1 \documentclass[a4paper,12pt]{article}
2 \usepackage[T1]{fontenc}
5 \usepackage[english,french]{babel}
11 %\usepackage[scaled=.85]{beramono}
12 \usepackage{../iman,../pdfsetup}
15 %\evensidemargin=4.6mm
22 \def\Colon{\mathord{:\mkern-1.5mu:}}
23 %\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}}
24 %\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}}
25 \def\lparr{\mathopen{(\mkern-4mu\mid}}
26 \def\rparr{\mathclose{\mid\mkern-4mu)}}
28 \def\undef{\textit{undefined}}
30 %\def\unr{\textit{others}}
32 \def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
33 \def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
35 \hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
36 counter-example counter-examples data-type data-types co-data-type
37 co-data-types in-duc-tive co-in-duc-tive}
43 \selectlanguage{english}
45 \title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
46 Picking Nits \\[\smallskipamount]
47 \Large A User's Guide to Nitpick for Isabelle/HOL 2010}
49 Jasmin Christian Blanchette \\
50 {\normalsize Fakult\"at f\"ur Informatik, Technische Universit\"at M\"unchen} \\
57 \setlength{\parskip}{.7em plus .2em minus .1em}
58 \setlength{\parindent}{0pt}
59 \setlength{\abovedisplayskip}{\parskip}
60 \setlength{\abovedisplayshortskip}{.9\parskip}
61 \setlength{\belowdisplayskip}{\parskip}
62 \setlength{\belowdisplayshortskip}{.9\parskip}
64 % General-purpose enum environment with correct spacing
65 \newenvironment{enum}%
67 \setlength{\topsep}{.1\parskip}%
68 \setlength{\partopsep}{.1\parskip}%
69 \setlength{\itemsep}{\parskip}%
70 \advance\itemsep by-\parsep}}
73 \def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin
74 \advance\rightskip by\leftmargin}
75 \def\post{\vskip0pt plus1ex\endgroup}
77 \def\prew{\pre\advance\rightskip by-\leftmargin}
80 \section{Introduction}
83 Nitpick \cite{blanchette-nipkow-2009} is a counterexample generator for
84 Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
85 combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
86 quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
87 first-order relational model finder developed by the Software Design Group at
88 MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
89 borrows many ideas and code fragments, but it benefits from Kodkod's
90 optimizations and a new encoding scheme. The name Nitpick is shamelessly
91 appropriated from a now retired Alloy precursor.
93 Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
94 theorem and wait a few seconds. Nonetheless, there are situations where knowing
95 how it works under the hood and how it reacts to various options helps
96 increase the test coverage. This manual also explains how to install the tool on
97 your workstation. Should the motivation fail you, think of the many hours of
98 hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.
100 Another common use of Nitpick is to find out whether the axioms of a locale are
101 satisfiable, while the locale is being developed. To check this, it suffices to
105 \textbf{lemma}~``$\textit{False}$'' \\
106 \textbf{nitpick}~[\textit{show\_all}]
109 after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
110 must find a model for the axioms. If it finds no model, we have an indication
111 that the axioms might be unsatisfiable.
113 Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
114 machine called \texttt{java}. The examples presented in this manual can be found
115 in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.
117 Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
118 Nitpick also provides an automatic mode that can be enabled using the
119 ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof General. In this
120 mode, Nitpick is run on every newly entered theorem, much like Auto Quickcheck.
121 The collective time limit for Auto Nitpick and Auto Quickcheck can be set using
122 the ``Auto Counterexample Time Limit'' option.
125 \setbox\boxA=\hbox{\texttt{nospam}}
127 The known bugs and limitations at the time of writing are listed in
128 \S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
129 or this manual should be directed to
130 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
131 in.\allowbreak tum.\allowbreak de}.
133 \vskip2.5\smallskipamount
135 \textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
136 suggesting several textual improvements.
137 % and Perry James for reporting a typo.
139 \section{First Steps}
142 This section introduces Nitpick by presenting small examples. If possible, you
143 should try out the examples on your workstation. Your theory file should start
147 \textbf{theory}~\textit{Scratch} \\
148 \textbf{imports}~\textit{Main} \\
152 The results presented here were obtained using the JNI version of MiniSat and
153 with multithreading disabled to reduce nondeterminism. This was done by adding
157 \textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSatJNI}, \,\textit{max\_threads}~= 1]
160 after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
161 Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
162 be installed, as explained in \S\ref{optimizations}. If you have already
163 configured SAT solvers in Isabelle (e.g., for Refute), these will also be
164 available to Nitpick.
166 \subsection{Propositional Logic}
167 \label{propositional-logic}
169 Let's start with a trivial example from propositional logic:
172 \textbf{lemma}~``$P \longleftrightarrow Q$'' \\
176 You should get the following output:
180 Nitpick found a counterexample: \\[2\smallskipamount]
181 \hbox{}\qquad Free variables: \nopagebreak \\
182 \hbox{}\qquad\qquad $P = \textit{True}$ \\
183 \hbox{}\qquad\qquad $Q = \textit{False}$
186 Nitpick can also be invoked on individual subgoals, as in the example below:
189 \textbf{apply}~\textit{auto} \\[2\smallskipamount]
190 {\slshape goal (2 subgoals): \\
191 \ 1. $P\,\Longrightarrow\, Q$ \\
192 \ 2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
193 \textbf{nitpick}~1 \\[2\smallskipamount]
194 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
195 \hbox{}\qquad Free variables: \nopagebreak \\
196 \hbox{}\qquad\qquad $P = \textit{True}$ \\
197 \hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
198 \textbf{nitpick}~2 \\[2\smallskipamount]
199 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
200 \hbox{}\qquad Free variables: \nopagebreak \\
201 \hbox{}\qquad\qquad $P = \textit{False}$ \\
202 \hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
206 \subsection{Type Variables}
207 \label{type-variables}
209 If you are left unimpressed by the previous example, don't worry. The next
210 one is more mind- and computer-boggling:
213 \textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
215 \pagebreak[2] %% TYPESETTING
217 The putative lemma involves the definite description operator, {THE}, presented
218 in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
219 operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
220 lemma is merely asserting the indefinite description operator axiom with {THE}
221 substituted for {SOME}.
223 The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
224 containing type variables, Nitpick enumerates the possible domains for each type
225 variable, up to a given cardinality (8 by default), looking for a finite
229 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
231 Trying 8 scopes: \nopagebreak \\
232 \hbox{}\qquad \textit{card}~$'a$~= 1; \\
233 \hbox{}\qquad \textit{card}~$'a$~= 2; \\
234 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
235 \hbox{}\qquad \textit{card}~$'a$~= 8. \\[2\smallskipamount]
236 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
237 \hbox{}\qquad Free variables: \nopagebreak \\
238 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
239 \hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
243 Nitpick found a counterexample in which $'a$ has cardinality 3. (For
244 cardinalities 1 and 2, the formula holds.) In the counterexample, the three
245 values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
247 The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
248 \textit{verbose} is enabled. You can specify \textit{verbose} each time you
249 invoke \textbf{nitpick}, or you can set it globally using the command
252 \textbf{nitpick\_params} [\textit{verbose}]
255 This command also displays the current default values for all of the options
256 supported by Nitpick. The options are listed in \S\ref{option-reference}.
258 \subsection{Constants}
261 By just looking at Nitpick's output, it might not be clear why the
262 counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
263 this time telling it to show the values of the constants that occur in the
267 \textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
268 \textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
270 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
271 \hbox{}\qquad Free variables: \nopagebreak \\
272 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
273 \hbox{}\qquad\qquad $x = a_3$ \\
274 \hbox{}\qquad Constant: \nopagebreak \\
275 \hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
278 We can see more clearly now. Since the predicate $P$ isn't true for a unique
279 value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
280 $a_1$. Since $P~a_1$ is false, the entire formula is falsified.
282 As an optimization, Nitpick's preprocessor introduced the special constant
283 ``\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
284 $\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
285 satisfying $P~y$. We disable this optimization by passing the
286 \textit{full\_descrs} option:
289 \textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
291 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
292 \hbox{}\qquad Free variables: \nopagebreak \\
293 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
294 \hbox{}\qquad\qquad $x = a_3$ \\
295 \hbox{}\qquad Constant: \nopagebreak \\
296 \hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
299 As the result of another optimization, Nitpick directly assigned a value to the
300 subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
301 disable this second optimization by using the command
304 \textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
305 \textit{show\_consts}]
308 we finally get \textit{The}:
311 \slshape Constant: \nopagebreak \\
312 \hbox{}\qquad $\mathit{The} = \undef{}
313 (\!\begin{aligned}[t]%
314 & \{\} := a_3,\> \{a_3\} := a_3,\> \{a_2\} := a_2, \\[-2pt] %% TYPESETTING
315 & \{a_2, a_3\} := a_1,\> \{a_1\} := a_1,\> \{a_1, a_3\} := a_3, \\[-2pt]
316 & \{a_1, a_2\} := a_3,\> \{a_1, a_2, a_3\} := a_3)\end{aligned}$
319 Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
320 just like before.\footnote{The \undef{} symbol's presence is explained as
321 follows: In higher-order logic, any function can be built from the undefined
322 function using repeated applications of the function update operator $f(x :=
323 y)$, just like any list can be built from the empty list using $x \mathbin{\#}
326 Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
327 unique $x$ such that'') at the front of our putative lemma's assumption:
330 \textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
333 The fix appears to work:
336 \textbf{nitpick} \\[2\smallskipamount]
337 \slshape Nitpick found no counterexample.
340 We can further increase our confidence in the formula by exhausting all
341 cardinalities up to 50:
344 \textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
345 can be entered as \texttt{-} (hyphen) or
346 \texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
347 \slshape Nitpick found no counterexample.
350 Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
353 \textbf{sledgehammer} \\[2\smallskipamount]
354 {\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
355 $\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
356 Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
357 \textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
358 {\slshape No subgoals!}% \\[2\smallskipamount]
362 This must be our lucky day.
364 \subsection{Skolemization}
365 \label{skolemization}
367 Are all invertible functions onto? Let's find out:
370 \textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
371 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
372 \textbf{nitpick} \\[2\smallskipamount]
374 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
375 \hbox{}\qquad Free variable: \nopagebreak \\
376 \hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
377 \hbox{}\qquad Skolem constants: \nopagebreak \\
378 \hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
379 \hbox{}\qquad\qquad $y = a_2$
382 Although $f$ is the only free variable occurring in the formula, Nitpick also
383 displays values for the bound variables $g$ and $y$. These values are available
384 to Nitpick because it performs skolemization as a preprocessing step.
386 In the previous example, skolemization only affected the outermost quantifiers.
387 This is not always the case, as illustrated below:
390 \textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
391 \textbf{nitpick} \\[2\smallskipamount]
393 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
394 \hbox{}\qquad Skolem constant: \nopagebreak \\
395 \hbox{}\qquad\qquad $\lambda x.\; f =
396 \undef{}(\!\begin{aligned}[t]
397 & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
398 & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
401 The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
402 $x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
403 function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
404 maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
406 The source of the Skolem constants is sometimes more obscure:
409 \textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
410 \textbf{nitpick} \\[2\smallskipamount]
412 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
413 \hbox{}\qquad Free variable: \nopagebreak \\
414 \hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
415 \hbox{}\qquad Skolem constants: \nopagebreak \\
416 \hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
417 \hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
420 What happened here is that Nitpick expanded the \textit{sym} constant to its
424 $\mathit{sym}~r \,\equiv\,
425 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
428 As their names suggest, the Skolem constants $\mathit{sym}.x$ and
429 $\mathit{sym}.y$ are simply the bound variables $x$ and $y$
430 from \textit{sym}'s definition.
432 Although skolemization is a useful optimization, you can disable it by invoking
433 Nitpick with \textit{dont\_skolemize}. See \S\ref{optimizations} for details.
435 \subsection{Natural Numbers and Integers}
436 \label{natural-numbers-and-integers}
438 Because of the axiom of infinity, the type \textit{nat} does not admit any
439 finite models. To deal with this, Nitpick considers prefixes $\{0,\, 1,\,
440 \ldots,\, K - 1\}$ of \textit{nat} (where $K = \textit{card}~\textit{nat}$) and
441 maps all other numbers to the undefined value ($\unk$). The type \textit{int} is
442 handled in a similar way: If $K = \textit{card}~\textit{int}$, the subset of
443 \textit{int} known to Nitpick is $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor
444 K/2 \rfloor\}$. Undefined values lead to a three-valued logic.
446 Here is an example involving \textit{int}:
449 \textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
450 \textbf{nitpick} \\[2\smallskipamount]
451 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
452 \hbox{}\qquad Free variables: \nopagebreak \\
453 \hbox{}\qquad\qquad $i = 0$ \\
454 \hbox{}\qquad\qquad $j = 1$ \\
455 \hbox{}\qquad\qquad $m = 1$ \\
456 \hbox{}\qquad\qquad $n = 0$
459 With infinite types, we don't always have the luxury of a genuine counterexample
460 and must often content ourselves with a potential one. The tedious task of
461 finding out whether the potential counterexample is in fact genuine can be
462 outsourced to \textit{auto} by passing the option \textit{check\_potential}. For
466 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
467 \textbf{nitpick} [\textit{card~nat}~= 100,\, \textit{check\_potential}] \\[2\smallskipamount]
468 \slshape Nitpick found a potential counterexample: \\[2\smallskipamount]
469 \hbox{}\qquad Free variable: \nopagebreak \\
470 \hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
471 Confirmation by ``\textit{auto}'': The above counterexample is genuine.
474 You might wonder why the counterexample is first reported as potential. The root
475 of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
476 \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
477 that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
478 \textit{False}; but otherwise, it does not know anything about values of $n \ge
479 \textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
480 \textit{True}. Since the assumption can never be satisfied, the putative lemma
481 can never be falsified.
483 Incidentally, if you distrust the so-called genuine counterexamples, you can
484 enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
485 aware that \textit{auto} will often fail to prove that the counterexample is
488 Some conjectures involving elementary number theory make Nitpick look like a
489 giant with feet of clay:
492 \textbf{lemma} ``$P~\textit{Suc}$'' \\
493 \textbf{nitpick} [\textit{card} = 1--6] \\[2\smallskipamount]
495 Nitpick found no counterexample.
498 For any cardinality $k$, \textit{Suc} is the partial function $\{0 \mapsto 1,\,
499 1 \mapsto 2,\, \ldots,\, k - 1 \mapsto \unk\}$, which evaluates to $\unk$ when
500 it is passed as argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$.
501 The next example is similar:
504 \textbf{lemma} ``$P~(\textit{op}~{+}\Colon
505 \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
506 \textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
507 {\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
508 \hbox{}\qquad Free variable: \nopagebreak \\
509 \hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
510 \textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
511 {\slshape Nitpick found no counterexample.}
514 The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
515 $\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
518 Because numbers are infinite and are approximated using a three-valued logic,
519 there is usually no need to systematically enumerate domain sizes. If Nitpick
520 cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
521 unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
522 example above is an exception to this principle.) Nitpick nonetheless enumerates
523 all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
524 cardinalities are fast to handle and give rise to simpler counterexamples. This
525 is explained in more detail in \S\ref{scope-monotonicity}.
527 \subsection{Inductive Datatypes}
528 \label{inductive-datatypes}
530 Like natural numbers and integers, inductive datatypes with recursive
531 constructors admit no finite models and must be approximated by a subterm-closed
532 subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
533 Nitpick looks for all counterexamples that can be built using at most 10
536 Let's see with an example involving \textit{hd} (which returns the first element
537 of a list) and $@$ (which concatenates two lists):
540 \textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
541 \textbf{nitpick} \\[2\smallskipamount]
542 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
543 \hbox{}\qquad Free variables: \nopagebreak \\
544 \hbox{}\qquad\qquad $\textit{xs} = []$ \\
545 \hbox{}\qquad\qquad $\textit{y} = a_3$
548 To see why the counterexample is genuine, we enable \textit{show\_consts}
549 and \textit{show\_\allowbreak datatypes}:
552 {\slshape Datatype:} \\
553 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_3, a_3],\, [a_3],\, \unr\}$ \\
554 {\slshape Constants:} \\
555 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_3, a_3],\> [a_3, a_3] := \unk,\> [a_3] := \unk)$ \\
556 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_3, a_3] := a_3,\> [a_3] := a_3)$
559 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
562 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
563 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_3,
564 a_3]$ to $[a_3]$ would normally give $[a_3, a_3, a_3]$, but this value is not
565 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
566 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
567 appending $[a_3, a_3]$ to itself gives $\unk$.
569 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
570 considers the following subsets:
572 \kern-.5\smallskipamount %% TYPESETTING
576 $\{[],\, [a_1],\, [a_2]\}$; \\
577 $\{[],\, [a_1],\, [a_3]\}$; \\
578 $\{[],\, [a_2],\, [a_3]\}$; \\
579 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\
580 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\
581 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\
582 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\
583 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\
584 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\
585 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\
586 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\
587 $\{[],\, [a_3],\, [a_3, a_3]\}$.
591 \kern-2\smallskipamount %% TYPESETTING
593 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
594 are listed and only those. As an example of a non-subterm-closed subset,
595 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_3]\}$, and observe
596 that $[a_1, a_3]$ (i.e., $a_1 \mathbin{\#} [a_3]$) has $[a_3] \notin
597 \mathcal{S}$ as a subterm.
599 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
602 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
603 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
605 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
606 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
607 \hbox{}\qquad Free variables: \nopagebreak \\
608 \hbox{}\qquad\qquad $\textit{xs} = [a_2]$ \\
609 \hbox{}\qquad\qquad $\textit{ys} = [a_3]$ \\
610 \hbox{}\qquad Datatypes: \\
611 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
612 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_3],\, [a_2],\, \unr\}$
615 Because datatypes are approximated using a three-valued logic, there is usually
616 no need to systematically enumerate cardinalities: If Nitpick cannot find a
617 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
618 unlikely that one could be found for smaller cardinalities.
620 \subsection{Typedefs, Records, Rationals, and Reals}
621 \label{typedefs-records-rationals-and-reals}
623 Nitpick generally treats types declared using \textbf{typedef} as datatypes
624 whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
628 \textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
629 \textbf{by}~\textit{blast} \\[2\smallskipamount]
630 \textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
631 \textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
632 \textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
633 \textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
634 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
635 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
636 \hbox{}\qquad Free variables: \nopagebreak \\
637 \hbox{}\qquad\qquad $P = \{\Abs{1},\, \Abs{0}\}$ \\
638 \hbox{}\qquad\qquad $x = \Abs{2}$ \\
639 \hbox{}\qquad Datatypes: \\
640 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
641 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{2},\, \Abs{1},\, \Abs{0},\, \unr\}$
645 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
648 Records, which are implemented as \textbf{typedef}s behind the scenes, are
649 handled in much the same way:
652 \textbf{record} \textit{point} = \\
653 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
654 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
655 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
656 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
657 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
658 \hbox{}\qquad Free variables: \nopagebreak \\
659 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
660 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
661 \hbox{}\qquad Datatypes: \\
662 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
663 \hbox{}\qquad\qquad $\textit{point} = \{\lparr\textit{Xcoord} = 1,\>
664 \textit{Ycoord} = 1\rparr,\> \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr,\, \unr\}$\kern-1pt %% QUIET
667 Finally, Nitpick provides rudimentary support for rationals and reals using a
671 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
672 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
673 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
674 \hbox{}\qquad Free variables: \nopagebreak \\
675 \hbox{}\qquad\qquad $x = 1/2$ \\
676 \hbox{}\qquad\qquad $y = -1/2$ \\
677 \hbox{}\qquad Datatypes: \\
678 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
679 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
680 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
683 \subsection{Inductive and Coinductive Predicates}
684 \label{inductive-and-coinductive-predicates}
686 Inductively defined predicates (and sets) are particularly problematic for
687 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
688 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
689 the problem is that they are defined using a least fixed point construction.
691 Nitpick's philosophy is that not all inductive predicates are equal. Consider
692 the \textit{even} predicate below:
695 \textbf{inductive}~\textit{even}~\textbf{where} \\
696 ``\textit{even}~0'' $\,\mid$ \\
697 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
700 This predicate enjoys the desirable property of being well-founded, which means
701 that the introduction rules don't give rise to infinite chains of the form
704 $\cdots\,\Longrightarrow\, \textit{even}~k''
705 \,\Longrightarrow\, \textit{even}~k'
706 \,\Longrightarrow\, \textit{even}~k.$
709 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
713 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
714 \,\Longrightarrow\, \textit{even}~(k - 2)
715 \,\Longrightarrow\, \textit{even}~k.$
718 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
719 fixed point computation.%
720 \footnote{If an inductive predicate is
721 well-founded, then it has exactly one fixed point, which is simultaneously the
722 least and the greatest fixed point. In these circumstances, the computation of
723 the least fixed point amounts to the computation of an arbitrary fixed point,
724 which can be performed using a straightforward recursive equation.}
725 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
726 just as Isabelle's \textbf{function} package usually discharges termination
727 proof obligations automatically.
729 Let's try an example:
732 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
733 \textbf{nitpick}~[\textit{card nat}~= 100,\, \textit{verbose}] \\[2\smallskipamount]
734 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
735 Nitpick can compute it efficiently. \\[2\smallskipamount]
737 \hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
738 Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
739 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
740 Nitpick could not find a better counterexample. \\[2\smallskipamount]
744 No genuine counterexample is possible because Nitpick cannot rule out the
745 existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
746 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
747 existential quantifier:
750 \textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
751 \textbf{nitpick}~[\textit{card nat}~= 100] \\[2\smallskipamount]
752 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
753 \hbox{}\qquad Empty assignment
756 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
757 we use the following definition instead?
760 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
761 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
762 ``$\textit{even}'~2$'' $\,\mid$ \\
763 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
766 This definition is not well-founded: From $\textit{even}'~0$ and
767 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
768 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
770 Let's check a property involving $\textit{even}'$. To make up for the
771 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
772 \textit{nat}'s cardinality to a mere 10:
775 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
776 \lnot\;\textit{even}'~n$'' \\
777 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
779 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
780 Nitpick might need to unroll it. \\[2\smallskipamount]
782 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
783 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
784 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
785 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
786 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
787 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
788 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
789 \hbox{}\qquad Constant: \nopagebreak \\
790 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
791 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
792 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
793 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
797 Nitpick's output is very instructive. First, it tells us that the predicate is
798 unrolled, meaning that it is computed iteratively from the empty set. Then it
799 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
802 The output also shows how each iteration contributes to $\textit{even}'$. The
803 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
804 predicate depends on an iteration counter. Iteration 0 provides the basis
805 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
806 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
807 iterations would not contribute any new elements.
809 Some values are marked with superscripted question
810 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
811 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
812 \textit{True} or $\unk$, never \textit{False}.
814 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
815 iterations. However, these numbers are bounded by the cardinality of the
816 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
817 ever needed to compute the value of a \textit{nat} predicate. You can specify
818 the number of iterations using the \textit{iter} option, as explained in
819 \S\ref{scope-of-search}.
821 In the next formula, $\textit{even}'$ occurs both positively and negatively:
824 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
825 \textbf{nitpick} [\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
826 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
827 \hbox{}\qquad Free variable: \nopagebreak \\
828 \hbox{}\qquad\qquad $n = 1$ \\
829 \hbox{}\qquad Constants: \nopagebreak \\
830 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
831 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
832 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
835 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
836 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
837 fixed point (not necessarily the least one). It is used to falsify
838 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
839 $\textit{even}'~(n - 2)$.
841 Coinductive predicates are handled dually. For example:
844 \textbf{coinductive} \textit{nats} \textbf{where} \\
845 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
846 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
847 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
848 \slshape Nitpick found a counterexample:
849 \\[2\smallskipamount]
850 \hbox{}\qquad Constants: \nopagebreak \\
851 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
852 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
853 & \unr\})\end{aligned}$ \\
854 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
857 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
858 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
859 inductive predicates for which each the predicate occurs in at most one
860 assumption of each introduction rule. For example:
863 \textbf{inductive} \textit{odd} \textbf{where} \\
864 ``$\textit{odd}~1$'' $\,\mid$ \\
865 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
866 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
867 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
868 \slshape Nitpick found a counterexample:
869 \\[2\smallskipamount]
870 \hbox{}\qquad Free variable: \nopagebreak \\
871 \hbox{}\qquad\qquad $n = 1$ \\
872 \hbox{}\qquad Constants: \nopagebreak \\
873 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
874 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
875 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
877 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
878 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
880 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
881 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
882 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
886 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
887 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
888 elements from known ones. The set $\textit{odd}$ consists of all the values
889 reachable through the reflexive transitive closure of
890 $\textit{odd}_{\textrm{step}}$ starting with any element from
891 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
892 transitive closure to encode linear predicates is normally either more thorough
893 or more efficient than unrolling (depending on the value of \textit{iter}), but
894 for those cases where it isn't you can disable it by passing the
895 \textit{dont\_star\_linear\_preds} option.
897 \subsection{Coinductive Datatypes}
898 \label{coinductive-datatypes}
900 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
901 datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
902 list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
903 these lazy lists seamlessly and provides a hook, described in
904 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
907 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
908 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
909 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
910 1, 2, 3, \ldots]$ can be defined as lazy lists using the
911 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
912 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
913 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
915 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
916 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
920 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
921 \textbf{nitpick} \\[2\smallskipamount]
922 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
923 \hbox{}\qquad Free variables: \nopagebreak \\
924 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
925 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
928 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
929 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
930 infinite list $[a_1, a_1, a_1, \ldots]$.
932 The next example is more interesting:
935 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
936 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
937 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
938 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
939 some scopes. \\[2\smallskipamount]
941 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 1,
942 and \textit{bisim\_depth}~= 0. \\
943 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
944 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 8,
945 and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
946 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
947 \textit{card}~``\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
949 \\[2\smallskipamount]
950 \hbox{}\qquad Free variables: \nopagebreak \\
951 \hbox{}\qquad\qquad $\textit{a} = a_2$ \\
952 \hbox{}\qquad\qquad $\textit{b} = a_1$ \\
953 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
954 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_1~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega)$ \\[2\smallskipamount]
958 The lazy list $\textit{xs}$ is simply $[a_2, a_2, a_2, \ldots]$, whereas
959 $\textit{ys}$ is $[a_1, a_2, a_2, a_2, \ldots]$, i.e., a lasso-shaped list with
960 $[a_1]$ as its stem and $[a_2]$ as its cycle. In general, the list segment
961 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
962 the segment leading to the binder is the stem.
964 A salient property of coinductive datatypes is that two objects are considered
965 equal if and only if they lead to the same observations. For example, the lazy
966 lists $\textrm{THE}~\omega.\; \omega =
967 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
968 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
969 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
970 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
971 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
972 concept of equality for coinductive datatypes is called bisimulation and is
973 defined coinductively.
975 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
976 the Kodkod problem to ensure that distinct objects lead to different
977 observations. This precaution is somewhat expensive and often unnecessary, so it
978 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
979 bisimilarity check is then performed \textsl{after} the counterexample has been
980 found to ensure correctness. If this after-the-fact check fails, the
981 counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
982 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
983 check for the previous example saves approximately 150~milli\-seconds; the speed
984 gains can be more significant for larger scopes.
986 The next formula illustrates the need for bisimilarity (either as a Kodkod
987 predicate or as an after-the-fact check) to prevent spurious counterexamples:
990 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
991 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
992 \textbf{nitpick} [\textit{bisim\_depth} = $-1$,\, \textit{show\_datatypes}] \\[2\smallskipamount]
993 \slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
994 \hbox{}\qquad Free variables: \nopagebreak \\
995 \hbox{}\qquad\qquad $a = a_2$ \\
996 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
997 \textit{LCons}~a_2~\omega$ \\
998 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
999 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
1000 \hbox{}\qquad\qquad $'a~\textit{llist} =
1001 \{\!\begin{aligned}[t]
1002 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega, \\[-2pt]
1003 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega,\> \unr\}\end{aligned}$
1004 \\[2\smallskipamount]
1005 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1006 that the counterexample is genuine. \\[2\smallskipamount]
1007 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1008 \slshape Nitpick found no counterexample.
1011 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1012 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1014 A compromise between leaving out the bisimilarity predicate from the Kodkod
1015 problem and performing the after-the-fact check is to specify a lower
1016 nonnegative \textit{bisim\_depth} value than the default one provided by
1017 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1018 be distinguished from each other by their prefixes of length $K$. Be aware that
1019 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1020 finding any counterexamples.
1025 Nitpick normally maps function and product types directly to the corresponding
1026 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1027 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1028 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1029 off to treat these types in the same way as plain datatypes, by approximating
1030 them by a subset of a given cardinality. This technique is called ``boxing'' and
1031 is particularly useful for functions passed as arguments to other functions, for
1032 high-arity functions, and for large tuples. Under the hood, boxing involves
1033 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1034 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1036 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1037 using de Bruijn's notation:
1040 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1043 The $\textit{lift}~t~k$ function increments all variables with indices greater
1044 than or equal to $k$ by one:
1047 \textbf{primrec} \textit{lift} \textbf{where} \\
1048 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1049 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1050 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1053 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1054 term $t$ has a loose variable with index $k$ or more:
1057 \textbf{primrec}~\textit{loose} \textbf{where} \\
1058 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1059 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1060 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1063 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1067 \textbf{primrec}~\textit{subst} \textbf{where} \\
1068 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1069 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1070 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1071 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1074 A substitution is a function that maps variable indices to terms. Observe that
1075 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1076 away, because the recursive call for the \textit{Lam} case involves an altered
1077 version. Also notice the \textit{lift} call, which increments the variable
1078 indices when moving under a \textit{Lam}.
1080 A reasonable property to expect of substitution is that it should leave closed
1081 terms unchanged. Alas, even this simple property does not hold:
1084 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1085 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1087 Trying 8 scopes: \nopagebreak \\
1088 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1089 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1090 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1091 \hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
1092 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1093 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1094 \hbox{}\qquad Free variables: \nopagebreak \\
1095 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1096 & 0 := \textit{Var}~0,\>
1097 1 := \textit{Var}~0,\>
1098 2 := \textit{Var}~0, \\[-2pt]
1099 & 3 := \textit{Var}~0,\>
1100 4 := \textit{Var}~0,\>
1101 5 := \textit{Var}~0)\end{aligned}$ \\
1102 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1103 Total time: $4679$ ms.
1106 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1107 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1108 $\lambda$-term notation, $t$~is
1109 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1110 The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
1111 replaced with $\textit{lift}~(\sigma~m)~0$.
1113 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1114 cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1115 For the formula of interest, knowing 6 values of that type was enough to find
1116 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1117 considered, a hopeless undertaking:
1120 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1121 {\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
1125 Boxing can be enabled or disabled globally or on a per-type basis using the
1126 \textit{box} option. Moreover, setting the cardinality of a function or
1127 product type implicitly enables boxing for that type. Nitpick usually performs
1128 reasonable choices about which types should be boxed, but option tweaking
1133 \subsection{Scope Monotonicity}
1134 \label{scope-monotonicity}
1136 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1137 and \textit{max}) controls which scopes are actually tested. In general, to
1138 exhaust all models below a certain cardinality bound, the number of scopes that
1139 Nitpick must consider increases exponentially with the number of type variables
1140 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1141 cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
1142 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1144 Fortunately, many formulas exhibit a property called \textsl{scope
1145 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1146 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1148 Consider the formula
1151 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1154 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1155 $'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
1156 exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
1157 that any counterexample found with a small scope would still be a counterexample
1158 in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
1159 by the larger scope. Nitpick comes to the same conclusion after a careful
1160 inspection of the formula and the relevant definitions:
1163 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1165 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1166 Nitpick might be able to skip some scopes.
1167 \\[2\smallskipamount]
1169 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1170 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1171 \textit{list}''~= 1, \\
1172 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
1173 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
1174 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1175 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1176 \textit{list}''~= 2, \\
1177 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
1178 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
1179 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1180 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
1181 \textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
1182 \textit{list}''~= 8, \\
1183 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
1184 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
1185 \\[2\smallskipamount]
1186 Nitpick found a counterexample for
1187 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1188 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1189 \textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
1190 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
1191 \\[2\smallskipamount]
1192 \hbox{}\qquad Free variables: \nopagebreak \\
1193 \hbox{}\qquad\qquad $\textit{xs} = [a_4, a_5]$ \\
1194 \hbox{}\qquad\qquad $\textit{ys} = [b_3, b_3]$ \\[2\smallskipamount]
1195 Total time: 1636 ms.
1198 In theory, it should be sufficient to test a single scope:
1201 \textbf{nitpick}~[\textit{card}~= 8]
1204 However, this is often less efficient in practice and may lead to overly complex
1207 If the monotonicity check fails but we believe that the formula is monotonic (or
1208 we don't mind missing some counterexamples), we can pass the
1209 \textit{mono} option. To convince yourself that this option is risky,
1210 simply consider this example from \S\ref{skolemization}:
1213 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1214 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1215 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1216 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1217 \textbf{nitpick} \\[2\smallskipamount]
1219 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1220 \hbox{}\qquad $\vdots$
1223 (It turns out the formula holds if and only if $\textit{card}~'a \le
1224 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1225 monotonicity checks can be disabled by passing \textit{non\_mono}
1226 (\S\ref{optimizations}).
1228 As insinuated in \S\ref{natural-numbers-and-integers} and
1229 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1230 are normally monotonic and treated as such. The same is true for record types,
1231 \textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
1232 cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
1233 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1234 consider only 8~scopes instead of $32\,768$.
1236 \section{Case Studies}
1237 \label{case-studies}
1239 As a didactic device, the previous section focused mostly on toy formulas whose
1240 validity can easily be assessed just by looking at the formula. We will now
1241 review two somewhat more realistic case studies that are within Nitpick's
1242 reach:\ a context-free grammar modeled by mutually inductive sets and a
1243 functional implementation of AA trees. The results presented in this
1244 section were produced with the following settings:
1247 \textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
1250 \subsection{A Context-Free Grammar}
1251 \label{a-context-free-grammar}
1253 Our first case study is taken from section 7.4 in the Isabelle tutorial
1254 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1255 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1258 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1259 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1260 $A$ & $::=$ & $aS \mid bAA$ \\
1261 $B$ & $::=$ & $bS \mid aBB$
1265 The intuition behind the grammar is that $A$ generates all string with one more
1266 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1268 The alphabet consists exclusively of $a$'s and $b$'s:
1271 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1274 Strings over the alphabet are represented by \textit{alphabet list}s.
1275 Nonterminals in the grammar become sets of strings. The production rules
1276 presented above can be expressed as a mutually inductive definition:
1279 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1280 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1281 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1282 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1283 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1284 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1285 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1288 The conversion of the grammar into the inductive definition was done manually by
1289 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1292 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1293 d'\^etre}. A good approach is to state desirable properties of the specification
1294 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1295 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1296 counterexamples will point to bugs in the specification. For our grammar
1297 example, we will proceed in two steps, separating the soundness and the
1298 completeness of the set $S$. First, soundness:
1301 \textbf{theorem}~\textit{S\_sound}: \\
1302 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1303 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1304 \textbf{nitpick} \\[2\smallskipamount]
1305 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1306 \hbox{}\qquad Free variable: \nopagebreak \\
1307 \hbox{}\qquad\qquad $w = [b]$
1310 It would seem that $[b] \in S$. How could this be? An inspection of the
1311 introduction rules reveals that the only rule with a right-hand side of the form
1312 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1316 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1319 On closer inspection, we can see that this rule is wrong. To match the
1320 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1324 \textbf{nitpick} \\[2\smallskipamount]
1325 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1326 \hbox{}\qquad Free variable: \nopagebreak \\
1327 \hbox{}\qquad\qquad $w = [a, a, b]$
1330 Some detective work is necessary to find out what went wrong here. To get $[a,
1331 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1335 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1338 Now, this formula must be wrong: The same assumption occurs twice, and the
1339 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1340 the assumptions should have been a $w$.
1342 With the correction made, we don't get any counterexample from Nitpick. Let's
1343 move on and check completeness:
1346 \textbf{theorem}~\textit{S\_complete}: \\
1347 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1348 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1349 \longrightarrow w \in S$'' \\
1350 \textbf{nitpick} \\[2\smallskipamount]
1351 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1352 \hbox{}\qquad Free variable: \nopagebreak \\
1353 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1356 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1357 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1358 the introduction rules we have are probably correct. Perhaps we simply lack an
1359 introduction rule. Comparing the grammar with the inductive definition, our
1360 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1361 without which the grammar cannot generate two or more $b$'s in a row. So we add
1365 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1368 With this last change, we don't get any counterexamples from Nitpick for either
1369 soundness or completeness. We can even generalize our result to cover $A$ and
1373 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1374 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1375 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1376 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1377 \textbf{nitpick} \\[2\smallskipamount]
1378 \slshape Nitpick found no counterexample.
1381 \subsection{AA Trees}
1384 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1385 similar performance to red-black trees, but with a simpler implementation
1386 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1387 a total order $<$. We start by defining the datatype and some basic extractor
1391 \textbf{datatype} $'a$~\textit{tree} = $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{tree}'' ``\kern1pt$'a$ \textit{tree}'' \\[2\smallskipamount]
1392 \textbf{primrec} \textit{data} \textbf{where} \\
1393 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1394 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1395 \textbf{primrec} \textit{dataset} \textbf{where} \\
1396 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1397 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1398 \textbf{primrec} \textit{level} \textbf{where} \\
1399 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1400 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1401 \textbf{primrec} \textit{left} \textbf{where} \\
1402 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1403 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1404 \textbf{primrec} \textit{right} \textbf{where} \\
1405 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1406 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1409 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1410 as follows \cite{wikipedia-2009-aa-trees}:
1412 \kern.2\parskip %% TYPESETTING
1415 Each node has a level field, and the following invariants must remain true for
1416 the tree to be valid:
1420 \kern-.4\parskip %% TYPESETTING
1425 \item[1.] The level of a leaf node is one.
1426 \item[2.] The level of a left child is strictly less than that of its parent.
1427 \item[3.] The level of a right child is less than or equal to that of its parent.
1428 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1429 \item[5.] Every node of level greater than one must have two children.
1434 \kern.4\parskip %% TYPESETTING
1436 The \textit{wf} predicate formalizes this description:
1439 \textbf{primrec} \textit{wf} \textbf{where} \\
1440 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1441 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1442 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1443 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1444 \phantom{``$($}$\textrm{else}$ \\
1445 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1446 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1447 \mathrel{\land} \textit{level}~u \le k$ \\
1448 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1451 Rebalancing the tree upon insertion and removal of elements is performed by two
1452 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1455 \textbf{primrec} \textit{skew} \textbf{where} \\
1456 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1457 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1458 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1459 \textit{level}~t~\textrm{then}$ \\
1460 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1461 (\textit{right}~t)~u)$ \\
1462 \phantom{``(}$\textrm{else}$ \\
1463 \phantom{``(\quad}$N~x~k~t~u)$''
1467 \textbf{primrec} \textit{split} \textbf{where} \\
1468 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1469 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1470 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1471 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1472 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1473 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1474 \phantom{``(}$\textrm{else}$ \\
1475 \phantom{``(\quad}$N~x~k~t~u)$''
1478 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1479 of elements stored in the tree:
1482 \textbf{theorem}~\textit{dataset\_skew\_split}:\\
1483 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1484 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1485 \textbf{nitpick} \\[2\smallskipamount]
1486 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1489 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1490 should not alter the tree:
1493 \textbf{theorem}~\textit{wf\_skew\_split}:\\
1494 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1495 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1496 \textbf{nitpick} \\[2\smallskipamount]
1497 {\slshape Nitpick found no counterexample.}
1500 Insertion is implemented recursively. It preserves the sort order:
1503 \textbf{primrec}~\textit{insort} \textbf{where} \\
1504 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1505 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1506 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1507 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1510 Notice that we deliberately commented out the application of \textit{skew} and
1511 \textit{split}. Let's see if this causes any problems:
1514 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1515 \textbf{nitpick} \\[2\smallskipamount]
1516 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1517 \hbox{}\qquad Free variables: \nopagebreak \\
1518 \hbox{}\qquad\qquad $t = N~a_3~1~\Lambda~\Lambda$ \\
1519 \hbox{}\qquad\qquad $x = a_4$ \\[2\smallskipamount]
1520 Hint: Maybe you forgot a type constraint?
1523 It's hard to see why this is a counterexample. The hint is of no help here. To
1524 improve readability, we will restrict the theorem to \textit{nat}, so that we
1525 don't need to look up the value of the $\textit{op}~{<}$ constant to find out
1526 which element is smaller than the other. In addition, we will tell Nitpick to
1527 display the value of $\textit{insort}~t~x$ using the \textit{eval} option. This
1531 \textbf{theorem} \textit{wf\_insort\_nat}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1532 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1533 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1534 \hbox{}\qquad Free variables: \nopagebreak \\
1535 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1536 \hbox{}\qquad\qquad $x = 0$ \\
1537 \hbox{}\qquad Evaluated term: \\
1538 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1541 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1542 where both have a level of 1. This violates the second AA tree invariant, which
1543 states that a left child's level must be less than its parent's. This shouldn't
1544 come as a surprise, considering that we commented out the tree rebalancing code.
1545 Reintroducing the code seems to solve the problem:
1548 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1549 \textbf{nitpick} \\[2\smallskipamount]
1550 {\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
1553 Insertion should transform the set of elements represented by the tree in the
1557 \textbf{theorem} \textit{dataset\_insort}:\kern.4em
1558 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1559 \textbf{nitpick} \\[2\smallskipamount]
1560 {\slshape Nitpick ran out of time after checking 5 of 8 scopes.}
1563 We could continue like this and sketch a complete theory of AA trees without
1564 performing a single proof. Once the definitions and main theorems are in place
1565 and have been thoroughly tested using Nitpick, we could start working on the
1566 proofs. Developing theories this way usually saves time, because faulty theorems
1567 and definitions are discovered much earlier in the process.
1569 \section{Option Reference}
1570 \label{option-reference}
1572 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1573 \def\qty#1{$\left<\textit{#1}\right>$}
1574 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1575 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1576 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1577 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1578 \def\ops#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1579 \def\opt#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1580 \def\opu#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1581 \def\opusmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1583 Nitpick's behavior can be influenced by various options, which can be specified
1584 in brackets after the \textbf{nitpick} command. Default values can be set
1585 using \textbf{nitpick\_\allowbreak params}. For example:
1588 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1591 The options are categorized as follows:\ mode of operation
1592 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1593 format (\S\ref{output-format}), automatic counterexample checks
1594 (\S\ref{authentication}), optimizations
1595 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1597 You can instruct Nitpick to run automatically on newly entered theorems by
1598 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1599 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
1600 and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1601 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1602 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1603 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1604 \textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
1605 Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
1608 The number of options can be overwhelming at first glance. Do not let that worry
1609 you: Nitpick's defaults have been chosen so that it almost always does the right
1610 thing, and the most important options have been covered in context in
1611 \S\ref{first-steps}.
1613 The descriptions below refer to the following syntactic quantities:
1616 \item[$\bullet$] \qtybf{string}: A string.
1617 \item[$\bullet$] \qtybf{bool}: \textit{true} or \textit{false}.
1618 \item[$\bullet$] \qtybf{bool\_or\_smart}: \textit{true}, \textit{false}, or \textit{smart}.
1619 \item[$\bullet$] \qtybf{int}: An integer. Negative integers are prefixed with a hyphen.
1620 \item[$\bullet$] \qtybf{int\_or\_smart}: An integer or \textit{smart}.
1621 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1622 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1624 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1625 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1626 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1627 \item[$\bullet$] \qtybf{const}: The name of a HOL constant.
1628 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1629 \item[$\bullet$] \qtybf{term\_list}: A space-separated list of HOL terms (e.g.,
1630 ``$f~x$''~``$g~y$'').
1631 \item[$\bullet$] \qtybf{type}: A HOL type.
1634 Default values are indicated in square brackets. Boolean options have a negated
1635 counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
1636 Boolean options, ``= \textit{true}'' may be omitted.
1638 \subsection{Mode of Operation}
1639 \label{mode-of-operation}
1642 \optrue{blocking}{non\_blocking}
1643 Specifies whether the \textbf{nitpick} command should operate synchronously.
1644 The asynchronous (non-blocking) mode lets the user start proving the putative
1645 theorem while Nitpick looks for a counterexample, but it can also be more
1646 confusing. For technical reasons, automatic runs currently always block.
1648 \optrue{falsify}{satisfy}
1649 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1650 satisfying examples (models). This manual assumes throughout that
1651 \textit{falsify} is enabled.
1653 \opsmart{user\_axioms}{no\_user\_axioms}
1654 Specifies whether the user-defined axioms (specified using
1655 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1656 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1657 the constants that occur in the formula to falsify. The option is implicitly set
1658 to \textit{true} for automatic runs.
1660 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1661 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1662 these conditions are tagged as ``likely genuine.'' The \textit{debug}
1663 (\S\ref{output-format}) option can be used to find out which axioms were
1667 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1668 (\S\ref{output-format}).}
1670 \optrue{assms}{no\_assms}
1671 Specifies whether the relevant assumptions in structured proof should be
1672 considered. The option is implicitly enabled for automatic runs.
1675 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1677 \opfalse{overlord}{no\_overlord}
1678 Specifies whether Nitpick should put its temporary files in
1679 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1680 debugging Nitpick but also unsafe if several instances of the tool are run
1684 {\small See also \textit{debug} (\S\ref{output-format}).}
1687 \subsection{Scope of Search}
1688 \label{scope-of-search}
1691 \opu{card}{type}{int\_seq}
1692 Specifies the sequence of cardinalities to use for a given type. For
1693 \textit{nat} and \textit{int}, the cardinality fully specifies the subset used
1694 to approximate the type. For example:
1696 $$\hbox{\begin{tabular}{@{}rll@{}}%
1697 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
1698 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
1699 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
1704 $$\hbox{\begin{tabular}{@{}rll@{}}%
1705 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
1706 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
1709 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1710 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1711 Although function and product types are normally mapped directly to the
1712 corresponding Kodkod concepts, setting
1713 the cardinality of such types is also allowed and implicitly enables ``boxing''
1714 for them, as explained in the description of the \textit{box}~\qty{type}
1715 and \textit{box} (\S\ref{scope-of-search}) options.
1718 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1720 \opt{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
1721 Specifies the default sequence of cardinalities to use. This can be overridden
1722 on a per-type basis using the \textit{card}~\qty{type} option described above.
1724 \opu{max}{const}{int\_seq}
1725 Specifies the sequence of maximum multiplicities to use for a given
1726 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1727 number of distinct values that it can construct. Nonsensical values (e.g.,
1728 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1729 datatypes equipped with several constructors.
1732 Specifies the default sequence of maximum multiplicities to use for
1733 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
1734 basis using the \textit{max}~\qty{const} option described above.
1736 \opusmart{wf}{const}{non\_wf}
1737 Specifies whether the specified (co)in\-duc\-tively defined predicate is
1738 well-founded. The option can take the following values:
1741 \item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
1742 predicate as if it were well-founded. Since this is generally not sound when the
1743 predicate is not well-founded, the counterexamples are tagged as ``likely
1746 \item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
1747 as if it were not well-founded. The predicate is then unrolled as prescribed by
1748 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
1751 \item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
1752 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
1753 \textit{sizechange} tactics. If this succeeds (or the predicate occurs with an
1754 appropriate polarity in the formula to falsify), use an efficient fixed point
1755 equation as specification of the predicate; otherwise, unroll the predicates
1756 according to the \textit{iter}~\qty{const} and \textit{iter} options.
1760 {\small See also \textit{iter} (\S\ref{scope-of-search}),
1761 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
1762 (\S\ref{timeouts}).}
1764 \opsmart{wf}{non\_wf}
1765 Specifies the default wellfoundedness setting to use. This can be overridden on
1766 a per-predicate basis using the \textit{wf}~\qty{const} option above.
1768 \opu{iter}{const}{int\_seq}
1769 Specifies the sequence of iteration counts to use when unrolling a given
1770 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
1771 predicates that occur negatively and coinductive predicates that occur
1772 positively in the formula to falsify and that cannot be proved to be
1773 well-founded, but this behavior is influenced by the \textit{wf} option. The
1774 iteration counts are automatically bounded by the cardinality of the predicate's
1777 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
1778 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
1780 \opt{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
1781 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
1782 predicates. This can be overridden on a per-predicate basis using the
1783 \textit{iter} \qty{const} option above.
1785 \opt{bisim\_depth}{int\_seq}{$\mathbf{7}$}
1786 Specifies the sequence of iteration counts to use when unrolling the
1787 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
1788 of $-1$ means that no predicate is generated, in which case Nitpick performs an
1789 after-the-fact check to see if the known coinductive datatype values are
1790 bidissimilar. If two values are found to be bisimilar, the counterexample is
1791 tagged as ``likely genuine.'' The iteration counts are automatically bounded by
1792 the sum of the cardinalities of the coinductive datatypes occurring in the
1795 \opusmart{box}{type}{dont\_box}
1796 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
1797 product type in an isomorphic datatype internally. Boxing is an effective mean
1798 to reduce the search space and speed up Nitpick, because the isomorphic datatype
1799 is approximated by a subset of the possible function or pair values;
1800 like other drastic optimizations, it can also prevent the discovery of
1801 counterexamples. The option can take the following values:
1804 \item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
1806 \item[$\bullet$] \textbf{\textit{false}}: Never box the type.
1807 \item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
1808 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
1809 higher-order functions are good candidates for boxing.
1812 Setting the \textit{card}~\qty{type} option for a function or product type
1813 implicitly enables boxing for that type.
1816 {\small See also \textit{verbose} (\S\ref{output-format})
1817 and \textit{debug} (\S\ref{output-format}).}
1819 \opsmart{box}{dont\_box}
1820 Specifies the default boxing setting to use. This can be overridden on a
1821 per-type basis using the \textit{box}~\qty{type} option described above.
1823 \opusmart{mono}{type}{non\_mono}
1824 Specifies whether the specified type should be considered monotonic when
1825 enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a
1826 monotonicity check on the type. Setting this option to \textit{true} can reduce
1827 the number of scopes tried, but it also diminishes the theoretical chance of
1828 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
1831 {\small See also \textit{card} (\S\ref{scope-of-search}),
1832 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
1833 (\S\ref{output-format}).}
1835 \opsmart{mono}{non\_box}
1836 Specifies the default monotonicity setting to use. This can be overridden on a
1837 per-type basis using the \textit{mono}~\qty{type} option described above.
1839 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
1840 Specifies whether type variables with the same sort constraints should be
1841 merged. Setting this option to \textit{true} can reduce the number of scopes
1842 tried and the size of the generated Kodkod formulas, but it also diminishes the
1843 theoretical chance of finding a counterexample.
1845 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1848 \subsection{Output Format}
1849 \label{output-format}
1852 \opfalse{verbose}{quiet}
1853 Specifies whether the \textbf{nitpick} command should explain what it does. This
1854 option is useful to determine which scopes are tried or which SAT solver is
1855 used. This option is implicitly disabled for automatic runs.
1857 \opfalse{debug}{no\_debug}
1858 Specifies whether Nitpick should display additional debugging information beyond
1859 what \textit{verbose} already displays. Enabling \textit{debug} also enables
1860 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
1861 option is implicitly disabled for automatic runs.
1864 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
1865 \textit{batch\_size} (\S\ref{optimizations}).}
1867 \optrue{show\_skolems}{hide\_skolem}
1868 Specifies whether the values of Skolem constants should be displayed as part of
1869 counterexamples. Skolem constants correspond to bound variables in the original
1870 formula and usually help us to understand why the counterexample falsifies the
1874 {\small See also \textit{skolemize} (\S\ref{optimizations}).}
1876 \opfalse{show\_datatypes}{hide\_datatypes}
1877 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
1878 be displayed as part of counterexamples. Such subsets are sometimes helpful when
1879 investigating whether a potential counterexample is genuine or spurious, but
1880 their potential for clutter is real.
1882 \opfalse{show\_consts}{hide\_consts}
1883 Specifies whether the values of constants occurring in the formula (including
1884 its axioms) should be displayed along with any counterexample. These values are
1885 sometimes helpful when investigating why a counterexample is
1886 genuine, but they can clutter the output.
1888 \opfalse{show\_all}{dont\_show\_all}
1889 Enabling this option effectively enables \textit{show\_skolems},
1890 \textit{show\_datatypes}, and \textit{show\_consts}.
1892 \opt{max\_potential}{int}{$\mathbf{1}$}
1893 Specifies the maximum number of potential counterexamples to display. Setting
1894 this option to 0 speeds up the search for a genuine counterexample. This option
1895 is implicitly set to 0 for automatic runs. If you set this option to a value
1896 greater than 1, you will need an incremental SAT solver: For efficiency, it is
1897 recommended to install the JNI version of MiniSat and set \textit{sat\_solver} =
1898 \textit{MiniSatJNI}. Also be aware that many of the counterexamples may look
1899 identical, unless the \textit{show\_all} (\S\ref{output-format}) option is
1903 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
1904 \textit{sat\_solver} (\S\ref{optimizations}).}
1906 \opt{max\_genuine}{int}{$\mathbf{1}$}
1907 Specifies the maximum number of genuine counterexamples to display. If you set
1908 this option to a value greater than 1, you will need an incremental SAT solver:
1909 For efficiency, it is recommended to install the JNI version of MiniSat and set
1910 \textit{sat\_solver} = \textit{MiniSatJNI}. Also be aware that many of the
1911 counterexamples may look identical, unless the \textit{show\_all}
1912 (\S\ref{output-format}) option is enabled.
1915 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
1916 \textit{sat\_solver} (\S\ref{optimizations}).}
1918 \ops{eval}{term\_list}
1919 Specifies the list of terms whose values should be displayed along with
1920 counterexamples. This option suffers from an ``observer effect'': Nitpick might
1921 find different counterexamples for different values of this option.
1923 \opu{format}{term}{int\_seq}
1924 Specifies how to uncurry the value displayed for a variable or constant.
1925 Uncurrying sometimes increases the readability of the output for high-arity
1926 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
1927 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
1928 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
1929 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
1930 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
1931 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
1932 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
1933 arguments that are not accounted for are left alone, as if the specification had
1934 been $1,\ldots,1,n_1,\ldots,n_k$.
1937 {\small See also \textit{uncurry} (\S\ref{optimizations}).}
1939 \opt{format}{int\_seq}{$\mathbf{1}$}
1940 Specifies the default format to use. Irrespective of the default format, the
1941 extra arguments to a Skolem constant corresponding to the outer bound variables
1942 are kept separated from the remaining arguments, the \textbf{for} arguments of
1943 an inductive definitions are kept separated from the remaining arguments, and
1944 the iteration counter of an unrolled inductive definition is shown alone. The
1945 default format can be overridden on a per-variable or per-constant basis using
1946 the \textit{format}~\qty{term} option described above.
1949 %% MARK: Authentication
1950 \subsection{Authentication}
1951 \label{authentication}
1954 \opfalse{check\_potential}{trust\_potential}
1955 Specifies whether potential counterexamples should be given to Isabelle's
1956 \textit{auto} tactic to assess their validity. If a potential counterexample is
1957 shown to be genuine, Nitpick displays a message to this effect and terminates.
1960 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
1962 \opfalse{check\_genuine}{trust\_genuine}
1963 Specifies whether genuine and likely genuine counterexamples should be given to
1964 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
1965 counterexample is shown to be spurious, the user is kindly asked to send a bug
1966 report to the author at
1967 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
1970 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
1972 \ops{expect}{string}
1973 Specifies the expected outcome, which must be one of the following:
1976 \item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
1977 \item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a ``likely
1978 genuine'' counterexample (i.e., a counterexample that is genuine unless
1979 it contradicts a missing axiom or a dangerous option was used inappropriately).
1980 \item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
1981 \item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
1982 \item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
1983 Kodkod ran out of memory).
1986 Nitpick emits an error if the actual outcome differs from the expected outcome.
1987 This option is useful for regression testing.
1990 \subsection{Optimizations}
1991 \label{optimizations}
1993 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
1998 \opt{sat\_solver}{string}{smart}
1999 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2000 to be faster than their Java counterparts, but they can be more difficult to
2001 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2002 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2003 you will need an incremental SAT solver, such as \textit{MiniSatJNI}
2004 (recommended) or \textit{SAT4J}.
2006 The supported solvers are listed below:
2010 \item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
2011 written in \cpp{}. To use MiniSat, set the environment variable
2012 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2013 executable. The \cpp{} sources and executables for MiniSat are available at
2014 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2015 and 2.0 beta (2007-07-21).
2017 \item[$\bullet$] \textbf{\textit{MiniSatJNI}}: The JNI (Java Native Interface)
2018 version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which
2019 you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2020 version of MiniSat, the JNI version can be used incrementally.
2022 \item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
2023 written in C. It is bundled with Kodkodi and requires no further installation or
2024 configuration steps. Alternatively, you can install a standard version of
2025 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2026 that contains the \texttt{picosat} executable. The C sources for PicoSAT are
2027 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2028 Nitpick has been tested with version 913.
2030 \item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
2031 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2032 the directory that contains the \texttt{zchaff} executable. The \cpp{} sources
2033 and executables for zChaff are available at
2034 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2035 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2037 \item[$\bullet$] \textbf{\textit{zChaffJNI}}: The JNI version of zChaff is
2038 bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on
2039 Kodkod's web site \cite{kodkod-2009}.
2041 \item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
2042 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2043 directory that contains the \texttt{rsat} executable. The \cpp{} sources for
2044 RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been
2045 tested with version 2.01.
2047 \item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
2048 written in C. To use BerkMin, set the environment variable
2049 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2050 executable. The BerkMin executables are available at
2051 \url{http://eigold.tripod.com/BerkMin.html}.
2053 \item[$\bullet$] \textbf{\textit{BerkMinAlloy}}: Variant of BerkMin that is
2054 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2055 version of BerkMin, set the environment variable
2056 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2059 \item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
2060 written in C. To use Jerusat, set the environment variable
2061 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2062 executable. The C sources for Jerusat are available at
2063 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2065 \item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
2066 written in Java that can be used incrementally. It is bundled with Kodkodi and
2067 requires no further installation or configuration steps. Do not attempt to
2068 install the official SAT4J packages, because their API is incompatible with
2071 \item[$\bullet$] \textbf{\textit{SAT4JLight}}: Variant of SAT4J that is
2072 optimized for small problems. It can also be used incrementally.
2074 \item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
2075 experimental solver written in \cpp. To use HaifaSat, set the environment
2076 variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
2077 \texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at
2078 \url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
2080 \item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
2081 \textit{smart}, Nitpick selects the first solver among MiniSatJNI, MiniSat,
2082 PicoSAT, zChaffJNI, zChaff, RSat, BerkMin, BerkMinAlloy, and Jerusat that is
2083 recognized by Isabelle. If none is found, it falls back on SAT4J, which should
2084 always be available. If \textit{verbose} is enabled, Nitpick displays which SAT
2089 \opt{batch\_size}{int\_or\_smart}{smart}
2090 Specifies the maximum number of Kodkod problems that should be lumped together
2091 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2092 together ensures that Kodkodi is launched less often, but it makes the verbose
2093 output less readable and is sometimes detrimental to performance. If
2094 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2095 \textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
2097 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2098 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2099 be rewritten to use (automatically generated) discriminators and destructors.
2100 This optimization can drastically reduce the size of the Boolean formulas given
2104 {\small See also \textit{debug} (\S\ref{output-format}).}
2106 \optrue{specialize}{dont\_specialize}
2107 Specifies whether functions invoked with static arguments should be specialized.
2108 This optimization can drastically reduce the search space, especially for
2109 higher-order functions.
2112 {\small See also \textit{debug} (\S\ref{output-format}) and
2113 \textit{show\_consts} (\S\ref{output-format}).}
2115 \optrue{skolemize}{dont\_skolemize}
2116 Specifies whether the formula should be skolemized. For performance reasons,
2117 (positive) $\forall$-quanti\-fiers that occur in the scope of a higher-order
2118 (positive) $\exists$-quanti\-fier are left unchanged.
2121 {\small See also \textit{debug} (\S\ref{output-format}) and
2122 \textit{show\_skolems} (\S\ref{output-format}).}
2124 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2125 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2126 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2127 predicates for which each the predicate occurs in at most one assumption of each
2128 introduction rule. Using the reflexive transitive closure is in principle
2129 equivalent to setting \textit{iter} to the cardinality of the predicate's
2130 domain, but it is usually more efficient.
2132 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2133 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2135 \optrue{uncurry}{dont\_uncurry}
2136 Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no
2137 tangible effect on efficiency, but it creates opportunities for the boxing
2141 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug}
2142 (\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).}
2144 \optrue{fast\_descrs}{full\_descrs}
2145 Specifies whether Nitpick should optimize the definite and indefinite
2146 description operators (THE and SOME). The optimized versions usually help
2147 Nitpick generate more counterexamples or at least find them faster, but only the
2148 unoptimized versions are complete when all types occurring in the formula are
2151 {\small See also \textit{debug} (\S\ref{output-format}).}
2153 \optrue{peephole\_optim}{no\_peephole\_optim}
2154 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2155 peephole optimizer. These optimizations can make a significant difference.
2156 Unless you are tracking down a bug in Nitpick or distrust the peephole
2157 optimizer, you should leave this option enabled.
2159 \opt{sym\_break}{int}{20}
2160 Specifies an upper bound on the number of relations for which Kodkod generates
2161 symmetry breaking predicates. According to the Kodkod documentation
2162 \cite{kodkod-2009-options}, ``in general, the higher this value, the more
2163 symmetries will be broken, and the faster the formula will be solved. But,
2164 setting the value too high may have the opposite effect and slow down the
2167 \opt{sharing\_depth}{int}{3}
2168 Specifies the depth to which Kodkod should check circuits for equivalence during
2169 the translation to SAT. The default of 3 is the same as in Alloy. The minimum
2170 allowed depth is 1. Increasing the sharing may result in a smaller SAT problem,
2171 but can also slow down Kodkod.
2173 \opfalse{flatten\_props}{dont\_flatten\_props}
2174 Specifies whether Kodkod should try to eliminate intermediate Boolean variables.
2175 Although this might sound like a good idea, in practice it can drastically slow
2178 \opt{max\_threads}{int}{0}
2179 Specifies the maximum number of threads to use in Kodkod. If this option is set
2180 to 0, Kodkod will compute an appropriate value based on the number of processor
2184 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2185 \textit{timeout} (\S\ref{timeouts}).}
2188 \subsection{Timeouts}
2192 \opt{timeout}{time}{$\mathbf{30}$ s}
2193 Specifies the maximum amount of time that the \textbf{nitpick} command should
2194 spend looking for a counterexample. Nitpick tries to honor this constraint as
2195 well as it can but offers no guarantees. For automatic runs,
2196 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2197 a time slot whose length is specified by the ``Auto Counterexample Time
2198 Limit'' option in Proof General.
2201 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2203 \opt{tac\_timeout}{time}{$\mathbf{500}$\,ms}
2204 Specifies the maximum amount of time that the \textit{auto} tactic should use
2205 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2206 and \textit{sizechange} should use when checking whether a (co)in\-duc\-tive
2207 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2208 can but offers no guarantees.
2211 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2212 \textit{check\_potential} (\S\ref{authentication}),
2213 and \textit{check\_genuine} (\S\ref{authentication}).}
2216 \section{Attribute Reference}
2217 \label{attribute-reference}
2219 Nitpick needs to consider the definitions of all constants occurring in a
2220 formula in order to falsify it. For constants introduced using the
2221 \textbf{definition} command, the definition is simply the associated
2222 \textit{\_def} axiom. In contrast, instead of using the internal representation
2223 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2224 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2225 equational specification entered by the user.
2227 Behind the scenes, Isabelle's built-in packages and theories rely on the
2228 following attributes to affect Nitpick's behavior:
2231 \flushitem{\textit{nitpick\_def}}
2234 This attribute specifies an alternative definition of a constant. The
2235 alternative definition should be logically equivalent to the constant's actual
2236 axiomatic definition and should be of the form
2238 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2240 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2243 \flushitem{\textit{nitpick\_simp}}
2246 This attribute specifies the equations that constitute the specification of a
2247 constant. For functions defined using the \textbf{primrec}, \textbf{function},
2248 and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
2249 \textit{simps} rules. The equations must be of the form
2251 \qquad $c~t_1~\ldots\ t_n \,=\, u.$
2253 \flushitem{\textit{nitpick\_psimp}}
2256 This attribute specifies the equations that constitute the partial specification
2257 of a constant. For functions defined using the \textbf{function} package, this
2258 corresponds to the \textit{psimps} rules. The conditional equations must be of
2261 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
2263 \flushitem{\textit{nitpick\_intro}}
2266 This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
2267 For predicates defined using the \textbf{inductive} or \textbf{coinductive}
2268 command, this corresponds to the \textit{intros} rules. The introduction rules
2271 \qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
2272 \ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\
2275 where the $P_i$'s are side conditions that do not involve $c$ and $M$ is an
2276 optional monotonic operator. The order of the assumptions is irrelevant.
2280 When faced with a constant, Nitpick proceeds as follows:
2283 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2284 is not empty, Nitpick uses these rules as the specification of the constant.
2286 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2287 the constant is not empty, it uses these rules as the specification of the
2290 \item[3.] Otherwise, it looks up the definition of the constant:
2293 \item[1.] If the \textit{nitpick\_def} set associated with the constant
2294 is not empty, it uses the latest rule added to the set as the definition of the
2295 constant; otherwise it uses the actual definition axiom.
2296 \item[2.] If the definition is of the form
2298 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
2300 then Nitpick assumes that the definition was made using an inductive package and
2301 based on the introduction rules marked with \textit{nitpick\_\allowbreak
2302 ind\_\allowbreak intros} tries to determine whether the definition is
2307 As an illustration, consider the inductive definition
2310 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2311 ``\textit{odd}~1'' $\,\mid$ \\
2312 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2315 Isabelle automatically attaches the \textit{nitpick\_intro} attribute to
2316 the above rules. Nitpick then uses the \textit{lfp}-based definition in
2317 conjunction with these rules. To override this, we can specify an alternative
2318 definition as follows:
2321 \textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]: ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2324 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2325 = 1$. Alternatively, we can specify an equational specification of the constant:
2328 \textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]: ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2331 Such tweaks should be done with great care, because Nitpick will assume that the
2332 constant is completely defined by its equational specification. For example, if
2333 you make ``$\textit{odd}~(2 * k + 1)$'' a \textit{nitpick\_simp} rule and neglect to provide rules to handle the $2 * k$ case, Nitpick will define
2334 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2335 (\S\ref{output-format}) option is extremely useful to understand what is going
2336 on when experimenting with \textit{nitpick\_} attributes.
2338 \section{Standard ML Interface}
2339 \label{standard-ml-interface}
2341 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2342 and debugging. Among the most interesting functions exported by Nitpick are
2343 those that let you invoke the tool programmatically and those that let you
2344 register and unregister custom coinductive datatypes.
2346 \subsection{Invocation of Nitpick}
2347 \label{invocation-of-nitpick}
2349 The \textit{Nitpick} structure offers the following functions for invoking your
2350 favorite counterexample generator:
2353 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2354 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2355 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2356 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2357 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2360 The return value is a new proof state paired with an outcome string
2361 (``genuine'', ``likely\_genuine'', ``potential'', ``none'', or ``unknown''). The
2362 \textit{params} type is a large record that lets you set Nitpick's options. The
2363 current default options can be retrieved by calling the following function
2364 defined in the \textit{Nitpick\_Isar} structure:
2367 $\textbf{val}\,~\textit{default\_params} :\,
2368 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2371 The second argument lets you override option values before they are parsed and
2372 put into a \textit{params} record. Here is an example:
2375 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2376 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2377 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2378 & \textit{subgoal}\end{aligned}$
2383 \subsection{Registration of Coinductive Datatypes}
2384 \label{registration-of-coinductive-datatypes}
2386 If you have defined a custom coinductive datatype, you can tell Nitpick about
2387 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2388 uses for lazy lists. The interface for registering and unregistering coinductive
2389 datatypes consists of the following pair of functions defined in the
2390 \textit{Nitpick} structure:
2393 $\textbf{val}\,~\textit{register\_codatatype} :\,
2394 \textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2395 $\textbf{val}\,~\textit{unregister\_codatatype} :\,
2396 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2399 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2400 not been, you could have told Nitpick about it by adding the following line
2401 to your theory file:
2404 $\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
2405 & \textit{Nitpick.register\_codatatype} \\[-2pt]
2406 & \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
2407 & \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
2410 The \textit{register\_codatatype} function takes a coinductive type, its case
2411 function, and the list of its constructors. The case function must take its
2412 arguments in the order that the constructors are listed. If no case function
2413 with the correct signature is available, simply pass the empty string.
2415 On the other hand, if your goal is to cripple Nitpick, add the following line to
2416 your theory file and try to check a few conjectures about lazy lists:
2419 $\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
2420 \kern1pt'a~\textit{list}\textrm{''}\}\ \,{*}\}$
2423 Inductive datatypes can be registered as coinductive datatypes, given
2424 appropriate coinductive constructors. However, doing so precludes
2425 the use of the inductive constructors---Nitpick will generate an error if they
2428 \section{Known Bugs and Limitations}
2429 \label{known-bugs-and-limitations}
2431 Here are the known bugs and limitations in Nitpick at the time of writing:
2434 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2435 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2436 Nitpick to generate spurious counterexamples for theorems that refer to values
2437 for which the function is not defined. For example:
2440 \textbf{primrec} \textit{prec} \textbf{where} \\
2441 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2442 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2443 \textbf{nitpick} \\[2\smallskipamount]
2444 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2446 \\[2\smallskipamount]
2447 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2448 \textbf{by}~(\textit{auto simp}: \textit{prec\_def})
2451 Such theorems are considered bad style because they rely on the internal
2452 representation of functions synthesized by Isabelle, which is an implementation
2455 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2456 which can become invalid if you change the definition of an inductive predicate
2457 that is registered in the cache. To clear the cache,
2458 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2459 501$\,\textit{ms}$).
2461 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2462 \textbf{guess} command in a structured proof.
2464 \item[$\bullet$] The \textit{nitpick\_} attributes and the
2465 \textit{Nitpick.register\_} functions can cause havoc if used improperly.
2467 \item[$\bullet$] Although this has never been observed, arbitrary theorem
2468 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
2470 \item[$\bullet$] Local definitions are not supported and result in an error.
2472 \item[$\bullet$] All constants and types whose names start with
2473 \textit{Nitpick}{.} are reserved for internal use.
2477 \bibliography{../manual}{}
2478 \bibliographystyle{abbrv}