1 \documentclass[a4paper,12pt]{article}
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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}
49 Jasmin Christian Blanchette \\
50 {\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\
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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's approach is to consider finite
440 subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined
441 value (displayed as `$\unk$'). The type \textit{int} is handled similarly.
442 Internally, undefined values lead to a three-valued logic.
444 Here is an example involving \textit{int}:
447 \textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
448 \textbf{nitpick} \\[2\smallskipamount]
449 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
450 \hbox{}\qquad Free variables: \nopagebreak \\
451 \hbox{}\qquad\qquad $i = 0$ \\
452 \hbox{}\qquad\qquad $j = 1$ \\
453 \hbox{}\qquad\qquad $m = 1$ \\
454 \hbox{}\qquad\qquad $n = 0$
457 Internally, Nitpick uses either a unary or a binary representation of numbers.
458 The unary representation is more efficient but only suitable for numbers very
459 close to zero. By default, Nitpick attempts to choose the more appropriate
460 encoding by inspecting the formula at hand. This behavior can be overridden by
461 passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For
462 binary notation, the number of bits to use can be specified using
463 the \textit{bits} option. For example:
466 \textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$]
469 With infinite types, we don't always have the luxury of a genuine counterexample
470 and must often content ourselves with a potential one. The tedious task of
471 finding out whether the potential counterexample is in fact genuine can be
472 outsourced to \textit{auto} by passing \textit{check\_potential}. For example:
475 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
476 \textbf{nitpick} [\textit{card~nat}~= 100, \textit{check\_potential}] \\[2\smallskipamount]
477 \slshape Nitpick found a potential counterexample: \\[2\smallskipamount]
478 \hbox{}\qquad Free variable: \nopagebreak \\
479 \hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
480 Confirmation by ``\textit{auto}'': The above counterexample is genuine.
483 You might wonder why the counterexample is first reported as potential. The root
484 of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
485 \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
486 that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
487 \textit{False}; but otherwise, it does not know anything about values of $n \ge
488 \textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
489 \textit{True}. Since the assumption can never be satisfied, the putative lemma
490 can never be falsified.
492 Incidentally, if you distrust the so-called genuine counterexamples, you can
493 enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
494 aware that \textit{auto} will usually fail to prove that the counterexample is
497 Some conjectures involving elementary number theory make Nitpick look like a
498 giant with feet of clay:
501 \textbf{lemma} ``$P~\textit{Suc}$'' \\
502 \textbf{nitpick} [\textit{card} = 1--6] \\[2\smallskipamount]
504 Nitpick found no counterexample.
507 On any finite set $N$, \textit{Suc} is a partial function; for example, if $N =
508 \{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\,
509 \ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when passed as
510 argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$. The next
514 \textbf{lemma} ``$P~(\textit{op}~{+}\Colon
515 \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
516 \textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
517 {\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
518 \hbox{}\qquad Free variable: \nopagebreak \\
519 \hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
520 \textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
521 {\slshape Nitpick found no counterexample.}
524 The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
525 $\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
528 Because numbers are infinite and are approximated using a three-valued logic,
529 there is usually no need to systematically enumerate domain sizes. If Nitpick
530 cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
531 unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
532 example above is an exception to this principle.) Nitpick nonetheless enumerates
533 all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
534 cardinalities are fast to handle and give rise to simpler counterexamples. This
535 is explained in more detail in \S\ref{scope-monotonicity}.
537 \subsection{Inductive Datatypes}
538 \label{inductive-datatypes}
540 Like natural numbers and integers, inductive datatypes with recursive
541 constructors admit no finite models and must be approximated by a subterm-closed
542 subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
543 Nitpick looks for all counterexamples that can be built using at most 10
546 Let's see with an example involving \textit{hd} (which returns the first element
547 of a list) and $@$ (which concatenates two lists):
550 \textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
551 \textbf{nitpick} \\[2\smallskipamount]
552 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
553 \hbox{}\qquad Free variables: \nopagebreak \\
554 \hbox{}\qquad\qquad $\textit{xs} = []$ \\
555 \hbox{}\qquad\qquad $\textit{y} = a_3$
558 To see why the counterexample is genuine, we enable \textit{show\_consts}
559 and \textit{show\_\allowbreak datatypes}:
562 {\slshape Datatype:} \\
563 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_3, a_3],\, [a_3],\, \unr\}$ \\
564 {\slshape Constants:} \\
565 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_3, a_3],\> [a_3, a_3] := \unk,\> [a_3] := \unk)$ \\
566 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_3, a_3] := a_3,\> [a_3] := a_3)$
569 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
572 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
573 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_3,
574 a_3]$ to $[a_3]$ would normally give $[a_3, a_3, a_3]$, but this value is not
575 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
576 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
577 appending $[a_3, a_3]$ to itself gives $\unk$.
579 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
580 considers the following subsets:
582 \kern-.5\smallskipamount %% TYPESETTING
586 $\{[],\, [a_1],\, [a_2]\}$; \\
587 $\{[],\, [a_1],\, [a_3]\}$; \\
588 $\{[],\, [a_2],\, [a_3]\}$; \\
589 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\
590 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\
591 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\
592 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\
593 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\
594 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\
595 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\
596 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\
597 $\{[],\, [a_3],\, [a_3, a_3]\}$.
601 \kern-2\smallskipamount %% TYPESETTING
603 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
604 are listed and only those. As an example of a non-subterm-closed subset,
605 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_3]\}$, and observe
606 that $[a_1, a_3]$ (i.e., $a_1 \mathbin{\#} [a_3]$) has $[a_3] \notin
607 \mathcal{S}$ as a subterm.
609 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
612 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
613 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
615 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
616 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
617 \hbox{}\qquad Free variables: \nopagebreak \\
618 \hbox{}\qquad\qquad $\textit{xs} = [a_2]$ \\
619 \hbox{}\qquad\qquad $\textit{ys} = [a_3]$ \\
620 \hbox{}\qquad Datatypes: \\
621 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
622 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_3],\, [a_2],\, \unr\}$
625 Because datatypes are approximated using a three-valued logic, there is usually
626 no need to systematically enumerate cardinalities: If Nitpick cannot find a
627 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
628 unlikely that one could be found for smaller cardinalities.
630 \subsection{Typedefs, Records, Rationals, and Reals}
631 \label{typedefs-records-rationals-and-reals}
633 Nitpick generally treats types declared using \textbf{typedef} as datatypes
634 whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
638 \textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
639 \textbf{by}~\textit{blast} \\[2\smallskipamount]
640 \textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
641 \textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
642 \textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
643 \textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
644 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
645 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
646 \hbox{}\qquad Free variables: \nopagebreak \\
647 \hbox{}\qquad\qquad $P = \{\Abs{1},\, \Abs{0}\}$ \\
648 \hbox{}\qquad\qquad $x = \Abs{2}$ \\
649 \hbox{}\qquad Datatypes: \\
650 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
651 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{2},\, \Abs{1},\, \Abs{0},\, \unr\}$
655 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
658 Records, which are implemented as \textbf{typedef}s behind the scenes, are
659 handled in much the same way:
662 \textbf{record} \textit{point} = \\
663 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
664 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
665 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
666 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
667 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
668 \hbox{}\qquad Free variables: \nopagebreak \\
669 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
670 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
671 \hbox{}\qquad Datatypes: \\
672 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
673 \hbox{}\qquad\qquad $\textit{point} = \{\lparr\textit{Xcoord} = 1,\>
674 \textit{Ycoord} = 1\rparr,\> \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr,\, \unr\}$\kern-1pt %% QUIET
677 Finally, Nitpick provides rudimentary support for rationals and reals using a
681 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
682 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
683 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
684 \hbox{}\qquad Free variables: \nopagebreak \\
685 \hbox{}\qquad\qquad $x = 1/2$ \\
686 \hbox{}\qquad\qquad $y = -1/2$ \\
687 \hbox{}\qquad Datatypes: \\
688 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
689 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
690 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
693 \subsection{Inductive and Coinductive Predicates}
694 \label{inductive-and-coinductive-predicates}
696 Inductively defined predicates (and sets) are particularly problematic for
697 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
698 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
699 the problem is that they are defined using a least fixed point construction.
701 Nitpick's philosophy is that not all inductive predicates are equal. Consider
702 the \textit{even} predicate below:
705 \textbf{inductive}~\textit{even}~\textbf{where} \\
706 ``\textit{even}~0'' $\,\mid$ \\
707 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
710 This predicate enjoys the desirable property of being well-founded, which means
711 that the introduction rules don't give rise to infinite chains of the form
714 $\cdots\,\Longrightarrow\, \textit{even}~k''
715 \,\Longrightarrow\, \textit{even}~k'
716 \,\Longrightarrow\, \textit{even}~k.$
719 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
723 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
724 \,\Longrightarrow\, \textit{even}~(k - 2)
725 \,\Longrightarrow\, \textit{even}~k.$
728 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
729 fixed point computation.%
730 \footnote{If an inductive predicate is
731 well-founded, then it has exactly one fixed point, which is simultaneously the
732 least and the greatest fixed point. In these circumstances, the computation of
733 the least fixed point amounts to the computation of an arbitrary fixed point,
734 which can be performed using a straightforward recursive equation.}
735 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
736 just as Isabelle's \textbf{function} package usually discharges termination
737 proof obligations automatically.
739 Let's try an example:
742 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
743 \textbf{nitpick}~[\textit{card nat}~= 100,\, \textit{verbose}] \\[2\smallskipamount]
744 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
745 Nitpick can compute it efficiently. \\[2\smallskipamount]
747 \hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
748 Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
749 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
750 Nitpick could not find a better counterexample. \\[2\smallskipamount]
754 No genuine counterexample is possible because Nitpick cannot rule out the
755 existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
756 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
757 existential quantifier:
760 \textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
761 \textbf{nitpick}~[\textit{card nat}~= 100] \\[2\smallskipamount]
762 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
763 \hbox{}\qquad Empty assignment
766 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
767 we use the following definition instead?
770 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
771 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
772 ``$\textit{even}'~2$'' $\,\mid$ \\
773 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
776 This definition is not well-founded: From $\textit{even}'~0$ and
777 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
778 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
780 Let's check a property involving $\textit{even}'$. To make up for the
781 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
782 \textit{nat}'s cardinality to a mere 10:
785 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
786 \lnot\;\textit{even}'~n$'' \\
787 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
789 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
790 Nitpick might need to unroll it. \\[2\smallskipamount]
792 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
793 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
794 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
795 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
796 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
797 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
798 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
799 \hbox{}\qquad Constant: \nopagebreak \\
800 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
801 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
802 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
803 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
807 Nitpick's output is very instructive. First, it tells us that the predicate is
808 unrolled, meaning that it is computed iteratively from the empty set. Then it
809 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
812 The output also shows how each iteration contributes to $\textit{even}'$. The
813 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
814 predicate depends on an iteration counter. Iteration 0 provides the basis
815 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
816 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
817 iterations would not contribute any new elements.
819 Some values are marked with superscripted question
820 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
821 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
822 \textit{True} or $\unk$, never \textit{False}.
824 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
825 iterations. However, these numbers are bounded by the cardinality of the
826 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
827 ever needed to compute the value of a \textit{nat} predicate. You can specify
828 the number of iterations using the \textit{iter} option, as explained in
829 \S\ref{scope-of-search}.
831 In the next formula, $\textit{even}'$ occurs both positively and negatively:
834 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
835 \textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
836 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
837 \hbox{}\qquad Free variable: \nopagebreak \\
838 \hbox{}\qquad\qquad $n = 1$ \\
839 \hbox{}\qquad Constants: \nopagebreak \\
840 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
841 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
842 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
845 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
846 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
847 fixed point (not necessarily the least one). It is used to falsify
848 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
849 $\textit{even}'~(n - 2)$.
851 Coinductive predicates are handled dually. For example:
854 \textbf{coinductive} \textit{nats} \textbf{where} \\
855 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
856 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
857 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
858 \slshape Nitpick found a counterexample:
859 \\[2\smallskipamount]
860 \hbox{}\qquad Constants: \nopagebreak \\
861 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
862 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
863 & \unr\})\end{aligned}$ \\
864 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
867 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
868 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
869 inductive predicates for which each the predicate occurs in at most one
870 assumption of each introduction rule. For example:
873 \textbf{inductive} \textit{odd} \textbf{where} \\
874 ``$\textit{odd}~1$'' $\,\mid$ \\
875 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
876 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
877 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
878 \slshape Nitpick found a counterexample:
879 \\[2\smallskipamount]
880 \hbox{}\qquad Free variable: \nopagebreak \\
881 \hbox{}\qquad\qquad $n = 1$ \\
882 \hbox{}\qquad Constants: \nopagebreak \\
883 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
884 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
885 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
887 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
888 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
890 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
891 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
892 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
896 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
897 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
898 elements from known ones. The set $\textit{odd}$ consists of all the values
899 reachable through the reflexive transitive closure of
900 $\textit{odd}_{\textrm{step}}$ starting with any element from
901 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
902 transitive closure to encode linear predicates is normally either more thorough
903 or more efficient than unrolling (depending on the value of \textit{iter}), but
904 for those cases where it isn't you can disable it by passing the
905 \textit{dont\_star\_linear\_preds} option.
907 \subsection{Coinductive Datatypes}
908 \label{coinductive-datatypes}
910 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
911 datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
912 list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
913 these lazy lists seamlessly and provides a hook, described in
914 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
917 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
918 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
919 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
920 1, 2, 3, \ldots]$ can be defined as lazy lists using the
921 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
922 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
923 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
925 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
926 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
930 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
931 \textbf{nitpick} \\[2\smallskipamount]
932 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
933 \hbox{}\qquad Free variables: \nopagebreak \\
934 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
935 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
938 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
939 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
940 infinite list $[a_1, a_1, a_1, \ldots]$.
942 The next example is more interesting:
945 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
946 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
947 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
948 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
949 some scopes. \\[2\smallskipamount]
951 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 1,
952 and \textit{bisim\_depth}~= 0. \\
953 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
954 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 8,
955 and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
956 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
957 \textit{card}~``\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
959 \\[2\smallskipamount]
960 \hbox{}\qquad Free variables: \nopagebreak \\
961 \hbox{}\qquad\qquad $\textit{a} = a_2$ \\
962 \hbox{}\qquad\qquad $\textit{b} = a_1$ \\
963 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
964 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_1~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega)$ \\[2\smallskipamount]
968 The lazy list $\textit{xs}$ is simply $[a_2, a_2, a_2, \ldots]$, whereas
969 $\textit{ys}$ is $[a_1, a_2, a_2, a_2, \ldots]$, i.e., a lasso-shaped list with
970 $[a_1]$ as its stem and $[a_2]$ as its cycle. In general, the list segment
971 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
972 the segment leading to the binder is the stem.
974 A salient property of coinductive datatypes is that two objects are considered
975 equal if and only if they lead to the same observations. For example, the lazy
976 lists $\textrm{THE}~\omega.\; \omega =
977 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
978 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
979 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
980 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
981 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
982 concept of equality for coinductive datatypes is called bisimulation and is
983 defined coinductively.
985 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
986 the Kodkod problem to ensure that distinct objects lead to different
987 observations. This precaution is somewhat expensive and often unnecessary, so it
988 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
989 bisimilarity check is then performed \textsl{after} the counterexample has been
990 found to ensure correctness. If this after-the-fact check fails, the
991 counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
992 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
993 check for the previous example saves approximately 150~milli\-seconds; the speed
994 gains can be more significant for larger scopes.
996 The next formula illustrates the need for bisimilarity (either as a Kodkod
997 predicate or as an after-the-fact check) to prevent spurious counterexamples:
1000 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
1001 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1002 \textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
1003 \slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
1004 \hbox{}\qquad Free variables: \nopagebreak \\
1005 \hbox{}\qquad\qquad $a = a_2$ \\
1006 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
1007 \textit{LCons}~a_2~\omega$ \\
1008 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
1009 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
1010 \hbox{}\qquad\qquad $'a~\textit{llist} =
1011 \{\!\begin{aligned}[t]
1012 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega, \\[-2pt]
1013 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega,\> \unr\}\end{aligned}$
1014 \\[2\smallskipamount]
1015 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1016 that the counterexample is genuine. \\[2\smallskipamount]
1017 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1018 \slshape Nitpick found no counterexample.
1021 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1022 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1024 A compromise between leaving out the bisimilarity predicate from the Kodkod
1025 problem and performing the after-the-fact check is to specify a lower
1026 nonnegative \textit{bisim\_depth} value than the default one provided by
1027 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1028 be distinguished from each other by their prefixes of length $K$. Be aware that
1029 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1030 finding any counterexamples.
1035 Nitpick normally maps function and product types directly to the corresponding
1036 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1037 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1038 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1039 off to treat these types in the same way as plain datatypes, by approximating
1040 them by a subset of a given cardinality. This technique is called ``boxing'' and
1041 is particularly useful for functions passed as arguments to other functions, for
1042 high-arity functions, and for large tuples. Under the hood, boxing involves
1043 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1044 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1046 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1047 using de Bruijn's notation:
1050 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1053 The $\textit{lift}~t~k$ function increments all variables with indices greater
1054 than or equal to $k$ by one:
1057 \textbf{primrec} \textit{lift} \textbf{where} \\
1058 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1059 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1060 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1063 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1064 term $t$ has a loose variable with index $k$ or more:
1067 \textbf{primrec}~\textit{loose} \textbf{where} \\
1068 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1069 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1070 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1073 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1077 \textbf{primrec}~\textit{subst} \textbf{where} \\
1078 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1079 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1080 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1081 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1084 A substitution is a function that maps variable indices to terms. Observe that
1085 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1086 away, because the recursive call for the \textit{Lam} case involves an altered
1087 version. Also notice the \textit{lift} call, which increments the variable
1088 indices when moving under a \textit{Lam}.
1090 A reasonable property to expect of substitution is that it should leave closed
1091 terms unchanged. Alas, even this simple property does not hold:
1094 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1095 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1097 Trying 8 scopes: \nopagebreak \\
1098 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1099 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1100 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1101 \hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
1102 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1103 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1104 \hbox{}\qquad Free variables: \nopagebreak \\
1105 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1106 & 0 := \textit{Var}~0,\>
1107 1 := \textit{Var}~0,\>
1108 2 := \textit{Var}~0, \\[-2pt]
1109 & 3 := \textit{Var}~0,\>
1110 4 := \textit{Var}~0,\>
1111 5 := \textit{Var}~0)\end{aligned}$ \\
1112 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1113 Total time: $4679$ ms.
1116 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1117 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1118 $\lambda$-term notation, $t$~is
1119 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1120 The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
1121 replaced with $\textit{lift}~(\sigma~m)~0$.
1123 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1124 cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1125 For the formula of interest, knowing 6 values of that type was enough to find
1126 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1127 considered, a hopeless undertaking:
1130 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1131 {\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
1135 Boxing can be enabled or disabled globally or on a per-type basis using the
1136 \textit{box} option. Moreover, setting the cardinality of a function or
1137 product type implicitly enables boxing for that type. Nitpick usually performs
1138 reasonable choices about which types should be boxed, but option tweaking
1143 \subsection{Scope Monotonicity}
1144 \label{scope-monotonicity}
1146 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1147 and \textit{max}) controls which scopes are actually tested. In general, to
1148 exhaust all models below a certain cardinality bound, the number of scopes that
1149 Nitpick must consider increases exponentially with the number of type variables
1150 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1151 cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
1152 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1154 Fortunately, many formulas exhibit a property called \textsl{scope
1155 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1156 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1158 Consider the formula
1161 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1164 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1165 $'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
1166 exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
1167 that any counterexample found with a small scope would still be a counterexample
1168 in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
1169 by the larger scope. Nitpick comes to the same conclusion after a careful
1170 inspection of the formula and the relevant definitions:
1173 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1175 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1176 Nitpick might be able to skip some scopes.
1177 \\[2\smallskipamount]
1179 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1180 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1181 \textit{list}''~= 1, \\
1182 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
1183 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
1184 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1185 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1186 \textit{list}''~= 2, \\
1187 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
1188 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
1189 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1190 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
1191 \textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
1192 \textit{list}''~= 8, \\
1193 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
1194 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
1195 \\[2\smallskipamount]
1196 Nitpick found a counterexample for
1197 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1198 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1199 \textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
1200 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
1201 \\[2\smallskipamount]
1202 \hbox{}\qquad Free variables: \nopagebreak \\
1203 \hbox{}\qquad\qquad $\textit{xs} = [a_4, a_5]$ \\
1204 \hbox{}\qquad\qquad $\textit{ys} = [b_3, b_3]$ \\[2\smallskipamount]
1205 Total time: 1636 ms.
1208 In theory, it should be sufficient to test a single scope:
1211 \textbf{nitpick}~[\textit{card}~= 8]
1214 However, this is often less efficient in practice and may lead to overly complex
1217 If the monotonicity check fails but we believe that the formula is monotonic (or
1218 we don't mind missing some counterexamples), we can pass the
1219 \textit{mono} option. To convince yourself that this option is risky,
1220 simply consider this example from \S\ref{skolemization}:
1223 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1224 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1225 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1226 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1227 \textbf{nitpick} \\[2\smallskipamount]
1229 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1230 \hbox{}\qquad $\vdots$
1233 (It turns out the formula holds if and only if $\textit{card}~'a \le
1234 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1235 monotonicity checks can be disabled by passing \textit{non\_mono}
1236 (\S\ref{optimizations}).
1238 As insinuated in \S\ref{natural-numbers-and-integers} and
1239 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1240 are normally monotonic and treated as such. The same is true for record types,
1241 \textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
1242 cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
1243 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1244 consider only 8~scopes instead of $32\,768$.
1246 \section{Case Studies}
1247 \label{case-studies}
1249 As a didactic device, the previous section focused mostly on toy formulas whose
1250 validity can easily be assessed just by looking at the formula. We will now
1251 review two somewhat more realistic case studies that are within Nitpick's
1252 reach:\ a context-free grammar modeled by mutually inductive sets and a
1253 functional implementation of AA trees. The results presented in this
1254 section were produced with the following settings:
1257 \textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
1260 \subsection{A Context-Free Grammar}
1261 \label{a-context-free-grammar}
1263 Our first case study is taken from section 7.4 in the Isabelle tutorial
1264 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1265 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1268 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1269 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1270 $A$ & $::=$ & $aS \mid bAA$ \\
1271 $B$ & $::=$ & $bS \mid aBB$
1275 The intuition behind the grammar is that $A$ generates all string with one more
1276 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1278 The alphabet consists exclusively of $a$'s and $b$'s:
1281 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1284 Strings over the alphabet are represented by \textit{alphabet list}s.
1285 Nonterminals in the grammar become sets of strings. The production rules
1286 presented above can be expressed as a mutually inductive definition:
1289 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1290 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1291 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1292 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1293 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1294 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1295 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1298 The conversion of the grammar into the inductive definition was done manually by
1299 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1302 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1303 d'\^etre}. A good approach is to state desirable properties of the specification
1304 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1305 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1306 counterexamples will point to bugs in the specification. For our grammar
1307 example, we will proceed in two steps, separating the soundness and the
1308 completeness of the set $S$. First, soundness:
1311 \textbf{theorem}~\textit{S\_sound}: \\
1312 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1313 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1314 \textbf{nitpick} \\[2\smallskipamount]
1315 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1316 \hbox{}\qquad Free variable: \nopagebreak \\
1317 \hbox{}\qquad\qquad $w = [b]$
1320 It would seem that $[b] \in S$. How could this be? An inspection of the
1321 introduction rules reveals that the only rule with a right-hand side of the form
1322 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1326 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1329 On closer inspection, we can see that this rule is wrong. To match the
1330 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1334 \textbf{nitpick} \\[2\smallskipamount]
1335 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1336 \hbox{}\qquad Free variable: \nopagebreak \\
1337 \hbox{}\qquad\qquad $w = [a, a, b]$
1340 Some detective work is necessary to find out what went wrong here. To get $[a,
1341 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1345 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1348 Now, this formula must be wrong: The same assumption occurs twice, and the
1349 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1350 the assumptions should have been a $w$.
1352 With the correction made, we don't get any counterexample from Nitpick. Let's
1353 move on and check completeness:
1356 \textbf{theorem}~\textit{S\_complete}: \\
1357 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1358 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1359 \longrightarrow w \in S$'' \\
1360 \textbf{nitpick} \\[2\smallskipamount]
1361 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1362 \hbox{}\qquad Free variable: \nopagebreak \\
1363 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1366 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1367 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1368 the introduction rules we have are probably correct. Perhaps we simply lack an
1369 introduction rule. Comparing the grammar with the inductive definition, our
1370 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1371 without which the grammar cannot generate two or more $b$'s in a row. So we add
1375 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1378 With this last change, we don't get any counterexamples from Nitpick for either
1379 soundness or completeness. We can even generalize our result to cover $A$ and
1383 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1384 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1385 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1386 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1387 \textbf{nitpick} \\[2\smallskipamount]
1388 \slshape Nitpick found no counterexample.
1391 \subsection{AA Trees}
1394 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1395 similar performance to red-black trees, but with a simpler implementation
1396 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1397 a total order $<$. We start by defining the datatype and some basic extractor
1401 \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]
1402 \textbf{primrec} \textit{data} \textbf{where} \\
1403 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1404 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1405 \textbf{primrec} \textit{dataset} \textbf{where} \\
1406 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1407 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1408 \textbf{primrec} \textit{level} \textbf{where} \\
1409 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1410 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1411 \textbf{primrec} \textit{left} \textbf{where} \\
1412 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1413 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1414 \textbf{primrec} \textit{right} \textbf{where} \\
1415 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1416 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1419 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1420 as follows \cite{wikipedia-2009-aa-trees}:
1422 \kern.2\parskip %% TYPESETTING
1425 Each node has a level field, and the following invariants must remain true for
1426 the tree to be valid:
1430 \kern-.4\parskip %% TYPESETTING
1435 \item[1.] The level of a leaf node is one.
1436 \item[2.] The level of a left child is strictly less than that of its parent.
1437 \item[3.] The level of a right child is less than or equal to that of its parent.
1438 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1439 \item[5.] Every node of level greater than one must have two children.
1444 \kern.4\parskip %% TYPESETTING
1446 The \textit{wf} predicate formalizes this description:
1449 \textbf{primrec} \textit{wf} \textbf{where} \\
1450 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1451 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1452 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1453 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1454 \phantom{``$($}$\textrm{else}$ \\
1455 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1456 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1457 \mathrel{\land} \textit{level}~u \le k$ \\
1458 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1461 Rebalancing the tree upon insertion and removal of elements is performed by two
1462 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1465 \textbf{primrec} \textit{skew} \textbf{where} \\
1466 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1467 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1468 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1469 \textit{level}~t~\textrm{then}$ \\
1470 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1471 (\textit{right}~t)~u)$ \\
1472 \phantom{``(}$\textrm{else}$ \\
1473 \phantom{``(\quad}$N~x~k~t~u)$''
1477 \textbf{primrec} \textit{split} \textbf{where} \\
1478 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1479 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1480 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1481 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1482 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1483 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1484 \phantom{``(}$\textrm{else}$ \\
1485 \phantom{``(\quad}$N~x~k~t~u)$''
1488 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1489 of elements stored in the tree:
1492 \textbf{theorem}~\textit{dataset\_skew\_split}:\\
1493 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1494 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1495 \textbf{nitpick} \\[2\smallskipamount]
1496 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1499 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1500 should not alter the tree:
1503 \textbf{theorem}~\textit{wf\_skew\_split}:\\
1504 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1505 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1506 \textbf{nitpick} \\[2\smallskipamount]
1507 {\slshape Nitpick found no counterexample.}
1510 Insertion is implemented recursively. It preserves the sort order:
1513 \textbf{primrec}~\textit{insort} \textbf{where} \\
1514 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1515 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1516 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1517 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1520 Notice that we deliberately commented out the application of \textit{skew} and
1521 \textit{split}. Let's see if this causes any problems:
1524 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1525 \textbf{nitpick} \\[2\smallskipamount]
1526 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1527 \hbox{}\qquad Free variables: \nopagebreak \\
1528 \hbox{}\qquad\qquad $t = N~a_3~1~\Lambda~\Lambda$ \\
1529 \hbox{}\qquad\qquad $x = a_4$ \\[2\smallskipamount]
1532 It's hard to see why this is a counterexample. To improve readability, we will
1533 restrict the theorem to \textit{nat}, so that we don't need to look up the value
1534 of the $\textit{op}~{<}$ constant to find out which element is smaller than the
1535 other. In addition, we will tell Nitpick to display the value of
1536 $\textit{insort}~t~x$ using the \textit{eval} option. This gives
1539 \textbf{theorem} \textit{wf\_insort\_nat}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1540 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1541 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1542 \hbox{}\qquad Free variables: \nopagebreak \\
1543 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1544 \hbox{}\qquad\qquad $x = 0$ \\
1545 \hbox{}\qquad Evaluated term: \\
1546 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1549 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1550 where both have a level of 1. This violates the second AA tree invariant, which
1551 states that a left child's level must be less than its parent's. This shouldn't
1552 come as a surprise, considering that we commented out the tree rebalancing code.
1553 Reintroducing the code seems to solve the problem:
1556 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1557 \textbf{nitpick} \\[2\smallskipamount]
1558 {\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
1561 Insertion should transform the set of elements represented by the tree in the
1565 \textbf{theorem} \textit{dataset\_insort}:\kern.4em
1566 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1567 \textbf{nitpick} \\[2\smallskipamount]
1568 {\slshape Nitpick ran out of time after checking 5 of 8 scopes.}
1571 We could continue like this and sketch a complete theory of AA trees without
1572 performing a single proof. Once the definitions and main theorems are in place
1573 and have been thoroughly tested using Nitpick, we could start working on the
1574 proofs. Developing theories this way usually saves time, because faulty theorems
1575 and definitions are discovered much earlier in the process.
1577 \section{Option Reference}
1578 \label{option-reference}
1580 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1581 \def\qty#1{$\left<\textit{#1}\right>$}
1582 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1583 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1584 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1585 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1586 \def\ops#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1587 \def\opt#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1588 \def\opu#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1589 \def\opusmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1591 Nitpick's behavior can be influenced by various options, which can be specified
1592 in brackets after the \textbf{nitpick} command. Default values can be set
1593 using \textbf{nitpick\_\allowbreak params}. For example:
1596 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1599 The options are categorized as follows:\ mode of operation
1600 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1601 format (\S\ref{output-format}), automatic counterexample checks
1602 (\S\ref{authentication}), optimizations
1603 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1605 You can instruct Nitpick to run automatically on newly entered theorems by
1606 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1607 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
1608 and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1609 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1610 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1611 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1612 \textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
1613 Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
1616 The number of options can be overwhelming at first glance. Do not let that worry
1617 you: Nitpick's defaults have been chosen so that it almost always does the right
1618 thing, and the most important options have been covered in context in
1619 \S\ref{first-steps}.
1621 The descriptions below refer to the following syntactic quantities:
1624 \item[$\bullet$] \qtybf{string}: A string.
1625 \item[$\bullet$] \qtybf{bool}: \textit{true} or \textit{false}.
1626 \item[$\bullet$] \qtybf{bool\_or\_smart}: \textit{true}, \textit{false}, or \textit{smart}.
1627 \item[$\bullet$] \qtybf{int}: An integer. Negative integers are prefixed with a hyphen.
1628 \item[$\bullet$] \qtybf{int\_or\_smart}: An integer or \textit{smart}.
1629 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1630 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1632 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1633 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1634 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1635 \item[$\bullet$] \qtybf{const}: The name of a HOL constant.
1636 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1637 \item[$\bullet$] \qtybf{term\_list}: A space-separated list of HOL terms (e.g.,
1638 ``$f~x$''~``$g~y$'').
1639 \item[$\bullet$] \qtybf{type}: A HOL type.
1642 Default values are indicated in square brackets. Boolean options have a negated
1643 counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
1644 Boolean options, ``= \textit{true}'' may be omitted.
1646 \subsection{Mode of Operation}
1647 \label{mode-of-operation}
1650 \optrue{blocking}{non\_blocking}
1651 Specifies whether the \textbf{nitpick} command should operate synchronously.
1652 The asynchronous (non-blocking) mode lets the user start proving the putative
1653 theorem while Nitpick looks for a counterexample, but it can also be more
1654 confusing. For technical reasons, automatic runs currently always block.
1656 \optrue{falsify}{satisfy}
1657 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1658 satisfying examples (models). This manual assumes throughout that
1659 \textit{falsify} is enabled.
1661 \opsmart{user\_axioms}{no\_user\_axioms}
1662 Specifies whether the user-defined axioms (specified using
1663 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1664 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1665 the constants that occur in the formula to falsify. The option is implicitly set
1666 to \textit{true} for automatic runs.
1668 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1669 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1670 these conditions are tagged as ``likely genuine.'' The \textit{debug}
1671 (\S\ref{output-format}) option can be used to find out which axioms were
1675 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1676 (\S\ref{output-format}).}
1678 \optrue{assms}{no\_assms}
1679 Specifies whether the relevant assumptions in structured proof should be
1680 considered. The option is implicitly enabled for automatic runs.
1683 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1685 \opfalse{overlord}{no\_overlord}
1686 Specifies whether Nitpick should put its temporary files in
1687 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1688 debugging Nitpick but also unsafe if several instances of the tool are run
1692 {\small See also \textit{debug} (\S\ref{output-format}).}
1695 \subsection{Scope of Search}
1696 \label{scope-of-search}
1699 \opu{card}{type}{int\_seq}
1700 Specifies the sequence of cardinalities to use for a given type.
1701 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1702 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1703 Although function and product types are normally mapped directly to the
1704 corresponding Kodkod concepts, setting
1705 the cardinality of such types is also allowed and implicitly enables ``boxing''
1706 for them, as explained in the description of the \textit{box}~\qty{type}
1707 and \textit{box} (\S\ref{scope-of-search}) options.
1710 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1712 \opt{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
1713 Specifies the default sequence of cardinalities to use. This can be overridden
1714 on a per-type basis using the \textit{card}~\qty{type} option described above.
1716 \opu{max}{const}{int\_seq}
1717 Specifies the sequence of maximum multiplicities to use for a given
1718 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1719 number of distinct values that it can construct. Nonsensical values (e.g.,
1720 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1721 datatypes equipped with several constructors.
1724 Specifies the default sequence of maximum multiplicities to use for
1725 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
1726 basis using the \textit{max}~\qty{const} option described above.
1728 \opsmart{binary\_ints}{unary\_ints}
1729 Specifies whether natural numbers and integers should be encoded using a unary
1730 or binary notation. In unary mode, the cardinality fully specifies the subset
1731 used to approximate the type. For example:
1733 $$\hbox{\begin{tabular}{@{}rll@{}}%
1734 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
1735 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
1736 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
1741 $$\hbox{\begin{tabular}{@{}rll@{}}%
1742 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
1743 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
1746 In binary mode, the cardinality specifies the number of distinct values that can
1747 be constructed. Each of these value is represented by a bit pattern whose length
1748 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
1749 Nitpick attempts to choose the more appropriate encoding by inspecting the
1750 formula at hand, preferring the binary notation for problems involving
1751 multiplicative operators or large constants.
1753 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
1754 problems that refer to the types \textit{rat} or \textit{real} or the constants
1755 \textit{gcd} or \textit{lcm}.
1757 {\small See also \textit{bits} (\S\ref{scope-of-search}) and
1758 \textit{show\_datatypes} (\S\ref{output-format}).}
1760 \opt{bits}{int\_seq}{$\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{8},\mathbf{10},\mathbf{12}$}
1761 Specifies the number of bits to use to represent natural numbers and integers in
1762 binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
1764 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
1766 \opusmart{wf}{const}{non\_wf}
1767 Specifies whether the specified (co)in\-duc\-tively defined predicate is
1768 well-founded. The option can take the following values:
1771 \item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
1772 predicate as if it were well-founded. Since this is generally not sound when the
1773 predicate is not well-founded, the counterexamples are tagged as ``likely
1776 \item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
1777 as if it were not well-founded. The predicate is then unrolled as prescribed by
1778 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
1781 \item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
1782 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
1783 \textit{sizechange} tactics. If this succeeds (or the predicate occurs with an
1784 appropriate polarity in the formula to falsify), use an efficient fixed point
1785 equation as specification of the predicate; otherwise, unroll the predicates
1786 according to the \textit{iter}~\qty{const} and \textit{iter} options.
1790 {\small See also \textit{iter} (\S\ref{scope-of-search}),
1791 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
1792 (\S\ref{timeouts}).}
1794 \opsmart{wf}{non\_wf}
1795 Specifies the default wellfoundedness setting to use. This can be overridden on
1796 a per-predicate basis using the \textit{wf}~\qty{const} option above.
1798 \opu{iter}{const}{int\_seq}
1799 Specifies the sequence of iteration counts to use when unrolling a given
1800 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
1801 predicates that occur negatively and coinductive predicates that occur
1802 positively in the formula to falsify and that cannot be proved to be
1803 well-founded, but this behavior is influenced by the \textit{wf} option. The
1804 iteration counts are automatically bounded by the cardinality of the predicate's
1807 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
1808 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
1810 \opt{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
1811 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
1812 predicates. This can be overridden on a per-predicate basis using the
1813 \textit{iter} \qty{const} option above.
1815 \opt{bisim\_depth}{int\_seq}{$\mathbf{7}$}
1816 Specifies the sequence of iteration counts to use when unrolling the
1817 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
1818 of $-1$ means that no predicate is generated, in which case Nitpick performs an
1819 after-the-fact check to see if the known coinductive datatype values are
1820 bidissimilar. If two values are found to be bisimilar, the counterexample is
1821 tagged as ``likely genuine.'' The iteration counts are automatically bounded by
1822 the sum of the cardinalities of the coinductive datatypes occurring in the
1825 \opusmart{box}{type}{dont\_box}
1826 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
1827 product type in an isomorphic datatype internally. Boxing is an effective mean
1828 to reduce the search space and speed up Nitpick, because the isomorphic datatype
1829 is approximated by a subset of the possible function or pair values;
1830 like other drastic optimizations, it can also prevent the discovery of
1831 counterexamples. The option can take the following values:
1834 \item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
1836 \item[$\bullet$] \textbf{\textit{false}}: Never box the type.
1837 \item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
1838 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
1839 higher-order functions are good candidates for boxing.
1842 Setting the \textit{card}~\qty{type} option for a function or product type
1843 implicitly enables boxing for that type.
1846 {\small See also \textit{verbose} (\S\ref{output-format})
1847 and \textit{debug} (\S\ref{output-format}).}
1849 \opsmart{box}{dont\_box}
1850 Specifies the default boxing setting to use. This can be overridden on a
1851 per-type basis using the \textit{box}~\qty{type} option described above.
1853 \opusmart{mono}{type}{non\_mono}
1854 Specifies whether the specified type should be considered monotonic when
1855 enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a
1856 monotonicity check on the type. Setting this option to \textit{true} can reduce
1857 the number of scopes tried, but it also diminishes the theoretical chance of
1858 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
1861 {\small See also \textit{card} (\S\ref{scope-of-search}),
1862 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
1863 (\S\ref{output-format}).}
1865 \opsmart{mono}{non\_box}
1866 Specifies the default monotonicity setting to use. This can be overridden on a
1867 per-type basis using the \textit{mono}~\qty{type} option described above.
1869 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
1870 Specifies whether type variables with the same sort constraints should be
1871 merged. Setting this option to \textit{true} can reduce the number of scopes
1872 tried and the size of the generated Kodkod formulas, but it also diminishes the
1873 theoretical chance of finding a counterexample.
1875 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1878 \subsection{Output Format}
1879 \label{output-format}
1882 \opfalse{verbose}{quiet}
1883 Specifies whether the \textbf{nitpick} command should explain what it does. This
1884 option is useful to determine which scopes are tried or which SAT solver is
1885 used. This option is implicitly disabled for automatic runs.
1887 \opfalse{debug}{no\_debug}
1888 Specifies whether Nitpick should display additional debugging information beyond
1889 what \textit{verbose} already displays. Enabling \textit{debug} also enables
1890 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
1891 option is implicitly disabled for automatic runs.
1894 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
1895 \textit{batch\_size} (\S\ref{optimizations}).}
1897 \optrue{show\_skolems}{hide\_skolem}
1898 Specifies whether the values of Skolem constants should be displayed as part of
1899 counterexamples. Skolem constants correspond to bound variables in the original
1900 formula and usually help us to understand why the counterexample falsifies the
1904 {\small See also \textit{skolemize} (\S\ref{optimizations}).}
1906 \opfalse{show\_datatypes}{hide\_datatypes}
1907 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
1908 be displayed as part of counterexamples. Such subsets are sometimes helpful when
1909 investigating whether a potential counterexample is genuine or spurious, but
1910 their potential for clutter is real.
1912 \opfalse{show\_consts}{hide\_consts}
1913 Specifies whether the values of constants occurring in the formula (including
1914 its axioms) should be displayed along with any counterexample. These values are
1915 sometimes helpful when investigating why a counterexample is
1916 genuine, but they can clutter the output.
1918 \opfalse{show\_all}{dont\_show\_all}
1919 Enabling this option effectively enables \textit{show\_skolems},
1920 \textit{show\_datatypes}, and \textit{show\_consts}.
1922 \opt{max\_potential}{int}{$\mathbf{1}$}
1923 Specifies the maximum number of potential counterexamples to display. Setting
1924 this option to 0 speeds up the search for a genuine counterexample. This option
1925 is implicitly set to 0 for automatic runs. If you set this option to a value
1926 greater than 1, you will need an incremental SAT solver: For efficiency, it is
1927 recommended to install the JNI version of MiniSat and set \textit{sat\_solver} =
1928 \textit{MiniSatJNI}. Also be aware that many of the counterexamples may look
1929 identical, unless the \textit{show\_all} (\S\ref{output-format}) option is
1933 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
1934 \textit{sat\_solver} (\S\ref{optimizations}).}
1936 \opt{max\_genuine}{int}{$\mathbf{1}$}
1937 Specifies the maximum number of genuine counterexamples to display. If you set
1938 this option to a value greater than 1, you will need an incremental SAT solver:
1939 For efficiency, it is recommended to install the JNI version of MiniSat and set
1940 \textit{sat\_solver} = \textit{MiniSatJNI}. Also be aware that many of the
1941 counterexamples may look identical, unless the \textit{show\_all}
1942 (\S\ref{output-format}) option is enabled.
1945 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
1946 \textit{sat\_solver} (\S\ref{optimizations}).}
1948 \ops{eval}{term\_list}
1949 Specifies the list of terms whose values should be displayed along with
1950 counterexamples. This option suffers from an ``observer effect'': Nitpick might
1951 find different counterexamples for different values of this option.
1953 \opu{format}{term}{int\_seq}
1954 Specifies how to uncurry the value displayed for a variable or constant.
1955 Uncurrying sometimes increases the readability of the output for high-arity
1956 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
1957 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
1958 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
1959 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
1960 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
1961 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
1962 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
1963 arguments that are not accounted for are left alone, as if the specification had
1964 been $1,\ldots,1,n_1,\ldots,n_k$.
1967 {\small See also \textit{uncurry} (\S\ref{optimizations}).}
1969 \opt{format}{int\_seq}{$\mathbf{1}$}
1970 Specifies the default format to use. Irrespective of the default format, the
1971 extra arguments to a Skolem constant corresponding to the outer bound variables
1972 are kept separated from the remaining arguments, the \textbf{for} arguments of
1973 an inductive definitions are kept separated from the remaining arguments, and
1974 the iteration counter of an unrolled inductive definition is shown alone. The
1975 default format can be overridden on a per-variable or per-constant basis using
1976 the \textit{format}~\qty{term} option described above.
1979 %% MARK: Authentication
1980 \subsection{Authentication}
1981 \label{authentication}
1984 \opfalse{check\_potential}{trust\_potential}
1985 Specifies whether potential counterexamples should be given to Isabelle's
1986 \textit{auto} tactic to assess their validity. If a potential counterexample is
1987 shown to be genuine, Nitpick displays a message to this effect and terminates.
1990 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
1992 \opfalse{check\_genuine}{trust\_genuine}
1993 Specifies whether genuine and likely genuine counterexamples should be given to
1994 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
1995 counterexample is shown to be spurious, the user is kindly asked to send a bug
1996 report to the author at
1997 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
2000 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
2002 \ops{expect}{string}
2003 Specifies the expected outcome, which must be one of the following:
2006 \item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
2007 \item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a ``likely
2008 genuine'' counterexample (i.e., a counterexample that is genuine unless
2009 it contradicts a missing axiom or a dangerous option was used inappropriately).
2010 \item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
2011 \item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
2012 \item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
2013 Kodkod ran out of memory).
2016 Nitpick emits an error if the actual outcome differs from the expected outcome.
2017 This option is useful for regression testing.
2020 \subsection{Optimizations}
2021 \label{optimizations}
2023 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
2028 \opt{sat\_solver}{string}{smart}
2029 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2030 to be faster than their Java counterparts, but they can be more difficult to
2031 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2032 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2033 you will need an incremental SAT solver, such as \textit{MiniSatJNI}
2034 (recommended) or \textit{SAT4J}.
2036 The supported solvers are listed below:
2040 \item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
2041 written in \cpp{}. To use MiniSat, set the environment variable
2042 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2043 executable. The \cpp{} sources and executables for MiniSat are available at
2044 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2045 and 2.0 beta (2007-07-21).
2047 \item[$\bullet$] \textbf{\textit{MiniSatJNI}}: The JNI (Java Native Interface)
2048 version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which
2049 you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2050 version of MiniSat, the JNI version can be used incrementally.
2053 %%% "It is bundled with Kodkodi and requires no further installation or
2054 %%% configuration steps. Alternatively,"
2055 \item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
2056 written in C. You can install a standard version of
2057 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2058 that contains the \texttt{picosat} executable. The C sources for PicoSAT are
2059 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2060 Nitpick has been tested with version 913.
2062 \item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
2063 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2064 the directory that contains the \texttt{zchaff} executable. The \cpp{} sources
2065 and executables for zChaff are available at
2066 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2067 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2069 \item[$\bullet$] \textbf{\textit{zChaffJNI}}: The JNI version of zChaff is
2070 bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on
2071 Kodkod's web site \cite{kodkod-2009}.
2073 \item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
2074 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2075 directory that contains the \texttt{rsat} executable. The \cpp{} sources for
2076 RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been
2077 tested with version 2.01.
2079 \item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
2080 written in C. To use BerkMin, set the environment variable
2081 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2082 executable. The BerkMin executables are available at
2083 \url{http://eigold.tripod.com/BerkMin.html}.
2085 \item[$\bullet$] \textbf{\textit{BerkMinAlloy}}: Variant of BerkMin that is
2086 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2087 version of BerkMin, set the environment variable
2088 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2091 \item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
2092 written in C. To use Jerusat, set the environment variable
2093 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2094 executable. The C sources for Jerusat are available at
2095 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2097 \item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
2098 written in Java that can be used incrementally. It is bundled with Kodkodi and
2099 requires no further installation or configuration steps. Do not attempt to
2100 install the official SAT4J packages, because their API is incompatible with
2103 \item[$\bullet$] \textbf{\textit{SAT4JLight}}: Variant of SAT4J that is
2104 optimized for small problems. It can also be used incrementally.
2106 \item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
2107 experimental solver written in \cpp. To use HaifaSat, set the environment
2108 variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
2109 \texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at
2110 \url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
2112 \item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
2113 \textit{smart}, Nitpick selects the first solver among MiniSat,
2114 PicoSAT, zChaff, RSat, BerkMin, BerkMinAlloy, Jerusat, MiniSatJNI, and zChaffJNI
2115 that is recognized by Isabelle. If none is found, it falls back on SAT4J, which
2116 should always be available. If \textit{verbose} (\S\ref{output-format}) is
2117 enabled, Nitpick displays which SAT solver was chosen.
2121 \opt{batch\_size}{int\_or\_smart}{smart}
2122 Specifies the maximum number of Kodkod problems that should be lumped together
2123 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2124 together ensures that Kodkodi is launched less often, but it makes the verbose
2125 output less readable and is sometimes detrimental to performance. If
2126 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2127 \textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
2129 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2130 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2131 be rewritten to use (automatically generated) discriminators and destructors.
2132 This optimization can drastically reduce the size of the Boolean formulas given
2136 {\small See also \textit{debug} (\S\ref{output-format}).}
2138 \optrue{specialize}{dont\_specialize}
2139 Specifies whether functions invoked with static arguments should be specialized.
2140 This optimization can drastically reduce the search space, especially for
2141 higher-order functions.
2144 {\small See also \textit{debug} (\S\ref{output-format}) and
2145 \textit{show\_consts} (\S\ref{output-format}).}
2147 \optrue{skolemize}{dont\_skolemize}
2148 Specifies whether the formula should be skolemized. For performance reasons,
2149 (positive) $\forall$-quanti\-fiers that occur in the scope of a higher-order
2150 (positive) $\exists$-quanti\-fier are left unchanged.
2153 {\small See also \textit{debug} (\S\ref{output-format}) and
2154 \textit{show\_skolems} (\S\ref{output-format}).}
2156 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2157 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2158 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2159 predicates for which each the predicate occurs in at most one assumption of each
2160 introduction rule. Using the reflexive transitive closure is in principle
2161 equivalent to setting \textit{iter} to the cardinality of the predicate's
2162 domain, but it is usually more efficient.
2164 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2165 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2167 \optrue{uncurry}{dont\_uncurry}
2168 Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no
2169 tangible effect on efficiency, but it creates opportunities for the boxing
2173 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug}
2174 (\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).}
2176 \optrue{fast\_descrs}{full\_descrs}
2177 Specifies whether Nitpick should optimize the definite and indefinite
2178 description operators (THE and SOME). The optimized versions usually help
2179 Nitpick generate more counterexamples or at least find them faster, but only the
2180 unoptimized versions are complete when all types occurring in the formula are
2183 {\small See also \textit{debug} (\S\ref{output-format}).}
2185 \optrue{peephole\_optim}{no\_peephole\_optim}
2186 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2187 peephole optimizer. These optimizations can make a significant difference.
2188 Unless you are tracking down a bug in Nitpick or distrust the peephole
2189 optimizer, you should leave this option enabled.
2191 \opt{sym\_break}{int}{20}
2192 Specifies an upper bound on the number of relations for which Kodkod generates
2193 symmetry breaking predicates. According to the Kodkod documentation
2194 \cite{kodkod-2009-options}, ``in general, the higher this value, the more
2195 symmetries will be broken, and the faster the formula will be solved. But,
2196 setting the value too high may have the opposite effect and slow down the
2199 \opt{sharing\_depth}{int}{3}
2200 Specifies the depth to which Kodkod should check circuits for equivalence during
2201 the translation to SAT. The default of 3 is the same as in Alloy. The minimum
2202 allowed depth is 1. Increasing the sharing may result in a smaller SAT problem,
2203 but can also slow down Kodkod.
2205 \opfalse{flatten\_props}{dont\_flatten\_props}
2206 Specifies whether Kodkod should try to eliminate intermediate Boolean variables.
2207 Although this might sound like a good idea, in practice it can drastically slow
2210 \opt{max\_threads}{int}{0}
2211 Specifies the maximum number of threads to use in Kodkod. If this option is set
2212 to 0, Kodkod will compute an appropriate value based on the number of processor
2216 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2217 \textit{timeout} (\S\ref{timeouts}).}
2220 \subsection{Timeouts}
2224 \opt{timeout}{time}{$\mathbf{30}$ s}
2225 Specifies the maximum amount of time that the \textbf{nitpick} command should
2226 spend looking for a counterexample. Nitpick tries to honor this constraint as
2227 well as it can but offers no guarantees. For automatic runs,
2228 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2229 a time slot whose length is specified by the ``Auto Counterexample Time
2230 Limit'' option in Proof General.
2233 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2235 \opt{tac\_timeout}{time}{$\mathbf{500}$\,ms}
2236 Specifies the maximum amount of time that the \textit{auto} tactic should use
2237 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2238 and \textit{sizechange} should use when checking whether a (co)in\-duc\-tive
2239 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2240 can but offers no guarantees.
2243 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2244 \textit{check\_potential} (\S\ref{authentication}),
2245 and \textit{check\_genuine} (\S\ref{authentication}).}
2248 \section{Attribute Reference}
2249 \label{attribute-reference}
2251 Nitpick needs to consider the definitions of all constants occurring in a
2252 formula in order to falsify it. For constants introduced using the
2253 \textbf{definition} command, the definition is simply the associated
2254 \textit{\_def} axiom. In contrast, instead of using the internal representation
2255 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2256 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2257 equational specification entered by the user.
2259 Behind the scenes, Isabelle's built-in packages and theories rely on the
2260 following attributes to affect Nitpick's behavior:
2263 \flushitem{\textit{nitpick\_def}}
2266 This attribute specifies an alternative definition of a constant. The
2267 alternative definition should be logically equivalent to the constant's actual
2268 axiomatic definition and should be of the form
2270 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2272 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2275 \flushitem{\textit{nitpick\_simp}}
2278 This attribute specifies the equations that constitute the specification of a
2279 constant. For functions defined using the \textbf{primrec}, \textbf{function},
2280 and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
2281 \textit{simps} rules. The equations must be of the form
2283 \qquad $c~t_1~\ldots\ t_n \,=\, u.$
2285 \flushitem{\textit{nitpick\_psimp}}
2288 This attribute specifies the equations that constitute the partial specification
2289 of a constant. For functions defined using the \textbf{function} package, this
2290 corresponds to the \textit{psimps} rules. The conditional equations must be of
2293 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
2295 \flushitem{\textit{nitpick\_intro}}
2298 This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
2299 For predicates defined using the \textbf{inductive} or \textbf{coinductive}
2300 command, this corresponds to the \textit{intros} rules. The introduction rules
2303 \qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
2304 \ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\
2307 where the $P_i$'s are side conditions that do not involve $c$ and $M$ is an
2308 optional monotonic operator. The order of the assumptions is irrelevant.
2312 When faced with a constant, Nitpick proceeds as follows:
2315 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2316 is not empty, Nitpick uses these rules as the specification of the constant.
2318 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2319 the constant is not empty, it uses these rules as the specification of the
2322 \item[3.] Otherwise, it looks up the definition of the constant:
2325 \item[1.] If the \textit{nitpick\_def} set associated with the constant
2326 is not empty, it uses the latest rule added to the set as the definition of the
2327 constant; otherwise it uses the actual definition axiom.
2328 \item[2.] If the definition is of the form
2330 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
2332 then Nitpick assumes that the definition was made using an inductive package and
2333 based on the introduction rules marked with \textit{nitpick\_\allowbreak
2334 ind\_\allowbreak intros} tries to determine whether the definition is
2339 As an illustration, consider the inductive definition
2342 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2343 ``\textit{odd}~1'' $\,\mid$ \\
2344 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2347 Isabelle automatically attaches the \textit{nitpick\_intro} attribute to
2348 the above rules. Nitpick then uses the \textit{lfp}-based definition in
2349 conjunction with these rules. To override this, we can specify an alternative
2350 definition as follows:
2353 \textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]: ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2356 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2357 = 1$. Alternatively, we can specify an equational specification of the constant:
2360 \textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]: ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2363 Such tweaks should be done with great care, because Nitpick will assume that the
2364 constant is completely defined by its equational specification. For example, if
2365 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
2366 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2367 (\S\ref{output-format}) option is extremely useful to understand what is going
2368 on when experimenting with \textit{nitpick\_} attributes.
2370 \section{Standard ML Interface}
2371 \label{standard-ml-interface}
2373 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2374 and debugging. Among the most interesting functions exported by Nitpick are
2375 those that let you invoke the tool programmatically and those that let you
2376 register and unregister custom coinductive datatypes.
2378 \subsection{Invocation of Nitpick}
2379 \label{invocation-of-nitpick}
2381 The \textit{Nitpick} structure offers the following functions for invoking your
2382 favorite counterexample generator:
2385 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2386 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2387 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2388 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2389 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2392 The return value is a new proof state paired with an outcome string
2393 (``genuine'', ``likely\_genuine'', ``potential'', ``none'', or ``unknown''). The
2394 \textit{params} type is a large record that lets you set Nitpick's options. The
2395 current default options can be retrieved by calling the following function
2396 defined in the \textit{Nitpick\_Isar} structure:
2399 $\textbf{val}\,~\textit{default\_params} :\,
2400 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2403 The second argument lets you override option values before they are parsed and
2404 put into a \textit{params} record. Here is an example:
2407 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2408 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2409 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2410 & \textit{subgoal}\end{aligned}$
2415 \subsection{Registration of Coinductive Datatypes}
2416 \label{registration-of-coinductive-datatypes}
2418 If you have defined a custom coinductive datatype, you can tell Nitpick about
2419 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2420 uses for lazy lists. The interface for registering and unregistering coinductive
2421 datatypes consists of the following pair of functions defined in the
2422 \textit{Nitpick} structure:
2425 $\textbf{val}\,~\textit{register\_codatatype} :\,
2426 \textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2427 $\textbf{val}\,~\textit{unregister\_codatatype} :\,
2428 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2431 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2432 not been, you could have told Nitpick about it by adding the following line
2433 to your theory file:
2436 $\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
2437 & \textit{Nitpick.register\_codatatype} \\[-2pt]
2438 & \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
2439 & \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
2442 The \textit{register\_codatatype} function takes a coinductive type, its case
2443 function, and the list of its constructors. The case function must take its
2444 arguments in the order that the constructors are listed. If no case function
2445 with the correct signature is available, simply pass the empty string.
2447 On the other hand, if your goal is to cripple Nitpick, add the following line to
2448 your theory file and try to check a few conjectures about lazy lists:
2451 $\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
2452 \kern1pt'a~\textit{list}\textrm{''}\}\ \,{*}\}$
2455 Inductive datatypes can be registered as coinductive datatypes, given
2456 appropriate coinductive constructors. However, doing so precludes
2457 the use of the inductive constructors---Nitpick will generate an error if they
2460 \section{Known Bugs and Limitations}
2461 \label{known-bugs-and-limitations}
2463 Here are the known bugs and limitations in Nitpick at the time of writing:
2466 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2467 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2468 Nitpick to generate spurious counterexamples for theorems that refer to values
2469 for which the function is not defined. For example:
2472 \textbf{primrec} \textit{prec} \textbf{where} \\
2473 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2474 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2475 \textbf{nitpick} \\[2\smallskipamount]
2476 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2478 \\[2\smallskipamount]
2479 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2480 \textbf{by}~(\textit{auto simp}: \textit{prec\_def})
2483 Such theorems are considered bad style because they rely on the internal
2484 representation of functions synthesized by Isabelle, which is an implementation
2487 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2488 which can become invalid if you change the definition of an inductive predicate
2489 that is registered in the cache. To clear the cache,
2490 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2491 501$\,\textit{ms}$).
2493 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2494 \textbf{guess} command in a structured proof.
2496 \item[$\bullet$] The \textit{nitpick\_} attributes and the
2497 \textit{Nitpick.register\_} functions can cause havoc if used improperly.
2499 \item[$\bullet$] Although this has never been observed, arbitrary theorem
2500 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
2502 \item[$\bullet$] Local definitions are not supported and result in an error.
2504 %\item[$\bullet$] All constants and types whose names start with
2505 %\textit{Nitpick}{.} are reserved for internal use.
2509 \bibliography{../manual}{}
2510 \bibliographystyle{abbrv}