+ − 893
s) \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> injval (ALT r\<^sub>1 r\<^sub>2) c (Left v')"} as needed. Similarly
330
+ − 894
in subcase \<open>(b)\<close> where, however, in addition we have to use
272
+ − 895
Proposition~\ref{derprop}(2) in order to infer @{term "c # s \<notin> L r\<^sub>1"} from @{term
+ − 896
"s \<notin> L (der c r\<^sub>1)"}.\smallskip
218
+ − 897
+ − 898
\item[$\bullet$] Case @{term "r = SEQ r\<^sub>1 r\<^sub>2"}. There are three subcases:
+ − 899
+ − 900
\begin{quote}
+ − 901
\begin{description}
330
+ − 902
\item[\<open>(a)\<close>] @{term "v = Left (Seq v\<^sub>1 v\<^sub>2)"} and @{term "nullable r\<^sub>1"}
+ − 903
\item[\<open>(b)\<close>] @{term "v = Right v\<^sub>1"} and @{term "nullable r\<^sub>1"}
+ − 904
\item[\<open>(c)\<close>] @{term "v = Seq v\<^sub>1 v\<^sub>2"} and @{term "\<not> nullable r\<^sub>1"}
218
+ − 905
\end{description}
+ − 906
\end{quote}
+ − 907
330
+ − 908
\noindent For \<open>(a)\<close> we know @{term "s\<^sub>1 \<in> der c r\<^sub>1 \<rightarrow> v\<^sub>1"} and
218
+ − 909
@{term "s\<^sub>2 \<in> r\<^sub>2 \<rightarrow> v\<^sub>2"} as well as
+ − 910
%
+ − 911
\[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> s\<^sub>1 @ s\<^sub>3 \<in> L (der c r\<^sub>1) \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]
+ − 912
272
+ − 913
\noindent From the latter we can infer by Proposition~\ref{derprop}(2):
218
+ − 914
%
+ − 915
\[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> (c # s\<^sub>1) @ s\<^sub>3 \<in> L r\<^sub>1 \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]
+ − 916
330
+ − 917
\noindent We can use the induction hypothesis for \<open>r\<^sub>1\<close> to obtain
218
+ − 918
@{term "(c # s\<^sub>1) \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v\<^sub>1"}. Putting this all together allows us to infer
330
+ − 919
@{term "((c # s\<^sub>1) @ s\<^sub>2) \<in> SEQ r\<^sub>1 r\<^sub>2 \<rightarrow> Seq (injval r\<^sub>1 c v\<^sub>1) v\<^sub>2"}. The case \<open>(c)\<close>
218
+ − 920
is similar.
+ − 921
330
+ − 922
For \<open>(b)\<close> we know @{term "s \<in> der c r\<^sub>2 \<rightarrow> v\<^sub>1"} and
218
+ − 923
@{term "s\<^sub>1 @ s\<^sub>2 \<notin> L (SEQ (der c r\<^sub>1) r\<^sub>2)"}. From the former
+ − 924
we have @{term "(c # s) \<in> r\<^sub>2 \<rightarrow> (injval r\<^sub>2 c v\<^sub>1)"} by induction hypothesis
+ − 925
for @{term "r\<^sub>2"}. From the latter we can infer
+ − 926
%
+ − 927
\[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> s\<^sub>3 \<in> L r\<^sub>1 \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]
+ − 928
272
+ − 929
\noindent By Lemma~\ref{lemmkeps} we know @{term "[] \<in> r\<^sub>1 \<rightarrow> (mkeps r\<^sub>1)"}
218
+ − 930
holds. Putting this all together, we can conclude with @{term "(c #
+ − 931
s) \<in> SEQ r\<^sub>1 r\<^sub>2 \<rightarrow> Seq (mkeps r\<^sub>1) (injval r\<^sub>2 c v\<^sub>1)"}, as required.
+ − 932
+ − 933
Finally suppose @{term "r = STAR r\<^sub>1"}. This case is very similar to the
+ − 934
sequence case, except that we need to also ensure that @{term "flat (injval r\<^sub>1
+ − 935
c v\<^sub>1) \<noteq> []"}. This follows from @{term "(c # s\<^sub>1)
+ − 936
\<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v\<^sub>1"} (which in turn follows from @{term "s\<^sub>1 \<in> der c
+ − 937
r\<^sub>1 \<rightarrow> v\<^sub>1"} and the induction hypothesis).\qed
+ − 938
\end{itemize}
+ − 939
\end{proof}
+ − 940
+ − 941
\noindent
272
+ − 942
With Lemma~\ref{Posix2} in place, it is completely routine to establish
218
+ − 943
that the Sulzmann and Lu lexer satisfies our specification (returning
+ − 944
the null value @{term "None"} iff the string is not in the language of the regular expression,
+ − 945
and returning a unique POSIX value iff the string \emph{is} in the language):
+ − 946
+ − 947
\begin{theorem}\mbox{}\smallskip\\\label{lexercorrect}
+ − 948
\begin{tabular}{ll}
+ − 949
(1) & @{thm (lhs) lexer_correct_None} if and only if @{thm (rhs) lexer_correct_None}\\
+ − 950
(2) & @{thm (lhs) lexer_correct_Some} if and only if @{thm (rhs) lexer_correct_Some}\\
+ − 951
\end{tabular}
+ − 952
\end{theorem}
+ − 953
+ − 954
\begin{proof}
272
+ − 955
By induction on @{term s} using Lemma~\ref{lemmkeps} and \ref{Posix2}.\qed
218
+ − 956
\end{proof}
+ − 957
273
+ − 958
\noindent In \textit{(2)} we further know by Theorem~\ref{posixdeterm} that the
218
+ − 959
value returned by the lexer must be unique. A simple corollary
+ − 960
of our two theorems is:
+ − 961
+ − 962
\begin{corollary}\mbox{}\smallskip\\\label{lexercorrectcor}
+ − 963
\begin{tabular}{ll}
+ − 964
(1) & @{thm (lhs) lexer_correctness(2)} if and only if @{thm (rhs) lexer_correctness(2)}\\
+ − 965
(2) & @{thm (lhs) lexer_correctness(1)} if and only if @{thm (rhs) lexer_correctness(1)}\\
+ − 966
\end{tabular}
+ − 967
\end{corollary}
+ − 968
272
+ − 969
\noindent This concludes our correctness proof. Note that we have
+ − 970
not changed the algorithm of Sulzmann and Lu,\footnote{All
+ − 971
deviations we introduced are harmless.} but introduced our own
+ − 972
specification for what a correct result---a POSIX value---should be.
+ − 973
In the next section we show that our specification coincides with
+ − 974
another one given by Okui and Suzuki using a different technique.
218
+ − 975
330
+ − 976
\<close>
218
+ − 977
330
+ − 978
section \<open>Ordering of Values according to Okui and Suzuki\<close>
268
+ − 979
330
+ − 980
text \<open>
268
+ − 981
+ − 982
While in the previous section we have defined POSIX values directly
+ − 983
in terms of a ternary relation (see inference rules in Figure~\ref{POSIXrules}),
+ − 984
Sulzmann and Lu took a different approach in \cite{Sulzmann2014}:
+ − 985
they introduced an ordering for values and identified POSIX values
+ − 986
as the maximal elements. An extended version of \cite{Sulzmann2014}
+ − 987
is available at the website of its first author; this includes more
+ − 988
details of their proofs, but which are evidently not in final form
+ − 989
yet. Unfortunately, we were not able to verify claims that their
+ − 990
ordering has properties such as being transitive or having maximal
273
+ − 991
elements.
268
+ − 992
+ − 993
Okui and Suzuki \cite{OkuiSuzuki2010,OkuiSuzukiTech} described
269
+ − 994
another ordering of values, which they use to establish the
+ − 995
correctness of their automata-based algorithm for POSIX matching.
+ − 996
Their ordering resembles some aspects of the one given by Sulzmann
273
+ − 997
and Lu, but overall is quite different. To begin with, Okui and
+ − 998
Suzuki identify POSIX values as minimal, rather than maximal,
+ − 999
elements in their ordering. A more substantial difference is that
+ − 1000
the ordering by Okui and Suzuki uses \emph{positions} in order to
+ − 1001
identify and compare subvalues. Positions are lists of natural
+ − 1002
numbers. This allows them to quite naturally formalise the Longest
+ − 1003
Match and Priority rules of the informal POSIX standard. Consider
+ − 1004
for example the value @{term v}
269
+ − 1005
+ − 1006
\begin{center}
+ − 1007
@{term "v == Stars [Seq (Char x) (Char y), Char z]"}
+ − 1008
\end{center}
+ − 1009
+ − 1010
\noindent
330
+ − 1011
At position \<open>[0,1]\<close> of this value is the
+ − 1012
subvalue \<open>Char y\<close> and at position \<open>[1]\<close> the
269
+ − 1013
subvalue @{term "Char z"}. At the `root' position, or empty list
330
+ − 1014
@{term "[]"}, is the whole value @{term v}. Positions such as \<open>[0,1,0]\<close> or \<open>[2]\<close> are outside of \<open>v\<close>. If it exists, the subvalue of @{term v} at a position \<open>p\<close>, written @{term "at v p"}, can be recursively defined by
268
+ − 1015
+ − 1016
\begin{center}
+ − 1017
\begin{tabular}{r@ {\hspace{0mm}}lcl}
330
+ − 1018
@{term v} & \<open>\<downharpoonleft>\<^bsub>[]\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(1)}\\
+ − 1019
@{term "Left v"} & \<open>\<downharpoonleft>\<^bsub>0::ps\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(2)}\\
+ − 1020
@{term "Right v"} & \<open>\<downharpoonleft>\<^bsub>1::ps\<^esub>\<close> & \<open>\<equiv>\<close> &
268
+ − 1021
@{thm (rhs) at.simps(3)[simplified Suc_0_fold]}\\
330
+ − 1022
@{term "Seq v\<^sub>1 v\<^sub>2"} & \<open>\<downharpoonleft>\<^bsub>0::ps\<^esub>\<close> & \<open>\<equiv>\<close> &
268
+ − 1023
@{thm (rhs) at.simps(4)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \\
330
+ − 1024
@{term "Seq v\<^sub>1 v\<^sub>2"} & \<open>\<downharpoonleft>\<^bsub>1::ps\<^esub>\<close>
+ − 1025
& \<open>\<equiv>\<close> &
268
+ − 1026
@{thm (rhs) at.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2", simplified Suc_0_fold]} \\
330
+ − 1027
@{term "Stars vs"} & \<open>\<downharpoonleft>\<^bsub>n::ps\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(6)}\\
268
+ − 1028
\end{tabular}
+ − 1029
\end{center}
+ − 1030
269
+ − 1031
\noindent In the last clause we use Isabelle's notation @{term "vs ! n"} for the
330
+ − 1032
\<open>n\<close>th element in a list. The set of positions inside a value \<open>v\<close>,
273
+ − 1033
written @{term "Pos v"}, is given by
268
+ − 1034
+ − 1035
\begin{center}
+ − 1036
\begin{tabular}{lcl}
330
+ − 1037
@{thm (lhs) Pos.simps(1)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(1)}\\
+ − 1038
@{thm (lhs) Pos.simps(2)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(2)}\\
+ − 1039
@{thm (lhs) Pos.simps(3)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(3)}\\
+ − 1040
@{thm (lhs) Pos.simps(4)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(4)}\\
268
+ − 1041
@{thm (lhs) Pos.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
330
+ − 1042
& \<open>\<equiv>\<close>
268
+ − 1043
& @{thm (rhs) Pos.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
330
+ − 1044
@{thm (lhs) Pos_stars} & \<open>\<equiv>\<close> & @{thm (rhs) Pos_stars}\\
268
+ − 1045
\end{tabular}
+ − 1046
\end{center}
+ − 1047
+ − 1048
\noindent
330
+ − 1049
whereby \<open>len\<close> in the last clause stands for the length of a list. Clearly
268
+ − 1050
for every position inside a value there exists a subvalue at that position.
+ − 1051
+ − 1052
+ − 1053
To help understanding the ordering of Okui and Suzuki, consider again
+ − 1054
the earlier value
330
+ − 1055
\<open>v\<close> and compare it with the following \<open>w\<close>:
268
+ − 1056
+ − 1057
\begin{center}
+ − 1058
\begin{tabular}{l}
+ − 1059
@{term "v == Stars [Seq (Char x) (Char y), Char z]"}\\
+ − 1060
@{term "w == Stars [Char x, Char y, Char z]"}
+ − 1061
\end{tabular}
+ − 1062
\end{center}
+ − 1063
330
+ − 1064
\noindent Both values match the string \<open>xyz\<close>, that means if
273
+ − 1065
we flatten these values at their respective root position, we obtain
330
+ − 1066
\<open>xyz\<close>. However, at position \<open>[0]\<close>, \<open>v\<close> matches
+ − 1067
\<open>xy\<close> whereas \<open>w\<close> matches only the shorter \<open>x\<close>. So
+ − 1068
according to the Longest Match Rule, we should prefer \<open>v\<close>,
+ − 1069
rather than \<open>w\<close> as POSIX value for string \<open>xyz\<close> (and
268
+ − 1070
corresponding regular expression). In order to
+ − 1071
formalise this idea, Okui and Suzuki introduce a measure for
330
+ − 1072
subvalues at position \<open>p\<close>, called the \emph{norm} of \<open>v\<close>
+ − 1073
at position \<open>p\<close>. We can define this measure in Isabelle as an
268
+ − 1074
integer as follows
+ − 1075
+ − 1076
\begin{center}
+ − 1077
@{thm pflat_len_def}
+ − 1078
\end{center}
+ − 1079
+ − 1080
\noindent where we take the length of the flattened value at
330
+ − 1081
position \<open>p\<close>, provided the position is inside \<open>v\<close>; if
+ − 1082
not, then the norm is \<open>-1\<close>. The default for outside
272
+ − 1083
positions is crucial for the POSIX requirement of preferring a
330
+ − 1084
\<open>Left\<close>-value over a \<open>Right\<close>-value (if they can match the
272
+ − 1085
same string---see the Priority Rule from the Introduction). For this
+ − 1086
consider
268
+ − 1087
+ − 1088
\begin{center}
+ − 1089
@{term "v == Left (Char x)"} \qquad and \qquad @{term "w == Right (Char x)"}
+ − 1090
\end{center}
+ − 1091
330
+ − 1092
\noindent Both values match \<open>x\<close>. At position \<open>[0]\<close>
+ − 1093
the norm of @{term v} is \<open>1\<close> (the subvalue matches \<open>x\<close>),
+ − 1094
but the norm of \<open>w\<close> is \<open>-1\<close> (the position is outside
+ − 1095
\<open>w\<close> according to how we defined the `inside' positions of
+ − 1096
\<open>Left\<close>- and \<open>Right\<close>-values). Of course at position
+ − 1097
\<open>[1]\<close>, the norms @{term "pflat_len v [1]"} and @{term
272
+ − 1098
"pflat_len w [1]"} are reversed, but the point is that subvalues
+ − 1099
will be analysed according to lexicographically ordered
330
+ − 1100
positions. According to this ordering, the position \<open>[0]\<close>
+ − 1101
takes precedence over \<open>[1]\<close> and thus also \<open>v\<close> will be
+ − 1102
preferred over \<open>w\<close>. The lexicographic ordering of positions, written
272
+ − 1103
@{term "DUMMY \<sqsubset>lex DUMMY"}, can be conveniently formalised
+ − 1104
by three inference rules
268
+ − 1105
+ − 1106
\begin{center}
+ − 1107
\begin{tabular}{ccc}
+ − 1108
@{thm [mode=Axiom] lex_list.intros(1)}\hspace{1cm} &
+ − 1109
@{thm [mode=Rule] lex_list.intros(3)[where ?p1.0="p\<^sub>1" and ?p2.0="p\<^sub>2" and
+ − 1110
?ps1.0="ps\<^sub>1" and ?ps2.0="ps\<^sub>2"]}\hspace{1cm} &
+ − 1111
@{thm [mode=Rule] lex_list.intros(2)[where ?ps1.0="ps\<^sub>1" and ?ps2.0="ps\<^sub>2"]}
+ − 1112
\end{tabular}
+ − 1113
\end{center}
+ − 1114
272
+ − 1115
With the norm and lexicographic order in place,
268
+ − 1116
we can state the key definition of Okui and Suzuki
330
+ − 1117
\cite{OkuiSuzuki2010}: a value @{term "v\<^sub>1"} is \emph{smaller at position \<open>p\<close>} than
273
+ − 1118
@{term "v\<^sub>2"}, written @{term "v\<^sub>1 \<sqsubset>val p v\<^sub>2"},
330
+ − 1119
if and only if $(i)$ the norm at position \<open>p\<close> is
268
+ − 1120
greater in @{term "v\<^sub>1"} (that is the string @{term "flat (at v\<^sub>1 p)"} is longer
+ − 1121
than @{term "flat (at v\<^sub>2 p)"}) and $(ii)$ all subvalues at
+ − 1122
positions that are inside @{term "v\<^sub>1"} or @{term "v\<^sub>2"} and that are
330
+ − 1123
lexicographically smaller than \<open>p\<close>, we have the same norm, namely
268
+ − 1124
+ − 1125
\begin{center}
+ − 1126
\begin{tabular}{c}
+ − 1127
@{thm (lhs) PosOrd_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
330
+ − 1128
\<open>\<equiv>\<close>
268
+ − 1129
$\begin{cases}
+ − 1130
(i) & @{term "pflat_len v\<^sub>1 p > pflat_len v\<^sub>2 p"} \quad\text{and}\smallskip \\
+ − 1131
(ii) & @{term "(\<forall>q \<in> Pos v\<^sub>1 \<union> Pos v\<^sub>2. q \<sqsubset>lex p --> pflat_len v\<^sub>1 q = pflat_len v\<^sub>2 q)"}
+ − 1132
\end{cases}$
+ − 1133
\end{tabular}
+ − 1134
\end{center}
+ − 1135
330
+ − 1136
\noindent The position \<open>p\<close> in this definition acts as the
+ − 1137
\emph{first distinct position} of \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close>, where both values match strings of different length
+ − 1138
\cite{OkuiSuzuki2010}. Since at \<open>p\<close> the values \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> match different strings, the
268
+ − 1139
ordering is irreflexive. Derived from the definition above
+ − 1140
are the following two orderings:
+ − 1141
+ − 1142
\begin{center}
+ − 1143
\begin{tabular}{l}
+ − 1144
@{thm PosOrd_ex_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+ − 1145
@{thm PosOrd_ex_eq_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
+ − 1146
\end{tabular}
+ − 1147
\end{center}
+ − 1148
272
+ − 1149
While we encountered a number of obstacles for establishing properties like
268
+ − 1150
transitivity for the ordering of Sulzmann and Lu (and which we failed
+ − 1151
to overcome), it is relatively straightforward to establish this
273
+ − 1152
property for the orderings
+ − 1153
@{term "DUMMY :\<sqsubset>val DUMMY"} and @{term "DUMMY :\<sqsubseteq>val DUMMY"}
+ − 1154
by Okui and Suzuki.
268
+ − 1155
+ − 1156
\begin{lemma}[Transitivity]\label{transitivity}
+ − 1157
@{thm [mode=IfThen] PosOrd_trans[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and ?v3.0="v\<^sub>3"]}
+ − 1158
\end{lemma}
+ − 1159
330
+ − 1160
\begin{proof} From the assumption we obtain two positions \<open>p\<close>
+ − 1161
and \<open>q\<close>, where the values \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> (respectively \<open>v\<^sub>2\<close> and \<open>v\<^sub>3\<close>) are `distinct'. Since \<open>\<prec>\<^bsub>lex\<^esub>\<close> is trichotomous, we need to consider
268
+ − 1162
three cases, namely @{term "p = q"}, @{term "p \<sqsubset>lex q"} and
273
+ − 1163
@{term "q \<sqsubset>lex p"}. Let us look at the first case. Clearly
+ − 1164
@{term "pflat_len v\<^sub>2 p < pflat_len v\<^sub>1 p"} and @{term
+ − 1165
"pflat_len v\<^sub>3 p < pflat_len v\<^sub>2 p"} imply @{term
+ − 1166
"pflat_len v\<^sub>3 p < pflat_len v\<^sub>1 p"}. It remains to show
+ − 1167
that for a @{term "p' \<in> Pos v\<^sub>1 \<union> Pos v\<^sub>3"}
+ − 1168
with @{term "p' \<sqsubset>lex p"} that @{term "pflat_len v\<^sub>1
+ − 1169
p' = pflat_len v\<^sub>3 p'"} holds. Suppose @{term "p' \<in> Pos
+ − 1170
v\<^sub>1"}, then we can infer from the first assumption that @{term
+ − 1171
"pflat_len v\<^sub>1 p' = pflat_len v\<^sub>2 p'"}. But this means
+ − 1172
that @{term "p'"} must be in @{term "Pos v\<^sub>2"} too (the norm
330
+ − 1173
cannot be \<open>-1\<close> given @{term "p' \<in> Pos v\<^sub>1"}).
273
+ − 1174
Hence we can use the second assumption and
+ − 1175
infer @{term "pflat_len v\<^sub>2 p' = pflat_len v\<^sub>3 p'"},
+ − 1176
which concludes this case with @{term "v\<^sub>1 :\<sqsubset>val
+ − 1177
v\<^sub>3"}. The reasoning in the other cases is similar.\qed
268
+ − 1178
\end{proof}
+ − 1179
273
+ − 1180
\noindent
+ − 1181
The proof for $\preccurlyeq$ is similar and omitted.
330
+ − 1182
It is also straightforward to show that \<open>\<prec>\<close> and
273
+ − 1183
$\preccurlyeq$ are partial orders. Okui and Suzuki furthermore show that they
+ − 1184
are linear orderings for lexical values \cite{OkuiSuzuki2010} of a given
+ − 1185
regular expression and given string, but we have not formalised this in Isabelle. It is
272
+ − 1186
not essential for our results. What we are going to show below is
330
+ − 1187
that for a given \<open>r\<close> and \<open>s\<close>, the orderings have a unique
269
+ − 1188
minimal element on the set @{term "LV r s"}, which is the POSIX value
273
+ − 1189
we defined in the previous section. We start with two properties that
330
+ − 1190
show how the length of a flattened value relates to the \<open>\<prec>\<close>-ordering.
273
+ − 1191
+ − 1192
\begin{proposition}\mbox{}\smallskip\\\label{ordlen}
+ − 1193
\begin{tabular}{@ {}ll}
+ − 1194
(1) &
+ − 1195
@{thm [mode=IfThen] PosOrd_shorterE[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+ − 1196
(2) &
+ − 1197
@{thm [mode=IfThen] PosOrd_shorterI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
+ − 1198
\end{tabular}
+ − 1199
\end{proposition}
+ − 1200
+ − 1201
\noindent Both properties follow from the definition of the ordering. Note that
+ − 1202
\textit{(2)} entails that a value, say @{term "v\<^sub>2"}, whose underlying
+ − 1203
string is a strict prefix of another flattened value, say @{term "v\<^sub>1"}, then
+ − 1204
@{term "v\<^sub>1"} must be smaller than @{term "v\<^sub>2"}. For our proofs it
+ − 1205
will be useful to have the following properties---in each case the underlying strings
+ − 1206
of the compared values are the same:
268
+ − 1207
273
+ − 1208
\begin{proposition}\mbox{}\smallskip\\\label{ordintros}
+ − 1209
\begin{tabular}{ll}
+ − 1210
\textit{(1)} &
+ − 1211
@{thm [mode=IfThen] PosOrd_Left_Right[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+ − 1212
\textit{(2)} & If
+ − 1213
@{thm (prem 1) PosOrd_Left_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;then\;
+ − 1214
@{thm (lhs) PosOrd_Left_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;iff\;
+ − 1215
@{thm (rhs) PosOrd_Left_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+ − 1216
\textit{(3)} & If
+ − 1217
@{thm (prem 1) PosOrd_Right_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;then\;
+ − 1218
@{thm (lhs) PosOrd_Right_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;iff\;
+ − 1219
@{thm (rhs) PosOrd_Right_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
+ − 1220
\textit{(4)} & If
+ − 1221
@{thm (prem 1) PosOrd_Seq_eq[where ?v2.0="v\<^sub>2" and ?w2.0="w\<^sub>2"]} \;then\;
+ − 1222
@{thm (lhs) PosOrd_Seq_eq[where ?v2.0="v\<^sub>2" and ?w2.0="w\<^sub>2"]} \;iff\;
+ − 1223
@{thm (rhs) PosOrd_Seq_eq[where ?v2.0="v\<^sub>2" and ?w2.0="w\<^sub>2"]}\\
+ − 1224
\textit{(5)} & If
+ − 1225
@{thm (prem 2) PosOrd_SeqI1[simplified, where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+ − 1226
?w1.0="w\<^sub>1" and ?w2.0="w\<^sub>2"]} \;and\;
+ − 1227
@{thm (prem 1) PosOrd_SeqI1[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+ − 1228
?w1.0="w\<^sub>1" and ?w2.0="w\<^sub>2"]} \;then\;
+ − 1229
@{thm (concl) PosOrd_SeqI1[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+ − 1230
?w1.0="w\<^sub>1" and ?w2.0="w\<^sub>2"]}\\
+ − 1231
\textit{(6)} & If
+ − 1232
@{thm (prem 1) PosOrd_Stars_append_eq[where ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;then\;
+ − 1233
@{thm (lhs) PosOrd_Stars_append_eq[where ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;iff\;
+ − 1234
@{thm (rhs) PosOrd_Stars_append_eq[where ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]}\\
+ − 1235
+ − 1236
\textit{(7)} & If
+ − 1237
@{thm (prem 2) PosOrd_StarsI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+ − 1238
?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;and\;
+ − 1239
@{thm (prem 1) PosOrd_StarsI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+ − 1240
?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;then\;
+ − 1241
@{thm (concl) PosOrd_StarsI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
+ − 1242
?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]}\\
+ − 1243
\end{tabular}
+ − 1244
\end{proposition}
268
+ − 1245
273
+ − 1246
\noindent One might prefer that statements \textit{(4)} and \textit{(5)}
+ − 1247
(respectively \textit{(6)} and \textit{(7)})
330
+ − 1248
are combined into a single \textit{iff}-statement (like the ones for \<open>Left\<close> and \<open>Right\<close>). Unfortunately this cannot be done easily: such
273
+ − 1249
a single statement would require an additional assumption about the
+ − 1250
two values @{term "Seq v\<^sub>1 v\<^sub>2"} and @{term "Seq w\<^sub>1 w\<^sub>2"}
+ − 1251
being inhabited by the same regular expression. The
+ − 1252
complexity of the proofs involved seems to not justify such a
+ − 1253
`cleaner' single statement. The statements given are just the properties that
275
+ − 1254
allow us to establish our theorems without any difficulty. The proofs
+ − 1255
for Proposition~\ref{ordintros} are routine.
268
+ − 1256
+ − 1257
273
+ − 1258
Next we establish how Okui and Suzuki's orderings relate to our
330
+ − 1259
definition of POSIX values. Given a \<open>POSIX\<close> value \<open>v\<^sub>1\<close>
+ − 1260
for \<open>r\<close> and \<open>s\<close>, then any other lexical value \<open>v\<^sub>2\<close> in @{term "LV r s"} is greater or equal than \<open>v\<^sub>1\<close>, namely:
268
+ − 1261
+ − 1262
272
+ − 1263
\begin{theorem}\label{orderone}
268
+ − 1264
@{thm [mode=IfThen] Posix_PosOrd[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
+ − 1265
\end{theorem}
+ − 1266
270
+ − 1267
\begin{proof} By induction on our POSIX rules. By
272
+ − 1268
Theorem~\ref{posixdeterm} and the definition of @{const LV}, it is clear
330
+ − 1269
that \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> have the same
270
+ − 1270
underlying string @{term s}. The three base cases are
+ − 1271
straightforward: for example for @{term "v\<^sub>1 = Void"}, we have
+ − 1272
that @{term "v\<^sub>2 \<in> LV ONE []"} must also be of the form
+ − 1273
\mbox{@{term "v\<^sub>2 = Void"}}. Therefore we have @{term
+ − 1274
"v\<^sub>1 :\<sqsubseteq>val v\<^sub>2"}. The inductive cases for
330
+ − 1275
\<open>r\<close> being of the form @{term "ALT r\<^sub>1 r\<^sub>2"} and
272
+ − 1276
@{term "SEQ r\<^sub>1 r\<^sub>2"} are as follows:
269
+ − 1277
270
+ − 1278
+ − 1279
\begin{itemize}
268
+ − 1280
330
+ − 1281
\item[$\bullet$] Case \<open>P+L\<close> with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
273
+ − 1282
\<rightarrow> (Left w\<^sub>1)"}: In this case the value
+ − 1283
@{term "v\<^sub>2"} is either of the
270
+ − 1284
form @{term "Left w\<^sub>2"} or @{term "Right w\<^sub>2"}. In the
273
+ − 1285
latter case we can immediately conclude with \mbox{@{term "v\<^sub>1
330
+ − 1286
:\<sqsubseteq>val v\<^sub>2"}} since a \<open>Left\<close>-value with the
+ − 1287
same underlying string \<open>s\<close> is always smaller than a
+ − 1288
\<open>Right\<close>-value by Proposition~\ref{ordintros}\textit{(1)}.
273
+ − 1289
In the former case we have @{term "w\<^sub>2
270
+ − 1290
\<in> LV r\<^sub>1 s"} and can use the induction hypothesis to infer
+ − 1291
@{term "w\<^sub>1 :\<sqsubseteq>val w\<^sub>2"}. Because @{term
+ − 1292
"w\<^sub>1"} and @{term "w\<^sub>2"} have the same underlying string
330
+ − 1293
\<open>s\<close>, we can conclude with @{term "Left w\<^sub>1
273
+ − 1294
:\<sqsubseteq>val Left w\<^sub>2"} using
+ − 1295
Proposition~\ref{ordintros}\textit{(2)}.\smallskip
268
+ − 1296
330
+ − 1297
\item[$\bullet$] Case \<open>P+R\<close> with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
272
+ − 1298
\<rightarrow> (Right w\<^sub>1)"}: This case similar to the previous
+ − 1299
case, except that we additionally know @{term "s \<notin> L
270
+ − 1300
r\<^sub>1"}. This is needed when @{term "v\<^sub>2"} is of the form
273
+ − 1301
\mbox{@{term "Left w\<^sub>2"}}. Since \mbox{@{term "flat v\<^sub>2 = flat
330
+ − 1302
w\<^sub>2"} \<open>= s\<close>} and @{term "\<Turnstile> w\<^sub>2 :
272
+ − 1303
r\<^sub>1"}, we can derive a contradiction for \mbox{@{term "s \<notin> L
+ − 1304
r\<^sub>1"}} using
+ − 1305
Proposition~\ref{inhabs}. So also in this case \mbox{@{term "v\<^sub>1
270
+ − 1306
:\<sqsubseteq>val v\<^sub>2"}}.\smallskip
268
+ − 1307
330
+ − 1308
\item[$\bullet$] Case \<open>PS\<close> with @{term "(s\<^sub>1 @
273
+ − 1309
s\<^sub>2) \<in> (SEQ r\<^sub>1 r\<^sub>2) \<rightarrow> (Seq
+ − 1310
w\<^sub>1 w\<^sub>2)"}: We can assume @{term "v\<^sub>2 = Seq
+ − 1311
(u\<^sub>1) (u\<^sub>2)"} with @{term "\<Turnstile> u\<^sub>1 :
+ − 1312
r\<^sub>1"} and \mbox{@{term "\<Turnstile> u\<^sub>2 :
+ − 1313
r\<^sub>2"}}. We have @{term "s\<^sub>1 @ s\<^sub>2 = (flat
+ − 1314
u\<^sub>1) @ (flat u\<^sub>2)"}. By the side-condition of the
330
+ − 1315
\<open>PS\<close>-rule we know that either @{term "s\<^sub>1 = flat
273
+ − 1316
u\<^sub>1"} or that @{term "flat u\<^sub>1"} is a strict prefix of
+ − 1317
@{term "s\<^sub>1"}. In the latter case we can infer @{term
+ − 1318
"w\<^sub>1 :\<sqsubset>val u\<^sub>1"} by
+ − 1319
Proposition~\ref{ordlen}\textit{(2)} and from this @{term "v\<^sub>1
+ − 1320
:\<sqsubseteq>val v\<^sub>2"} by Proposition~\ref{ordintros}\textit{(5)}
+ − 1321
(as noted above @{term "v\<^sub>1"} and @{term "v\<^sub>2"} must have the
+ − 1322
same underlying string).
+ − 1323
In the former case we know
+ − 1324
@{term "u\<^sub>1 \<in> LV r\<^sub>1 s\<^sub>1"} and @{term
+ − 1325
"u\<^sub>2 \<in> LV r\<^sub>2 s\<^sub>2"}. With this we can use the
+ − 1326
induction hypotheses to infer @{term "w\<^sub>1 :\<sqsubseteq>val
+ − 1327
u\<^sub>1"} and @{term "w\<^sub>2 :\<sqsubseteq>val u\<^sub>2"}. By
+ − 1328
Proposition~\ref{ordintros}\textit{(4,5)} we can again infer
+ − 1329
@{term "v\<^sub>1 :\<sqsubseteq>val
+ − 1330
v\<^sub>2"}.
270
+ − 1331
268
+ − 1332
\end{itemize}
270
+ − 1333
330
+ − 1334
\noindent The case for \<open>P\<star>\<close> is similar to the \<open>PS\<close>-case and omitted.\qed
268
+ − 1335
\end{proof}
+ − 1336
330
+ − 1337
\noindent This theorem shows that our \<open>POSIX\<close> value for a
+ − 1338
regular expression \<open>r\<close> and string @{term s} is in fact a
+ − 1339
minimal element of the values in \<open>LV r s\<close>. By
273
+ − 1340
Proposition~\ref{ordlen}\textit{(2)} we also know that any value in
330
+ − 1341
\<open>LV r s'\<close>, with @{term "s'"} being a strict prefix, cannot be
+ − 1342
smaller than \<open>v\<^sub>1\<close>. The next theorem shows the
273
+ − 1343
opposite---namely any minimal element in @{term "LV r s"} must be a
330
+ − 1344
\<open>POSIX\<close> value. This can be established by induction on \<open>r\<close>, but the proof can be drastically simplified by using the fact
+ − 1345
from the previous section about the existence of a \<open>POSIX\<close> value
273
+ − 1346
whenever a string @{term "s \<in> L r"}.
+ − 1347
268
+ − 1348
+ − 1349
\begin{theorem}
272
+ − 1350
@{thm [mode=IfThen] PosOrd_Posix[where ?v1.0="v\<^sub>1"]}
268
+ − 1351
\end{theorem}
+ − 1352
272
+ − 1353
\begin{proof}
+ − 1354
If @{thm (prem 1) PosOrd_Posix[where ?v1.0="v\<^sub>1"]} then
+ − 1355
@{term "s \<in> L r"} by Proposition~\ref{inhabs}. Hence by Theorem~\ref{lexercorrect}(2)
+ − 1356
there exists a
330
+ − 1357
\<open>POSIX\<close> value @{term "v\<^sub>P"} with @{term "s \<in> r \<rightarrow> v\<^sub>P"}
273
+ − 1358
and by Lemma~\ref{LVposix} we also have \mbox{@{term "v\<^sub>P \<in> LV r s"}}.
272
+ − 1359
By Theorem~\ref{orderone} we therefore have
+ − 1360
@{term "v\<^sub>P :\<sqsubseteq>val v\<^sub>1"}. If @{term "v\<^sub>P = v\<^sub>1"} then
273
+ − 1361
we are done. Otherwise we have @{term "v\<^sub>P :\<sqsubset>val v\<^sub>1"}, which
+ − 1362
however contradicts the second assumption about @{term "v\<^sub>1"} being the smallest
+ − 1363
element in @{term "LV r s"}. So we are done in this case too.\qed
272
+ − 1364
\end{proof}
270
+ − 1365
273
+ − 1366
\noindent
+ − 1367
From this we can also show
+ − 1368
that if @{term "LV r s"} is non-empty (or equivalently @{term "s \<in> L r"}) then
+ − 1369
it has a unique minimal element:
+ − 1370
272
+ − 1371
\begin{corollary}
+ − 1372
@{thm [mode=IfThen] Least_existence1}
+ − 1373
\end{corollary}
270
+ − 1374
+ − 1375
+ − 1376
273
+ − 1377
\noindent To sum up, we have shown that the (unique) minimal elements
330
+ − 1378
of the ordering by Okui and Suzuki are exactly the \<open>POSIX\<close>
273
+ − 1379
values we defined inductively in Section~\ref{posixsec}. This provides
330
+ − 1380
an independent confirmation that our ternary relation formalises the
273
+ − 1381
informal POSIX rules.
+ − 1382
330
+ − 1383
\<close>
268
+ − 1384
330
+ − 1385
section \<open>Bitcoded Lexing\<close>
289
+ − 1386
+ − 1387
330
+ − 1388
text \<open>
289
+ − 1389
+ − 1390
Incremental calculation of the value. To simplify the proof we first define the function
+ − 1391
@{const flex} which calculates the ``iterated'' injection function. With this we can
+ − 1392
rewrite the lexer as
+ − 1393
+ − 1394
\begin{center}
+ − 1395
@{thm lexer_flex}
+ − 1396
\end{center}
+ − 1397
+ − 1398
\begin{center}
+ − 1399
\begin{tabular}{lcl}
+ − 1400
@{thm (lhs) code.simps(1)} & $\dn$ & @{thm (rhs) code.simps(1)}\\
+ − 1401
@{thm (lhs) code.simps(2)} & $\dn$ & @{thm (rhs) code.simps(2)}\\
+ − 1402
@{thm (lhs) code.simps(3)} & $\dn$ & @{thm (rhs) code.simps(3)}\\
+ − 1403
@{thm (lhs) code.simps(4)} & $\dn$ & @{thm (rhs) code.simps(4)}\\
+ − 1404
@{thm (lhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]} & $\dn$ & @{thm (rhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]}\\
+ − 1405
@{thm (lhs) code.simps(6)} & $\dn$ & @{thm (rhs) code.simps(6)}\\
+ − 1406
@{thm (lhs) code.simps(7)} & $\dn$ & @{thm (rhs) code.simps(7)}
+ − 1407
\end{tabular}
+ − 1408
\end{center}
+ − 1409
+ − 1410
\begin{center}
+ − 1411
\begin{tabular}{lcl}
+ − 1412
@{term areg} & $::=$ & @{term "AZERO"}\\
+ − 1413
& $\mid$ & @{term "AONE bs"}\\
+ − 1414
& $\mid$ & @{term "ACHAR bs c"}\\
+ − 1415
& $\mid$ & @{term "AALT bs r\<^sub>1 r\<^sub>2"}\\
+ − 1416
& $\mid$ & @{term "ASEQ bs r\<^sub>1 r\<^sub>2"}\\
+ − 1417
& $\mid$ & @{term "ASTAR bs r"}
+ − 1418
\end{tabular}
+ − 1419
\end{center}
+ − 1420
+ − 1421
\begin{center}
+ − 1422
\begin{tabular}{lcl}
+ − 1423
@{thm (lhs) intern.simps(1)} & $\dn$ & @{thm (rhs) intern.simps(1)}\\
+ − 1424
@{thm (lhs) intern.simps(2)} & $\dn$ & @{thm (rhs) intern.simps(2)}\\
+ − 1425
@{thm (lhs) intern.simps(3)} & $\dn$ & @{thm (rhs) intern.simps(3)}\\
+ − 1426
@{thm (lhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1427
@{thm (lhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1428
@{thm (lhs) intern.simps(6)} & $\dn$ & @{thm (rhs) intern.simps(6)}\\
+ − 1429
\end{tabular}
+ − 1430
\end{center}
+ − 1431
+ − 1432
\begin{center}
+ − 1433
\begin{tabular}{lcl}
+ − 1434
@{thm (lhs) erase.simps(1)} & $\dn$ & @{thm (rhs) erase.simps(1)}\\
+ − 1435
@{thm (lhs) erase.simps(2)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(2)[of bs]}\\
+ − 1436
@{thm (lhs) erase.simps(3)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(3)[of bs]}\\
+ − 1437
@{thm (lhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1438
@{thm (lhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1439
@{thm (lhs) erase.simps(6)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(6)[of bs]}\\
+ − 1440
\end{tabular}
+ − 1441
\end{center}
+ − 1442
+ − 1443
Some simple facts about erase
+ − 1444
+ − 1445
\begin{lemma}\mbox{}\\
+ − 1446
@{thm erase_bder}\\
+ − 1447
@{thm erase_intern}
+ − 1448
\end{lemma}
+ − 1449
+ − 1450
\begin{center}
+ − 1451
\begin{tabular}{lcl}
+ − 1452
@{thm (lhs) bnullable.simps(1)} & $\dn$ & @{thm (rhs) bnullable.simps(1)}\\
+ − 1453
@{thm (lhs) bnullable.simps(2)} & $\dn$ & @{thm (rhs) bnullable.simps(2)}\\
+ − 1454
@{thm (lhs) bnullable.simps(3)} & $\dn$ & @{thm (rhs) bnullable.simps(3)}\\
+ − 1455
@{thm (lhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1456
@{thm (lhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1457
@{thm (lhs) bnullable.simps(6)} & $\dn$ & @{thm (rhs) bnullable.simps(6)}\medskip\\
+ − 1458
+ − 1459
% \end{tabular}
+ − 1460
% \end{center}
+ − 1461
+ − 1462
% \begin{center}
+ − 1463
% \begin{tabular}{lcl}
+ − 1464
+ − 1465
@{thm (lhs) bder.simps(1)} & $\dn$ & @{thm (rhs) bder.simps(1)}\\
+ − 1466
@{thm (lhs) bder.simps(2)} & $\dn$ & @{thm (rhs) bder.simps(2)}\\
+ − 1467
@{thm (lhs) bder.simps(3)} & $\dn$ & @{thm (rhs) bder.simps(3)}\\
+ − 1468
@{thm (lhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1469
@{thm (lhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1470
@{thm (lhs) bder.simps(6)} & $\dn$ & @{thm (rhs) bder.simps(6)}
+ − 1471
\end{tabular}
+ − 1472
\end{center}
+ − 1473
+ − 1474
+ − 1475
\begin{center}
+ − 1476
\begin{tabular}{lcl}
+ − 1477
@{thm (lhs) bmkeps.simps(1)} & $\dn$ & @{thm (rhs) bmkeps.simps(1)}\\
+ − 1478
@{thm (lhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1479
@{thm (lhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1480
@{thm (lhs) bmkeps.simps(4)} & $\dn$ & @{thm (rhs) bmkeps.simps(4)}\medskip\\
+ − 1481
\end{tabular}
+ − 1482
\end{center}
+ − 1483
+ − 1484
+ − 1485
@{thm [mode=IfThen] bder_retrieve}
+ − 1486
330
+ − 1487
By induction on \<open>r\<close>
289
+ − 1488
+ − 1489
\begin{theorem}[Main Lemma]\mbox{}\\
+ − 1490
@{thm [mode=IfThen] MAIN_decode}
+ − 1491
\end{theorem}
+ − 1492
+ − 1493
\noindent
+ − 1494
Definition of the bitcoded lexer
+ − 1495
+ − 1496
@{thm blexer_def}
+ − 1497
+ − 1498
+ − 1499
\begin{theorem}
+ − 1500
@{thm blexer_correctness}
+ − 1501
\end{theorem}
+ − 1502
330
+ − 1503
\<close>
289
+ − 1504
330
+ − 1505
section \<open>Optimisations\<close>
218
+ − 1506
330
+ − 1507
text \<open>
218
+ − 1508
+ − 1509
Derivatives as calculated by \Brz's method are usually more complex
+ − 1510
regular expressions than the initial one; the result is that the
+ − 1511
derivative-based matching and lexing algorithms are often abysmally slow.
+ − 1512
However, various optimisations are possible, such as the simplifications
+ − 1513
of @{term "ALT ZERO r"}, @{term "ALT r ZERO"}, @{term "SEQ ONE r"} and
+ − 1514
@{term "SEQ r ONE"} to @{term r}. These simplifications can speed up the
+ − 1515
algorithms considerably, as noted in \cite{Sulzmann2014}. One of the
+ − 1516
advantages of having a simple specification and correctness proof is that
+ − 1517
the latter can be refined to prove the correctness of such simplification
+ − 1518
steps. While the simplification of regular expressions according to
+ − 1519
rules like
+ − 1520
+ − 1521
\begin{equation}\label{Simpl}
+ − 1522
\begin{array}{lcllcllcllcl}
330
+ − 1523
@{term "ALT ZERO r"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+ − 1524
@{term "ALT r ZERO"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+ − 1525
@{term "SEQ ONE r"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+ − 1526
@{term "SEQ r ONE"} & \<open>\<Rightarrow>\<close> & @{term r}
218
+ − 1527
\end{array}
+ − 1528
\end{equation}
+ − 1529
+ − 1530
\noindent is well understood, there is an obstacle with the POSIX value
+ − 1531
calculation algorithm by Sulzmann and Lu: if we build a derivative regular
+ − 1532
expression and then simplify it, we will calculate a POSIX value for this
+ − 1533
simplified derivative regular expression, \emph{not} for the original (unsimplified)
+ − 1534
derivative regular expression. Sulzmann and Lu \cite{Sulzmann2014} overcome this obstacle by
+ − 1535
not just calculating a simplified regular expression, but also calculating
+ − 1536
a \emph{rectification function} that ``repairs'' the incorrect value.
+ − 1537
+ − 1538
The rectification functions can be (slightly clumsily) implemented in
+ − 1539
Isabelle/HOL as follows using some auxiliary functions:
+ − 1540
+ − 1541
\begin{center}
+ − 1542
\begin{tabular}{lcl}
330
+ − 1543
@{thm (lhs) F_RIGHT.simps(1)} & $\dn$ & \<open>Right (f v)\<close>\\
+ − 1544
@{thm (lhs) F_LEFT.simps(1)} & $\dn$ & \<open>Left (f v)\<close>\\
218
+ − 1545
330
+ − 1546
@{thm (lhs) F_ALT.simps(1)} & $\dn$ & \<open>Right (f\<^sub>2 v)\<close>\\
+ − 1547
@{thm (lhs) F_ALT.simps(2)} & $\dn$ & \<open>Left (f\<^sub>1 v)\<close>\\
218
+ − 1548
330
+ − 1549
@{thm (lhs) F_SEQ1.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 ()) (f\<^sub>2 v)\<close>\\
+ − 1550
@{thm (lhs) F_SEQ2.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v) (f\<^sub>2 ())\<close>\\
+ − 1551
@{thm (lhs) F_SEQ.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)\<close>\medskip\\
218
+ − 1552
%\end{tabular}
+ − 1553
%
+ − 1554
%\begin{tabular}{lcl}
+ − 1555
@{term "simp_ALT (ZERO, DUMMY) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_RIGHT f\<^sub>2)"}\\
+ − 1556
@{term "simp_ALT (r\<^sub>1, f\<^sub>1) (ZERO, DUMMY)"} & $\dn$ & @{term "(r\<^sub>1, F_LEFT f\<^sub>1)"}\\
+ − 1557
@{term "simp_ALT (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(ALT r\<^sub>1 r\<^sub>2, F_ALT f\<^sub>1 f\<^sub>2)"}\\
+ − 1558
@{term "simp_SEQ (ONE, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_SEQ1 f\<^sub>1 f\<^sub>2)"}\\
+ − 1559
@{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (ONE, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>1, F_SEQ2 f\<^sub>1 f\<^sub>2)"}\\
+ − 1560
@{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(SEQ r\<^sub>1 r\<^sub>2, F_SEQ f\<^sub>1 f\<^sub>2)"}\\
+ − 1561
\end{tabular}
+ − 1562
\end{center}
+ − 1563
+ − 1564
\noindent
330
+ − 1565
The functions \<open>simp\<^bsub>Alt\<^esub>\<close> and \<open>simp\<^bsub>Seq\<^esub>\<close> encode the simplification rules
218
+ − 1566
in \eqref{Simpl} and compose the rectification functions (simplifications can occur
+ − 1567
deep inside the regular expression). The main simplification function is then
+ − 1568
+ − 1569
\begin{center}
+ − 1570
\begin{tabular}{lcl}
+ − 1571
@{term "simp (ALT r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_ALT (simp r\<^sub>1) (simp r\<^sub>2)"}\\
+ − 1572
@{term "simp (SEQ r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_SEQ (simp r\<^sub>1) (simp r\<^sub>2)"}\\
+ − 1573
@{term "simp r"} & $\dn$ & @{term "(r, id)"}\\
+ − 1574
\end{tabular}
+ − 1575
\end{center}
+ − 1576
+ − 1577
\noindent where @{term "id"} stands for the identity function. The
+ − 1578
function @{const simp} returns a simplified regular expression and a corresponding
+ − 1579
rectification function. Note that we do not simplify under stars: this
+ − 1580
seems to slow down the algorithm, rather than speed it up. The optimised
+ − 1581
lexer is then given by the clauses:
+ − 1582
+ − 1583
\begin{center}
+ − 1584
\begin{tabular}{lcl}
+ − 1585
@{thm (lhs) slexer.simps(1)} & $\dn$ & @{thm (rhs) slexer.simps(1)}\\
+ − 1586
@{thm (lhs) slexer.simps(2)} & $\dn$ &
330
+ − 1587
\<open>let (r\<^sub>s, f\<^sub>r) = simp (r \<close>$\backslash$\<open> c) in\<close>\\
+ − 1588
& & \<open>case\<close> @{term "slexer r\<^sub>s s"} \<open>of\<close>\\
+ − 1589
& & \phantom{$|$} @{term "None"} \<open>\<Rightarrow>\<close> @{term None}\\
+ − 1590
& & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> \<open>Some (inj r c (f\<^sub>r v))\<close>
218
+ − 1591
\end{tabular}
+ − 1592
\end{center}
+ − 1593
+ − 1594
\noindent
+ − 1595
In the second clause we first calculate the derivative @{term "der c r"}
+ − 1596
and then simplify the result. This gives us a simplified derivative
330
+ − 1597
\<open>r\<^sub>s\<close> and a rectification function \<open>f\<^sub>r\<close>. The lexer
218
+ − 1598
is then recursively called with the simplified derivative, but before
+ − 1599
we inject the character @{term c} into the value @{term v}, we need to rectify
+ − 1600
@{term v} (that is construct @{term "f\<^sub>r v"}). Before we can establish the correctness
+ − 1601
of @{term "slexer"}, we need to show that simplification preserves the language
+ − 1602
and simplification preserves our POSIX relation once the value is rectified
+ − 1603
(recall @{const "simp"} generates a (regular expression, rectification function) pair):
+ − 1604
+ − 1605
\begin{lemma}\mbox{}\smallskip\\\label{slexeraux}
+ − 1606
\begin{tabular}{ll}
+ − 1607
(1) & @{thm L_fst_simp[symmetric]}\\
+ − 1608
(2) & @{thm[mode=IfThen] Posix_simp}
+ − 1609
\end{tabular}
+ − 1610
\end{lemma}
+ − 1611
330
+ − 1612
\begin{proof} Both are by induction on \<open>r\<close>. There is no
218
+ − 1613
interesting case for the first statement. For the second statement,
+ − 1614
of interest are the @{term "r = ALT r\<^sub>1 r\<^sub>2"} and @{term "r = SEQ r\<^sub>1
+ − 1615
r\<^sub>2"} cases. In each case we have to analyse four subcases whether
+ − 1616
@{term "fst (simp r\<^sub>1)"} and @{term "fst (simp r\<^sub>2)"} equals @{const
+ − 1617
ZERO} (respectively @{const ONE}). For example for @{term "r = ALT
+ − 1618
r\<^sub>1 r\<^sub>2"}, consider the subcase @{term "fst (simp r\<^sub>1) = ZERO"} and
+ − 1619
@{term "fst (simp r\<^sub>2) \<noteq> ZERO"}. By assumption we know @{term "s \<in>
+ − 1620
fst (simp (ALT r\<^sub>1 r\<^sub>2)) \<rightarrow> v"}. From this we can infer @{term "s \<in> fst (simp r\<^sub>2) \<rightarrow> v"}
+ − 1621
and by IH also (*) @{term "s \<in> r\<^sub>2 \<rightarrow> (snd (simp r\<^sub>2) v)"}. Given @{term "fst (simp r\<^sub>1) = ZERO"}
+ − 1622
we know @{term "L (fst (simp r\<^sub>1)) = {}"}. By the first statement
+ − 1623
@{term "L r\<^sub>1"} is the empty set, meaning (**) @{term "s \<notin> L r\<^sub>1"}.
330
+ − 1624
Taking (*) and (**) together gives by the \mbox{\<open>P+R\<close>}-rule
218
+ − 1625
@{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> Right (snd (simp r\<^sub>2) v)"}. In turn this
+ − 1626
gives @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> snd (simp (ALT r\<^sub>1 r\<^sub>2)) v"} as we need to show.
+ − 1627
The other cases are similar.\qed
+ − 1628
\end{proof}
+ − 1629
+ − 1630
\noindent We can now prove relatively straightforwardly that the
+ − 1631
optimised lexer produces the expected result:
+ − 1632
+ − 1633
\begin{theorem}
+ − 1634
@{thm slexer_correctness}
+ − 1635
\end{theorem}
+ − 1636
+ − 1637
\begin{proof} By induction on @{term s} generalising over @{term
+ − 1638
r}. The case @{term "[]"} is trivial. For the cons-case suppose the
+ − 1639
string is of the form @{term "c # s"}. By induction hypothesis we
+ − 1640
know @{term "slexer r s = lexer r s"} holds for all @{term r} (in
+ − 1641
particular for @{term "r"} being the derivative @{term "der c
+ − 1642
r"}). Let @{term "r\<^sub>s"} be the simplified derivative regular expression, that is @{term
+ − 1643
"fst (simp (der c r))"}, and @{term "f\<^sub>r"} be the rectification
+ − 1644
function, that is @{term "snd (simp (der c r))"}. We distinguish the cases
+ − 1645
whether (*) @{term "s \<in> L (der c r)"} or not. In the first case we
272
+ − 1646
have by Theorem~\ref{lexercorrect}(2) a value @{term "v"} so that @{term
218
+ − 1647
"lexer (der c r) s = Some v"} and @{term "s \<in> der c r \<rightarrow> v"} hold.
272
+ − 1648
By Lemma~\ref{slexeraux}(1) we can also infer from~(*) that @{term "s
+ − 1649
\<in> L r\<^sub>s"} holds. Hence we know by Theorem~\ref{lexercorrect}(2) that
218
+ − 1650
there exists a @{term "v'"} with @{term "lexer r\<^sub>s s = Some v'"} and
+ − 1651
@{term "s \<in> r\<^sub>s \<rightarrow> v'"}. From the latter we know by
272
+ − 1652
Lemma~\ref{slexeraux}(2) that @{term "s \<in> der c r \<rightarrow> (f\<^sub>r v')"} holds.
+ − 1653
By the uniqueness of the POSIX relation (Theorem~\ref{posixdeterm}) we
218
+ − 1654
can infer that @{term v} is equal to @{term "f\<^sub>r v'"}---that is the
+ − 1655
rectification function applied to @{term "v'"}
+ − 1656
produces the original @{term "v"}. Now the case follows by the
+ − 1657
definitions of @{const lexer} and @{const slexer}.
+ − 1658
+ − 1659
In the second case where @{term "s \<notin> L (der c r)"} we have that
272
+ − 1660
@{term "lexer (der c r) s = None"} by Theorem~\ref{lexercorrect}(1). We
+ − 1661
also know by Lemma~\ref{slexeraux}(1) that @{term "s \<notin> L r\<^sub>s"}. Hence
+ − 1662
@{term "lexer r\<^sub>s s = None"} by Theorem~\ref{lexercorrect}(1) and
218
+ − 1663
by IH then also @{term "slexer r\<^sub>s s = None"}. With this we can
+ − 1664
conclude in this case too.\qed
+ − 1665
+ − 1666
\end{proof}
272
+ − 1667
330
+ − 1668
\<close>
272
+ − 1669
+ − 1670
330
+ − 1671
section \<open>HERE\<close>
318
+ − 1672
330
+ − 1673
text \<open>
318
+ − 1674
\begin{center}
+ − 1675
\begin{tabular}{llcl}
+ − 1676
1) & @{thm (lhs) erase.simps(1)} & $\dn$ & @{thm (rhs) erase.simps(1)}\\
+ − 1677
2) & @{thm (lhs) erase.simps(2)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(2)[of bs]}\\
+ − 1678
3) & @{thm (lhs) erase.simps(3)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(3)[of bs]}\\
+ − 1679
4a) & @{term "erase (AALTs bs [])"} & $\dn$ & @{term ZERO}\\
+ − 1680
4b) & @{term "erase (AALTs bs [r])"} & $\dn$ & @{term "erase r"}\\
+ − 1681
4c) & @{term "erase (AALTs bs (r\<^sub>1#r\<^sub>2#rs))"} & $\dn$ &
+ − 1682
@{term "ALT (erase r\<^sub>1) (erase (AALTs bs (r\<^sub>2#rs)))"}\\
+ − 1683
5) & @{thm (lhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
+ − 1684
6) & @{thm (lhs) erase.simps(6)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(6)[of bs]}\\
+ − 1685
\end{tabular}
+ − 1686
\end{center}
+ − 1687
+ − 1688
\begin{lemma}
+ − 1689
@{thm [mode=IfThen] bder_retrieve}
+ − 1690
\end{lemma}
+ − 1691
+ − 1692
\begin{proof}
+ − 1693
By induction on the definition of @{term "erase r"}. The cases for rule 1) and 2) are
+ − 1694
straightforward as @{term "der c ZERO"} and @{term "der c ONE"} are both equal to
+ − 1695
@{term ZERO}. This means @{term "\<Turnstile> v : ZERO"} cannot hold. Similarly in case of rule 3)
+ − 1696
where @{term r} is of the form @{term "ACHAR d"} with @{term "c = d"}. Then by assumption
+ − 1697
we know @{term "\<Turnstile> v : ONE"}, which implies @{term "v = Void"}. The equation follows by
+ − 1698
simplification of left- and right-hand side. In case @{term "c \<noteq> d"} we have again
+ − 1699
@{term "\<Turnstile> v : ZERO"}, which cannot hold.
+ − 1700
+ − 1701
For rule 4a) we have again @{term "\<Turnstile> v : ZERO"}. The property holds by IH for rule 4b).
+ − 1702
The induction hypothesis is
+ − 1703
\[
+ − 1704
@{term "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"}
+ − 1705
\]
+ − 1706
which is what left- and right-hand side simplify to. The slightly more interesting case
+ − 1707
is for 4c). By assumption we have
+ − 1708
@{term "\<Turnstile> v : ALT (der c (erase r\<^sub>1)) (der c (erase (AALTs bs (r\<^sub>2 # rs))))"}. This means we
+ − 1709
have either (*) @{term "\<Turnstile> v1 : der c (erase r\<^sub>1)"} with @{term "v = Left v1"} or
+ − 1710
(**) @{term "\<Turnstile> v2 : der c (erase (AALTs bs (r\<^sub>2 # rs)))"} with @{term "v = Right v2"}.
+ − 1711
The former case is straightforward by simplification. The second case is \ldots TBD.
+ − 1712
+ − 1713
Rule 5) TBD.
+ − 1714
+ − 1715
Finally for rule 6) the reasoning is as follows: By assumption we have
+ − 1716
@{term "\<Turnstile> v : SEQ (der c (erase r)) (STAR (erase r))"}. This means we also have
+ − 1717
@{term "v = Seq v1 v2"}, @{term "\<Turnstile> v1 : der c (erase r)"} and @{term "v2 = Stars vs"}.
+ − 1718
We want to prove
+ − 1719
\begin{align}
+ − 1720
& @{term "retrieve (ASEQ bs (fuse [Z] (bder c r)) (ASTAR [] r)) v"}\\
+ − 1721
&= @{term "retrieve (ASTAR bs r) (injval (STAR (erase r)) c v)"}
+ − 1722
\end{align}
+ − 1723
The right-hand side @{term inj}-expression is equal to
+ − 1724
@{term "Stars (injval (erase r) c v1 # vs)"}, which means the @{term retrieve}-expression
+ − 1725
simplifies to
+ − 1726
\[
+ − 1727
@{term "bs @ [Z] @ retrieve r (injval (erase r) c v1) @ retrieve (ASTAR [] r) (Stars vs)"}
+ − 1728
\]
+ − 1729
The left-hand side (3) above simplifies to
+ − 1730
\[
+ − 1731
@{term "bs @ retrieve (fuse [Z] (bder c r)) v1 @ retrieve (ASTAR [] r) (Stars vs)"}
+ − 1732
\]
+ − 1733
We can move out the @{term "fuse [Z]"} and then use the IH to show that left-hand side
+ − 1734
and right-hand side are equal. This completes the proof.
+ − 1735
\end{proof}
330
+ − 1736
\<close>
318
+ − 1737
218
+ − 1738
+ − 1739
+ − 1740
(*<*)
+ − 1741
end
+ − 1742
(*>*)