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 \title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
44 Picking Nits \\[\smallskipamount]
45 \Large A User's Guide to Nitpick for Isabelle/HOL 2010}
47 Jasmin Christian Blanchette \\
48 {\normalsize Fakult\"at f\"ur Informatik, Technische Universit\"at M\"unchen} \\
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78 \section{Introduction}
81 Nitpick \cite{blanchette-nipkow-2009} is a counterexample generator for
82 Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
83 combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
84 quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
85 first-order relational model finder developed by the Software Design Group at
86 MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
87 borrows many ideas and code fragments, but it benefits from Kodkod's
88 optimizations and a new encoding scheme. The name Nitpick is shamelessly
89 appropriated from a now retired Alloy precursor.
91 Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
92 theorem and wait a few seconds. Nonetheless, there are situations where knowing
93 how it works under the hood and how it reacts to various options helps
94 increase the test coverage. This manual also explains how to install the tool on
95 your workstation. Should the motivation fail you, think of the many hours of
96 hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.
98 Another common use of Nitpick is to find out whether the axioms of a locale are
99 satisfiable, while the locale is being developed. To check this, it suffices to
103 \textbf{lemma}~``$\textit{False}$'' \\
104 \textbf{nitpick}~[\textit{show\_all}]
107 after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
108 must find a model for the axioms. If it finds no model, we have an indication
109 that the axioms might be unsatisfiable.
112 \setbox\boxA=\hbox{\texttt{nospam}}
114 The known bugs and limitations at the time of writing are listed in
115 \S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
116 or this manual should be directed to
117 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
118 in.\allowbreak tum.\allowbreak de}.
120 \vskip2.5\smallskipamount
122 \textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
123 suggesting several textual improvements.
124 % and Perry James for reporting a typo.
126 \section{First Steps}
129 This section introduces Nitpick by presenting small examples. If possible, you
130 should try out the examples on your workstation. Your theory file should start
134 \textbf{theory}~\textit{Scratch} \\
135 \textbf{imports}~\textit{Main} \\
139 The results presented here were obtained using the JNI version of MiniSat and
140 with multithreading disabled to reduce nondeterminism. This was done by adding
144 \textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSatJNI}, \,\textit{max\_threads}~= 1]
147 after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
148 Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
149 be installed, as explained in \S\ref{optimizations}. If you have already
150 configured SAT solvers in Isabelle (e.g., for Refute), these will also be
151 available to Nitpick.
153 Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
154 Nitpick also provides an automatic mode that can be enabled by specifying
157 \textbf{nitpick\_params} [\textit{auto}]
160 at the beginning of the theory file. In this mode, Nitpick is run for up to 5
161 seconds (by default) on every newly entered theorem, much like Auto Quickcheck.
163 \subsection{Propositional Logic}
164 \label{propositional-logic}
166 Let's start with a trivial example from propositional logic:
169 \textbf{lemma}~``$P \longleftrightarrow Q$'' \\
173 You should get the following output:
177 Nitpick found a counterexample: \\[2\smallskipamount]
178 \hbox{}\qquad Free variables: \nopagebreak \\
179 \hbox{}\qquad\qquad $P = \textit{True}$ \\
180 \hbox{}\qquad\qquad $Q = \textit{False}$
183 Nitpick can also be invoked on individual subgoals, as in the example below:
186 \textbf{apply}~\textit{auto} \\[2\smallskipamount]
187 {\slshape goal (2 subgoals): \\
188 \ 1. $P\,\Longrightarrow\, Q$ \\
189 \ 2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
190 \textbf{nitpick}~1 \\[2\smallskipamount]
191 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
192 \hbox{}\qquad Free variables: \nopagebreak \\
193 \hbox{}\qquad\qquad $P = \textit{True}$ \\
194 \hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
195 \textbf{nitpick}~2 \\[2\smallskipamount]
196 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
197 \hbox{}\qquad Free variables: \nopagebreak \\
198 \hbox{}\qquad\qquad $P = \textit{False}$ \\
199 \hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
203 \subsection{Type Variables}
204 \label{type-variables}
206 If you are left unimpressed by the previous example, don't worry. The next
207 one is more mind- and computer-boggling:
210 \textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
212 \pagebreak[2] %% TYPESETTING
214 The putative lemma involves the definite description operator, {THE}, presented
215 in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
216 operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
217 lemma is merely asserting the indefinite description operator axiom with {THE}
218 substituted for {SOME}.
220 The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
221 containing type variables, Nitpick enumerates the possible domains for each type
222 variable, up to a given cardinality (8 by default), looking for a finite
226 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
228 Trying 8 scopes: \nopagebreak \\
229 \hbox{}\qquad \textit{card}~$'a$~= 1; \\
230 \hbox{}\qquad \textit{card}~$'a$~= 2; \\
231 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
232 \hbox{}\qquad \textit{card}~$'a$~= 8. \\[2\smallskipamount]
233 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
234 \hbox{}\qquad Free variables: \nopagebreak \\
235 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
236 \hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
240 Nitpick found a counterexample in which $'a$ has cardinality 3. (For
241 cardinalities 1 and 2, the formula holds.) In the counterexample, the three
242 values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
244 The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
245 \textit{verbose} is enabled. You can specify \textit{verbose} each time you
246 invoke \textbf{nitpick}, or you can set it globally using the command
249 \textbf{nitpick\_params} [\textit{verbose}]
252 This command also displays the current default values for all of the options
253 supported by Nitpick. The options are listed in \S\ref{option-reference}.
255 \subsection{Constants}
258 By just looking at Nitpick's output, it might not be clear why the
259 counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
260 this time telling it to show the values of the constants that occur in the
264 \textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
265 \textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
267 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
268 \hbox{}\qquad Free variables: \nopagebreak \\
269 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
270 \hbox{}\qquad\qquad $x = a_3$ \\
271 \hbox{}\qquad Constant: \nopagebreak \\
272 \hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
275 We can see more clearly now. Since the predicate $P$ isn't true for a unique
276 value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
277 $a_1$. Since $P~a_1$ is false, the entire formula is falsified.
279 As an optimization, Nitpick's preprocessor introduced the special constant
280 ``\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
281 $\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
282 satisfying $P~y$. We disable this optimization by passing the
283 \textit{full\_descrs} option:
286 \textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
288 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
289 \hbox{}\qquad Free variables: \nopagebreak \\
290 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
291 \hbox{}\qquad\qquad $x = a_3$ \\
292 \hbox{}\qquad Constant: \nopagebreak \\
293 \hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
296 As the result of another optimization, Nitpick directly assigned a value to the
297 subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
298 disable this second optimization by using the command
301 \textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
302 \textit{show\_consts}]
305 we finally get \textit{The}:
308 \slshape Constant: \nopagebreak \\
309 \hbox{}\qquad $\mathit{The} = \undef{}
310 (\!\begin{aligned}[t]%
311 & \{\} := a_3,\> \{a_3\} := a_3,\> \{a_2\} := a_2, \\[-2pt] %% TYPESETTING
312 & \{a_2, a_3\} := a_1,\> \{a_1\} := a_1,\> \{a_1, a_3\} := a_3, \\[-2pt]
313 & \{a_1, a_2\} := a_3,\> \{a_1, a_2, a_3\} := a_3)\end{aligned}$
316 Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
317 just like before.\footnote{The \undef{} symbol's presence is explained as
318 follows: In higher-order logic, any function can be built from the undefined
319 function using repeated applications of the function update operator $f(x :=
320 y)$, just like any list can be built from the empty list using $x \mathbin{\#}
323 Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
324 unique $x$ such that'') at the front of our putative lemma's assumption:
327 \textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
330 The fix appears to work:
333 \textbf{nitpick} \\[2\smallskipamount]
334 \slshape Nitpick found no counterexample.
337 We can further increase our confidence in the formula by exhausting all
338 cardinalities up to 50:
341 \textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
342 can be entered as \texttt{-} (hyphen) or
343 \texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
344 \slshape Nitpick found no counterexample.
347 Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
350 \textbf{sledgehammer} \\[2\smallskipamount]
351 {\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
352 $\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
353 Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
354 \textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
355 {\slshape No subgoals!}% \\[2\smallskipamount]
359 This must be our lucky day.
361 \subsection{Skolemization}
362 \label{skolemization}
364 Are all invertible functions onto? Let's find out:
367 \textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
368 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
369 \textbf{nitpick} \\[2\smallskipamount]
371 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
372 \hbox{}\qquad Free variable: \nopagebreak \\
373 \hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
374 \hbox{}\qquad Skolem constants: \nopagebreak \\
375 \hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
376 \hbox{}\qquad\qquad $y = a_2$
379 Although $f$ is the only free variable occurring in the formula, Nitpick also
380 displays values for the bound variables $g$ and $y$. These values are available
381 to Nitpick because it performs skolemization as a preprocessing step.
383 In the previous example, skolemization only affected the outermost quantifiers.
384 This is not always the case, as illustrated below:
387 \textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
388 \textbf{nitpick} \\[2\smallskipamount]
390 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
391 \hbox{}\qquad Skolem constant: \nopagebreak \\
392 \hbox{}\qquad\qquad $\lambda x.\; f =
393 \undef{}(\!\begin{aligned}[t]
394 & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
395 & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
398 The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
399 $x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
400 function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
401 maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
403 The source of the Skolem constants is sometimes more obscure:
406 \textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
407 \textbf{nitpick} \\[2\smallskipamount]
409 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
410 \hbox{}\qquad Free variable: \nopagebreak \\
411 \hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
412 \hbox{}\qquad Skolem constants: \nopagebreak \\
413 \hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
414 \hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
417 What happened here is that Nitpick expanded the \textit{sym} constant to its
421 $\mathit{sym}~r \,\equiv\,
422 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
425 As their names suggest, the Skolem constants $\mathit{sym}.x$ and
426 $\mathit{sym}.y$ are simply the bound variables $x$ and $y$
427 from \textit{sym}'s definition.
429 Although skolemization is a useful optimization, you can disable it by invoking
430 Nitpick with \textit{dont\_skolemize}. See \S\ref{optimizations} for details.
432 \subsection{Natural Numbers and Integers}
433 \label{natural-numbers-and-integers}
435 Because of the axiom of infinity, the type \textit{nat} does not admit any
436 finite models. To deal with this, Nitpick considers prefixes $\{0,\, 1,\,
437 \ldots,\, K - 1\}$ of \textit{nat} (where $K = \textit{card}~\textit{nat}$) and
438 maps all other numbers to the undefined value ($\unk$). The type \textit{int} is
439 handled in a similar way: If $K = \textit{card}~\textit{int}$, the subset of
440 \textit{int} known to Nitpick is $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor
441 K/2 \rfloor\}$. Undefined values lead to a three-valued logic.
443 Here is an example involving \textit{int}:
446 \textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
447 \textbf{nitpick} \\[2\smallskipamount]
448 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
449 \hbox{}\qquad Free variables: \nopagebreak \\
450 \hbox{}\qquad\qquad $i = 0$ \\
451 \hbox{}\qquad\qquad $j = 1$ \\
452 \hbox{}\qquad\qquad $m = 1$ \\
453 \hbox{}\qquad\qquad $n = 0$
456 With infinite types, we don't always have the luxury of a genuine counterexample
457 and must often content ourselves with a potential one. The tedious task of
458 finding out whether the potential counterexample is in fact genuine can be
459 outsourced to \textit{auto} by passing the option \textit{check\_potential}. For
463 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
464 \textbf{nitpick} [\textit{card~nat}~= 100,\, \textit{check\_potential}] \\[2\smallskipamount]
465 \slshape Nitpick found a potential counterexample: \\[2\smallskipamount]
466 \hbox{}\qquad Free variable: \nopagebreak \\
467 \hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
468 Confirmation by ``\textit{auto}'': The above counterexample is genuine.
471 You might wonder why the counterexample is first reported as potential. The root
472 of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
473 \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
474 that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
475 \textit{False}; but otherwise, it does not know anything about values of $n \ge
476 \textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
477 \textit{True}. Since the assumption can never be satisfied, the putative lemma
478 can never be falsified.
480 Incidentally, if you distrust the so-called genuine counterexamples, you can
481 enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
482 aware that \textit{auto} will often fail to prove that the counterexample is
485 Some conjectures involving elementary number theory make Nitpick look like a
486 giant with feet of clay:
489 \textbf{lemma} ``$P~\textit{Suc}$'' \\
490 \textbf{nitpick} [\textit{card} = 1--6] \\[2\smallskipamount]
492 Nitpick found no counterexample.
495 For any cardinality $k$, \textit{Suc} is the partial function $\{0 \mapsto 1,\,
496 1 \mapsto 2,\, \ldots,\, k - 1 \mapsto \unk\}$, which evaluates to $\unk$ when
497 it is passed as argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$.
498 The next example is similar:
501 \textbf{lemma} ``$P~(\textit{op}~{+}\Colon
502 \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
503 \textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
504 {\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
505 \hbox{}\qquad Free variable: \nopagebreak \\
506 \hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
507 \textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
508 {\slshape Nitpick found no counterexample.}
511 The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
512 $\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
515 Because numbers are infinite and are approximated using a three-valued logic,
516 there is usually no need to systematically enumerate domain sizes. If Nitpick
517 cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
518 unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
519 example above is an exception to this principle.) Nitpick nonetheless enumerates
520 all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
521 cardinalities are fast to handle and give rise to simpler counterexamples. This
522 is explained in more detail in \S\ref{scope-monotonicity}.
524 \subsection{Inductive Datatypes}
525 \label{inductive-datatypes}
527 Like natural numbers and integers, inductive datatypes with recursive
528 constructors admit no finite models and must be approximated by a subterm-closed
529 subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
530 Nitpick looks for all counterexamples that can be built using at most 10
533 Let's see with an example involving \textit{hd} (which returns the first element
534 of a list) and $@$ (which concatenates two lists):
537 \textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
538 \textbf{nitpick} \\[2\smallskipamount]
539 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
540 \hbox{}\qquad Free variables: \nopagebreak \\
541 \hbox{}\qquad\qquad $\textit{xs} = []$ \\
542 \hbox{}\qquad\qquad $\textit{y} = a_3$
545 To see why the counterexample is genuine, we enable \textit{show\_consts}
546 and \textit{show\_\allowbreak datatypes}:
549 {\slshape Datatype:} \\
550 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_3, a_3],\, [a_3],\, \unr\}$ \\
551 {\slshape Constants:} \\
552 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_3, a_3],\> [a_3, a_3] := \unk,\> [a_3] := \unk)$ \\
553 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_3, a_3] := a_3,\> [a_3] := a_3)$
556 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
559 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
560 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_3,
561 a_3]$ to $[a_3]$ would normally give $[a_3, a_3, a_3]$, but this value is not
562 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
563 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
564 appending $[a_3, a_3]$ to itself gives $\unk$.
566 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
567 considers the following subsets:
569 \kern-.5\smallskipamount %% TYPESETTING
573 $\{[],\, [a_1],\, [a_2]\}$; \\
574 $\{[],\, [a_1],\, [a_3]\}$; \\
575 $\{[],\, [a_2],\, [a_3]\}$; \\
576 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\
577 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\
578 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\
579 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\
580 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\
581 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\
582 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\
583 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\
584 $\{[],\, [a_3],\, [a_3, a_3]\}$.
588 \kern-2\smallskipamount %% TYPESETTING
590 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
591 are listed and only those. As an example of a non-subterm-closed subset,
592 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_3]\}$, and observe
593 that $[a_1, a_3]$ (i.e., $a_1 \mathbin{\#} [a_3]$) has $[a_3] \notin
594 \mathcal{S}$ as a subterm.
596 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
599 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
600 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
602 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
603 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
604 \hbox{}\qquad Free variables: \nopagebreak \\
605 \hbox{}\qquad\qquad $\textit{xs} = [a_2]$ \\
606 \hbox{}\qquad\qquad $\textit{ys} = [a_3]$ \\
607 \hbox{}\qquad Datatypes: \\
608 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
609 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_3],\, [a_2],\, \unr\}$
612 Because datatypes are approximated using a three-valued logic, there is usually
613 no need to systematically enumerate cardinalities: If Nitpick cannot find a
614 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
615 unlikely that one could be found for smaller cardinalities.
617 \subsection{Typedefs, Records, Rationals, and Reals}
618 \label{typedefs-records-rationals-and-reals}
620 Nitpick generally treats types declared using \textbf{typedef} as datatypes
621 whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
625 \textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
626 \textbf{by}~\textit{blast} \\[2\smallskipamount]
627 \textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
628 \textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
629 \textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
630 \textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
631 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
632 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
633 \hbox{}\qquad Free variables: \nopagebreak \\
634 \hbox{}\qquad\qquad $P = \{\Abs{1},\, \Abs{0}\}$ \\
635 \hbox{}\qquad\qquad $x = \Abs{2}$ \\
636 \hbox{}\qquad Datatypes: \\
637 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
638 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{2},\, \Abs{1},\, \Abs{0},\, \unr\}$
642 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
645 Records, which are implemented as \textbf{typedef}s behind the scenes, are
646 handled in much the same way:
649 \textbf{record} \textit{point} = \\
650 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
651 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
652 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
653 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
654 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
655 \hbox{}\qquad Free variables: \nopagebreak \\
656 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
657 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
658 \hbox{}\qquad Datatypes: \\
659 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
660 \hbox{}\qquad\qquad $\textit{point} = \{\lparr\textit{Xcoord} = 1,\>
661 \textit{Ycoord} = 1\rparr,\> \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr,\, \unr\}$\kern-1pt %% QUIET
664 Finally, Nitpick provides rudimentary support for rationals and reals using a
668 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
669 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
670 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
671 \hbox{}\qquad Free variables: \nopagebreak \\
672 \hbox{}\qquad\qquad $x = 1/2$ \\
673 \hbox{}\qquad\qquad $y = -1/2$ \\
674 \hbox{}\qquad Datatypes: \\
675 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
676 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
677 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
680 \subsection{Inductive and Coinductive Predicates}
681 \label{inductive-and-coinductive-predicates}
683 Inductively defined predicates (and sets) are particularly problematic for
684 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
685 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
686 the problem is that they are defined using a least fixed point construction.
688 Nitpick's philosophy is that not all inductive predicates are equal. Consider
689 the \textit{even} predicate below:
692 \textbf{inductive}~\textit{even}~\textbf{where} \\
693 ``\textit{even}~0'' $\,\mid$ \\
694 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
697 This predicate enjoys the desirable property of being well-founded, which means
698 that the introduction rules don't give rise to infinite chains of the form
701 $\cdots\,\Longrightarrow\, \textit{even}~k''
702 \,\Longrightarrow\, \textit{even}~k'
703 \,\Longrightarrow\, \textit{even}~k.$
706 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
710 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
711 \,\Longrightarrow\, \textit{even}~(k - 2)
712 \,\Longrightarrow\, \textit{even}~k.$
715 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
716 fixed point computation.%
717 \footnote{If an inductive predicate is
718 well-founded, then it has exactly one fixed point, which is simultaneously the
719 least and the greatest fixed point. In these circumstances, the computation of
720 the least fixed point amounts to the computation of an arbitrary fixed point,
721 which can be performed using a straightforward recursive equation.}
722 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
723 just as Isabelle's \textbf{function} package usually discharges termination
724 proof obligations automatically.
726 Let's try an example:
729 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
730 \textbf{nitpick}~[\textit{card nat}~= 100,\, \textit{verbose}] \\[2\smallskipamount]
731 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
732 Nitpick can compute it efficiently. \\[2\smallskipamount]
734 \hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
735 Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
736 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
737 Nitpick could not find a better counterexample. \\[2\smallskipamount]
741 No genuine counterexample is possible because Nitpick cannot rule out the
742 existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
743 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
744 existential quantifier:
747 \textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
748 \textbf{nitpick}~[\textit{card nat}~= 100] \\[2\smallskipamount]
749 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
750 \hbox{}\qquad Empty assignment
753 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
754 we use the following definition instead?
757 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
758 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
759 ``$\textit{even}'~2$'' $\,\mid$ \\
760 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
763 This definition is not well-founded: From $\textit{even}'~0$ and
764 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
765 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
767 Let's check a property involving $\textit{even}'$. To make up for the
768 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
769 \textit{nat}'s cardinality to a mere 10:
772 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
773 \lnot\;\textit{even}'~n$'' \\
774 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
776 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
777 Nitpick might need to unroll it. \\[2\smallskipamount]
779 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
780 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
781 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
782 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
783 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
784 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
785 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
786 \hbox{}\qquad Constant: \nopagebreak \\
787 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
788 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
789 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
790 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
794 Nitpick's output is very instructive. First, it tells us that the predicate is
795 unrolled, meaning that it is computed iteratively from the empty set. Then it
796 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
799 The output also shows how each iteration contributes to $\textit{even}'$. The
800 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
801 predicate depends on an iteration counter. Iteration 0 provides the basis
802 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
803 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
804 iterations would not contribute any new elements.
806 Some values are marked with superscripted question
807 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
808 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
809 \textit{True} or $\unk$, never \textit{False}.
811 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
812 iterations. However, these numbers are bounded by the cardinality of the
813 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
814 ever needed to compute the value of a \textit{nat} predicate. You can specify
815 the number of iterations using the \textit{iter} option, as explained in
816 \S\ref{scope-of-search}.
818 In the next formula, $\textit{even}'$ occurs both positively and negatively:
821 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
822 \textbf{nitpick} [\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
823 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
824 \hbox{}\qquad Free variable: \nopagebreak \\
825 \hbox{}\qquad\qquad $n = 1$ \\
826 \hbox{}\qquad Constants: \nopagebreak \\
827 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
828 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
829 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
832 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
833 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
834 fixed point (not necessarily the least one). It is used to falsify
835 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
836 $\textit{even}'~(n - 2)$.
838 Coinductive predicates are handled dually. For example:
841 \textbf{coinductive} \textit{nats} \textbf{where} \\
842 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
843 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
844 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
845 \slshape Nitpick found a counterexample:
846 \\[2\smallskipamount]
847 \hbox{}\qquad Constants: \nopagebreak \\
848 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
849 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
850 & \unr\})\end{aligned}$ \\
851 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
854 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
855 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
856 inductive predicates for which each the predicate occurs in at most one
857 assumption of each introduction rule. For example:
860 \textbf{inductive} \textit{odd} \textbf{where} \\
861 ``$\textit{odd}~1$'' $\,\mid$ \\
862 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
863 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
864 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
865 \slshape Nitpick found a counterexample:
866 \\[2\smallskipamount]
867 \hbox{}\qquad Free variable: \nopagebreak \\
868 \hbox{}\qquad\qquad $n = 1$ \\
869 \hbox{}\qquad Constants: \nopagebreak \\
870 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
871 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
872 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
874 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
875 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
877 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
878 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
879 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
883 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
884 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
885 elements from known ones. The set $\textit{odd}$ consists of all the values
886 reachable through the reflexive transitive closure of
887 $\textit{odd}_{\textrm{step}}$ starting with any element from
888 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
889 transitive closure to encode linear predicates is normally either more thorough
890 or more efficient than unrolling (depending on the value of \textit{iter}), but
891 for those cases where it isn't you can disable it by passing the
892 \textit{dont\_star\_linear\_preds} option.
894 \subsection{Coinductive Datatypes}
895 \label{coinductive-datatypes}
897 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
898 datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
899 list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
900 these lazy lists seamlessly and provides a hook, described in
901 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
904 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
905 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
906 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
907 1, 2, 3, \ldots]$ can be defined as lazy lists using the
908 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
909 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
910 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
912 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
913 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
917 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
918 \textbf{nitpick} \\[2\smallskipamount]
919 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
920 \hbox{}\qquad Free variables: \nopagebreak \\
921 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
922 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
925 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
926 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
927 infinite list $[a_1, a_1, a_1, \ldots]$.
929 The next example is more interesting:
932 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
933 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
934 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
935 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
936 some scopes. \\[2\smallskipamount]
938 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 1,
939 and \textit{bisim\_depth}~= 0. \\
940 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
941 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 8,
942 and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
943 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
944 \textit{card}~``\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
946 \\[2\smallskipamount]
947 \hbox{}\qquad Free variables: \nopagebreak \\
948 \hbox{}\qquad\qquad $\textit{a} = a_2$ \\
949 \hbox{}\qquad\qquad $\textit{b} = a_1$ \\
950 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
951 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_1~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega)$ \\[2\smallskipamount]
955 The lazy list $\textit{xs}$ is simply $[a_2, a_2, a_2, \ldots]$, whereas
956 $\textit{ys}$ is $[a_1, a_2, a_2, a_2, \ldots]$, i.e., a lasso-shaped list with
957 $[a_1]$ as its stem and $[a_2]$ as its cycle. In general, the list segment
958 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
959 the segment leading to the binder is the stem.
961 A salient property of coinductive datatypes is that two objects are considered
962 equal if and only if they lead to the same observations. For example, the lazy
963 lists $\textrm{THE}~\omega.\; \omega =
964 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
965 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
966 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
967 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
968 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
969 concept of equality for coinductive datatypes is called bisimulation and is
970 defined coinductively.
972 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
973 the Kodkod problem to ensure that distinct objects lead to different
974 observations. This precaution is somewhat expensive and often unnecessary, so it
975 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
976 bisimilarity check is then performed \textsl{after} the counterexample has been
977 found to ensure correctness. If this after-the-fact check fails, the
978 counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
979 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
980 check for the previous example saves approximately 150~milli\-seconds; the speed
981 gains can be more significant for larger scopes.
983 The next formula illustrates the need for bisimilarity (either as a Kodkod
984 predicate or as an after-the-fact check) to prevent spurious counterexamples:
987 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
988 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
989 \textbf{nitpick} [\textit{bisim\_depth} = $-1$,\, \textit{show\_datatypes}] \\[2\smallskipamount]
990 \slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
991 \hbox{}\qquad Free variables: \nopagebreak \\
992 \hbox{}\qquad\qquad $a = a_2$ \\
993 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
994 \textit{LCons}~a_2~\omega$ \\
995 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
996 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
997 \hbox{}\qquad\qquad $'a~\textit{llist} =
998 \{\!\begin{aligned}[t]
999 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega, \\[-2pt]
1000 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega,\> \unr\}\end{aligned}$
1001 \\[2\smallskipamount]
1002 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1003 that the counterexample is genuine. \\[2\smallskipamount]
1004 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1005 \slshape Nitpick found no counterexample.
1008 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1009 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1011 A compromise between leaving out the bisimilarity predicate from the Kodkod
1012 problem and performing the after-the-fact check is to specify a lower
1013 nonnegative \textit{bisim\_depth} value than the default one provided by
1014 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1015 be distinguished from each other by their prefixes of length $K$. Be aware that
1016 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1017 finding any counterexamples.
1022 Nitpick normally maps function and product types directly to the corresponding
1023 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1024 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1025 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1026 off to treat these types in the same way as plain datatypes, by approximating
1027 them by a subset of a given cardinality. This technique is called ``boxing'' and
1028 is particularly useful for functions passed as arguments to other functions, for
1029 high-arity functions, and for large tuples. Under the hood, boxing involves
1030 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1031 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1033 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1034 using de Bruijn's notation:
1037 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1040 The $\textit{lift}~t~k$ function increments all variables with indices greater
1041 than or equal to $k$ by one:
1044 \textbf{primrec} \textit{lift} \textbf{where} \\
1045 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1046 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1047 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1050 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1051 term $t$ has a loose variable with index $k$ or more:
1054 \textbf{primrec}~\textit{loose} \textbf{where} \\
1055 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1056 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1057 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1060 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1064 \textbf{primrec}~\textit{subst} \textbf{where} \\
1065 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1066 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1067 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1068 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1071 A substitution is a function that maps variable indices to terms. Observe that
1072 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1073 away, because the recursive call for the \textit{Lam} case involves an altered
1074 version. Also notice the \textit{lift} call, which increments the variable
1075 indices when moving under a \textit{Lam}.
1077 A reasonable property to expect of substitution is that it should leave closed
1078 terms unchanged. Alas, even this simple property does not hold:
1081 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1082 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1084 Trying 8 scopes: \nopagebreak \\
1085 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1086 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1087 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1088 \hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
1089 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1090 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1091 \hbox{}\qquad Free variables: \nopagebreak \\
1092 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1093 & 0 := \textit{Var}~0,\>
1094 1 := \textit{Var}~0,\>
1095 2 := \textit{Var}~0, \\[-2pt]
1096 & 3 := \textit{Var}~0,\>
1097 4 := \textit{Var}~0,\>
1098 5 := \textit{Var}~0)\end{aligned}$ \\
1099 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1100 Total time: $4679$ ms.
1103 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1104 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1105 $\lambda$-term notation, $t$~is
1106 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1107 The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
1108 replaced with $\textit{lift}~(\sigma~m)~0$.
1110 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1111 cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1112 For the formula of interest, knowing 6 values of that type was enough to find
1113 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1114 considered, a hopeless undertaking:
1117 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1118 {\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
1122 Boxing can be enabled or disabled globally or on a per-type basis using the
1123 \textit{box} option. Moreover, setting the cardinality of a function or
1124 product type implicitly enables boxing for that type. Nitpick usually performs
1125 reasonable choices about which types should be boxed, but option tweaking
1130 \subsection{Scope Monotonicity}
1131 \label{scope-monotonicity}
1133 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1134 and \textit{max}) controls which scopes are actually tested. In general, to
1135 exhaust all models below a certain cardinality bound, the number of scopes that
1136 Nitpick must consider increases exponentially with the number of type variables
1137 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1138 cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
1139 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1141 Fortunately, many formulas exhibit a property called \textsl{scope
1142 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1143 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1145 Consider the formula
1148 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1151 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1152 $'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
1153 exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
1154 that any counterexample found with a small scope would still be a counterexample
1155 in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
1156 by the larger scope. Nitpick comes to the same conclusion after a careful
1157 inspection of the formula and the relevant definitions:
1160 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1162 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1163 Nitpick might be able to skip some scopes.
1164 \\[2\smallskipamount]
1166 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1167 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1168 \textit{list}''~= 1, \\
1169 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
1170 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
1171 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1172 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1173 \textit{list}''~= 2, \\
1174 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
1175 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
1176 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1177 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
1178 \textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
1179 \textit{list}''~= 8, \\
1180 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
1181 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
1182 \\[2\smallskipamount]
1183 Nitpick found a counterexample for
1184 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1185 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1186 \textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
1187 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
1188 \\[2\smallskipamount]
1189 \hbox{}\qquad Free variables: \nopagebreak \\
1190 \hbox{}\qquad\qquad $\textit{xs} = [a_4, a_5]$ \\
1191 \hbox{}\qquad\qquad $\textit{ys} = [b_3, b_3]$ \\[2\smallskipamount]
1192 Total time: 1636 ms.
1195 In theory, it should be sufficient to test a single scope:
1198 \textbf{nitpick}~[\textit{card}~= 8]
1201 However, this is often less efficient in practice and may lead to overly complex
1204 If the monotonicity check fails but we believe that the formula is monotonic (or
1205 we don't mind missing some counterexamples), we can pass the
1206 \textit{mono} option. To convince yourself that this option is risky,
1207 simply consider this example from \S\ref{skolemization}:
1210 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1211 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1212 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1213 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1214 \textbf{nitpick} \\[2\smallskipamount]
1216 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1217 \hbox{}\qquad $\vdots$
1220 (It turns out the formula holds if and only if $\textit{card}~'a \le
1221 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1222 monotonicity checks can be disabled by passing \textit{non\_mono}
1223 (\S\ref{optimizations}).
1225 As insinuated in \S\ref{natural-numbers-and-integers} and
1226 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1227 are normally monotonic and treated as such. The same is true for record types,
1228 \textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
1229 cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
1230 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1231 consider only 8~scopes instead of $32\,768$.
1233 \section{Case Studies}
1234 \label{case-studies}
1236 As a didactic device, the previous section focused mostly on toy formulas whose
1237 validity can easily be assessed just by looking at the formula. We will now
1238 review two somewhat more realistic case studies that are within Nitpick's
1239 reach:\ a context-free grammar modeled by mutually inductive sets and a
1240 functional implementation of AA trees. The results presented in this
1241 section were produced with the following settings:
1244 \textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
1247 \subsection{A Context-Free Grammar}
1248 \label{a-context-free-grammar}
1250 Our first case study is taken from section 7.4 in the Isabelle tutorial
1251 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1252 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1255 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1256 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1257 $A$ & $::=$ & $aS \mid bAA$ \\
1258 $B$ & $::=$ & $bS \mid aBB$
1262 The intuition behind the grammar is that $A$ generates all string with one more
1263 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1265 The alphabet consists exclusively of $a$'s and $b$'s:
1268 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1271 Strings over the alphabet are represented by \textit{alphabet list}s.
1272 Nonterminals in the grammar become sets of strings. The production rules
1273 presented above can be expressed as a mutually inductive definition:
1276 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1277 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1278 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1279 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1280 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1281 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1282 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1285 The conversion of the grammar into the inductive definition was done manually by
1286 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1289 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1290 d'\^etre}. A good approach is to state desirable properties of the specification
1291 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1292 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1293 counterexamples will point to bugs in the specification. For our grammar
1294 example, we will proceed in two steps, separating the soundness and the
1295 completeness of the set $S$. First, soundness:
1298 \textbf{theorem}~\textit{S\_sound}: \\
1299 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1300 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1301 \textbf{nitpick} \\[2\smallskipamount]
1302 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1303 \hbox{}\qquad Free variable: \nopagebreak \\
1304 \hbox{}\qquad\qquad $w = [b]$
1307 It would seem that $[b] \in S$. How could this be? An inspection of the
1308 introduction rules reveals that the only rule with a right-hand side of the form
1309 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1313 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1316 On closer inspection, we can see that this rule is wrong. To match the
1317 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1321 \textbf{nitpick} \\[2\smallskipamount]
1322 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1323 \hbox{}\qquad Free variable: \nopagebreak \\
1324 \hbox{}\qquad\qquad $w = [a, a, b]$
1327 Some detective work is necessary to find out what went wrong here. To get $[a,
1328 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1332 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1335 Now, this formula must be wrong: The same assumption occurs twice, and the
1336 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1337 the assumptions should have been a $w$.
1339 With the correction made, we don't get any counterexample from Nitpick. Let's
1340 move on and check completeness:
1343 \textbf{theorem}~\textit{S\_complete}: \\
1344 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1345 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1346 \longrightarrow w \in S$'' \\
1347 \textbf{nitpick} \\[2\smallskipamount]
1348 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1349 \hbox{}\qquad Free variable: \nopagebreak \\
1350 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1353 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1354 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1355 the introduction rules we have are probably correct. Perhaps we simply lack an
1356 introduction rule. Comparing the grammar with the inductive definition, our
1357 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1358 without which the grammar cannot generate two or more $b$'s in a row. So we add
1362 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1365 With this last change, we don't get any counterexamples from Nitpick for either
1366 soundness or completeness. We can even generalize our result to cover $A$ and
1370 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1371 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1372 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1373 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1374 \textbf{nitpick} \\[2\smallskipamount]
1375 \slshape Nitpick found no counterexample.
1378 \subsection{AA Trees}
1381 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1382 similar performance to red-black trees, but with a simpler implementation
1383 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1384 a total order $<$. We start by defining the datatype and some basic extractor
1388 \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]
1389 \textbf{primrec} \textit{data} \textbf{where} \\
1390 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1391 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1392 \textbf{primrec} \textit{dataset} \textbf{where} \\
1393 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1394 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1395 \textbf{primrec} \textit{level} \textbf{where} \\
1396 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1397 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1398 \textbf{primrec} \textit{left} \textbf{where} \\
1399 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1400 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1401 \textbf{primrec} \textit{right} \textbf{where} \\
1402 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1403 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1406 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1407 as follows \cite{wikipedia-2009-aa-trees}:
1409 \kern.2\parskip %% TYPESETTING
1412 Each node has a level field, and the following invariants must remain true for
1413 the tree to be valid:
1417 \kern-.4\parskip %% TYPESETTING
1422 \item[1.] The level of a leaf node is one.
1423 \item[2.] The level of a left child is strictly less than that of its parent.
1424 \item[3.] The level of a right child is less than or equal to that of its parent.
1425 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1426 \item[5.] Every node of level greater than one must have two children.
1431 \kern.4\parskip %% TYPESETTING
1433 The \textit{wf} predicate formalizes this description:
1436 \textbf{primrec} \textit{wf} \textbf{where} \\
1437 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1438 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1439 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1440 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1441 \phantom{``$($}$\textrm{else}$ \\
1442 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1443 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1444 \mathrel{\land} \textit{level}~u \le k$ \\
1445 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1448 Rebalancing the tree upon insertion and removal of elements is performed by two
1449 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1452 \textbf{primrec} \textit{skew} \textbf{where} \\
1453 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1454 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1455 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1456 \textit{level}~t~\textrm{then}$ \\
1457 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1458 (\textit{right}~t)~u)$ \\
1459 \phantom{``(}$\textrm{else}$ \\
1460 \phantom{``(\quad}$N~x~k~t~u)$''
1464 \textbf{primrec} \textit{split} \textbf{where} \\
1465 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1466 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1467 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1468 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1469 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1470 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1471 \phantom{``(}$\textrm{else}$ \\
1472 \phantom{``(\quad}$N~x~k~t~u)$''
1475 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1476 of elements stored in the tree:
1479 \textbf{theorem}~\textit{dataset\_skew\_split}:\\
1480 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1481 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1482 \textbf{nitpick} \\[2\smallskipamount]
1483 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1486 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1487 should not alter the tree:
1490 \textbf{theorem}~\textit{wf\_skew\_split}:\\
1491 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1492 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1493 \textbf{nitpick} \\[2\smallskipamount]
1494 {\slshape Nitpick found no counterexample.}
1497 Insertion is implemented recursively. It preserves the sort order:
1500 \textbf{primrec}~\textit{insort} \textbf{where} \\
1501 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1502 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1503 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1504 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1507 Notice that we deliberately commented out the application of \textit{skew} and
1508 \textit{split}. Let's see if this causes any problems:
1511 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1512 \textbf{nitpick} \\[2\smallskipamount]
1513 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1514 \hbox{}\qquad Free variables: \nopagebreak \\
1515 \hbox{}\qquad\qquad $t = N~a_3~1~\Lambda~\Lambda$ \\
1516 \hbox{}\qquad\qquad $x = a_4$ \\[2\smallskipamount]
1517 Hint: Maybe you forgot a type constraint?
1520 It's hard to see why this is a counterexample. The hint is of no help here. To
1521 improve readability, we will restrict the theorem to \textit{nat}, so that we
1522 don't need to look up the value of the $\textit{op}~{<}$ constant to find out
1523 which element is smaller than the other. In addition, we will tell Nitpick to
1524 display the value of $\textit{insort}~t~x$ using the \textit{eval} option. This
1528 \textbf{theorem} \textit{wf\_insort\_nat}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1529 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1530 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1531 \hbox{}\qquad Free variables: \nopagebreak \\
1532 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1533 \hbox{}\qquad\qquad $x = 0$ \\
1534 \hbox{}\qquad Evaluated term: \\
1535 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1538 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1539 where both have a level of 1. This violates the second AA tree invariant, which
1540 states that a left child's level must be less than its parent's. This shouldn't
1541 come as a surprise, considering that we commented out the tree rebalancing code.
1542 Reintroducing the code seems to solve the problem:
1545 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1546 \textbf{nitpick} \\[2\smallskipamount]
1547 {\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
1550 Insertion should transform the set of elements represented by the tree in the
1554 \textbf{theorem} \textit{dataset\_insort}:\kern.4em
1555 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1556 \textbf{nitpick} \\[2\smallskipamount]
1557 {\slshape Nitpick ran out of time after checking 5 of 8 scopes.}
1560 We could continue like this and sketch a complete theory of AA trees without
1561 performing a single proof. Once the definitions and main theorems are in place
1562 and have been thoroughly tested using Nitpick, we could start working on the
1563 proofs. Developing theories this way usually saves time, because faulty theorems
1564 and definitions are discovered much earlier in the process.
1566 \section{Option Reference}
1567 \label{option-reference}
1569 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1570 \def\qty#1{$\left<\textit{#1}\right>$}
1571 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1572 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1573 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1574 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1575 \def\ops#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1576 \def\opt#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1577 \def\opu#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1578 \def\opusmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1580 Nitpick's behavior can be influenced by various options, which can be specified
1581 in brackets after the \textbf{nitpick} command. Default values can be set
1582 using \textbf{nitpick\_\allowbreak params}. For example:
1585 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1588 The options are categorized as follows:\ mode of operation
1589 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1590 format (\S\ref{output-format}), automatic counterexample checks
1591 (\S\ref{authentication}), optimizations
1592 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1594 The number of options can be overwhelming at first glance. Do not let that worry
1595 you: Nitpick's defaults have been chosen so that it almost always does the right
1596 thing, and the most important options have been covered in context in
1597 \S\ref{first-steps}.
1599 The descriptions below refer to the following syntactic quantities:
1602 \item[$\bullet$] \qtybf{string}: A string.
1603 \item[$\bullet$] \qtybf{bool}: \textit{true} or \textit{false}.
1604 \item[$\bullet$] \qtybf{bool\_or\_smart}: \textit{true}, \textit{false}, or \textit{smart}.
1605 \item[$\bullet$] \qtybf{int}: An integer. Negative integers are prefixed with a hyphen.
1606 \item[$\bullet$] \qtybf{int\_or\_smart}: An integer or \textit{smart}.
1607 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1608 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1610 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1611 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1612 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1613 \item[$\bullet$] \qtybf{const}: The name of a HOL constant.
1614 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1615 \item[$\bullet$] \qtybf{term\_list}: A space-separated list of HOL terms (e.g.,
1616 ``$f~x$''~``$g~y$'').
1617 \item[$\bullet$] \qtybf{type}: A HOL type.
1620 Default values are indicated in square brackets. Boolean options have a negated
1621 counterpart (e.g., \textit{auto} vs.\ \textit{no\_auto}). When setting Boolean
1622 options, ``= \textit{true}'' may be omitted.
1624 \subsection{Mode of Operation}
1625 \label{mode-of-operation}
1628 \opfalse{auto}{no\_auto}
1629 Specifies whether Nitpick should be run automatically on newly entered theorems.
1630 For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation}) and
1631 \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1632 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1633 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1634 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1635 \textit{auto\_timeout} (\S\ref{timeouts}) is used as the time limit instead of
1636 \textit{timeout} (\S\ref{timeouts}). The output is also more concise.
1639 {\small See also \textit{auto\_timeout} (\S\ref{timeouts}).}
1641 \optrue{blocking}{non\_blocking}
1642 Specifies whether the \textbf{nitpick} command should operate synchronously.
1643 The asynchronous (non-blocking) mode lets the user start proving the putative
1644 theorem while Nitpick looks for a counterexample, but it can also be more
1645 confusing. For technical reasons, automatic runs currently always block.
1648 {\small See also \textit{auto} (\S\ref{mode-of-operation}).}
1650 \optrue{falsify}{satisfy}
1651 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1652 satisfying examples (models). This manual assumes throughout that
1653 \textit{falsify} is enabled.
1655 \opsmart{user\_axioms}{no\_user\_axioms}
1656 Specifies whether the user-defined axioms (specified using
1657 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1658 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1659 the constants that occur in the formula to falsify. The option is implicitly set
1660 to \textit{true} for automatic runs.
1662 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1663 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1664 these conditions are tagged as ``likely genuine.'' The \textit{debug}
1665 (\S\ref{output-format}) option can be used to find out which axioms were
1669 {\small See also \textit{auto} (\S\ref{mode-of-operation}), \textit{assms}
1670 (\S\ref{mode-of-operation}), and \textit{debug} (\S\ref{output-format}).}
1672 \optrue{assms}{no\_assms}
1673 Specifies whether the relevant assumptions in structured proof should be
1674 considered. The option is implicitly enabled for automatic runs.
1677 {\small See also \textit{auto} (\S\ref{mode-of-operation})
1678 and \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1680 \opfalse{overlord}{no\_overlord}
1681 Specifies whether Nitpick should put its temporary files in
1682 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1683 debugging Nitpick but also unsafe if several instances of the tool are run
1684 simultaneously. This option is disabled by default unless your home directory
1685 ends with \texttt{blanchet} or \texttt{blanchette}.
1686 %``I thought there was only one overlord.'' --- Tobias Nipkow
1689 {\small See also \textit{debug} (\S\ref{output-format}).}
1692 \subsection{Scope of Search}
1693 \label{scope-of-search}
1696 \opu{card}{type}{int\_seq}
1697 Specifies the sequence of cardinalities to use for a given type. For
1698 \textit{nat} and \textit{int}, the cardinality fully specifies the subset used
1699 to approximate the type. For example:
1701 $$\hbox{\begin{tabular}{@{}rll@{}}%
1702 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
1703 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
1704 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
1709 $$\hbox{\begin{tabular}{@{}rll@{}}%
1710 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
1711 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
1714 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1715 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1716 Although function and product types are normally mapped directly to the
1717 corresponding Kodkod concepts, setting
1718 the cardinality of such types is also allowed and implicitly enables ``boxing''
1719 for them, as explained in the description of the \textit{box}~\qty{type}
1720 and \textit{box} (\S\ref{scope-of-search}) options.
1723 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1725 \opt{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
1726 Specifies the default sequence of cardinalities to use. This can be overridden
1727 on a per-type basis using the \textit{card}~\qty{type} option described above.
1729 \opu{max}{const}{int\_seq}
1730 Specifies the sequence of maximum multiplicities to use for a given
1731 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1732 number of distinct values that it can construct. Nonsensical values (e.g.,
1733 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1734 datatypes equipped with several constructors.
1737 Specifies the default sequence of maximum multiplicities to use for
1738 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
1739 basis using the \textit{max}~\qty{const} option described above.
1741 \opusmart{wf}{const}{non\_wf}
1742 Specifies whether the specified (co)in\-duc\-tively defined predicate is
1743 well-founded. The option can take the following values:
1746 \item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
1747 predicate as if it were well-founded. Since this is generally not sound when the
1748 predicate is not well-founded, the counterexamples are tagged as ``likely
1751 \item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
1752 as if it were not well-founded. The predicate is then unrolled as prescribed by
1753 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
1756 \item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
1757 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
1758 \textit{sizechange} tactics. If this succeeds (or the predicate occurs with an
1759 appropriate polarity in the formula to falsify), use an efficient fixed point
1760 equation as specification of the predicate; otherwise, unroll the predicates
1761 according to the \textit{iter}~\qty{const} and \textit{iter} options.
1765 {\small See also \textit{iter} (\S\ref{scope-of-search}),
1766 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
1767 (\S\ref{timeouts}).}
1769 \opsmart{wf}{non\_wf}
1770 Specifies the default wellfoundedness setting to use. This can be overridden on
1771 a per-predicate basis using the \textit{wf}~\qty{const} option above.
1773 \opu{iter}{const}{int\_seq}
1774 Specifies the sequence of iteration counts to use when unrolling a given
1775 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
1776 predicates that occur negatively and coinductive predicates that occur
1777 positively in the formula to falsify and that cannot be proved to be
1778 well-founded, but this behavior is influenced by the \textit{wf} option. The
1779 iteration counts are automatically bounded by the cardinality of the predicate's
1782 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
1783 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
1785 \opt{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
1786 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
1787 predicates. This can be overridden on a per-predicate basis using the
1788 \textit{iter} \qty{const} option above.
1790 \opt{bisim\_depth}{int\_seq}{$\mathbf{7}$}
1791 Specifies the sequence of iteration counts to use when unrolling the
1792 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
1793 of $-1$ means that no predicate is generated, in which case Nitpick performs an
1794 after-the-fact check to see if the known coinductive datatype values are
1795 bidissimilar. If two values are found to be bisimilar, the counterexample is
1796 tagged as ``likely genuine.'' The iteration counts are automatically bounded by
1797 the sum of the cardinalities of the coinductive datatypes occurring in the
1800 \opusmart{box}{type}{dont\_box}
1801 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
1802 product type in an isomorphic datatype internally. Boxing is an effective mean
1803 to reduce the search space and speed up Nitpick, because the isomorphic datatype
1804 is approximated by a subset of the possible function or pair values;
1805 like other drastic optimizations, it can also prevent the discovery of
1806 counterexamples. The option can take the following values:
1809 \item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
1811 \item[$\bullet$] \textbf{\textit{false}}: Never box the type.
1812 \item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
1813 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
1814 higher-order functions are good candidates for boxing.
1817 Setting the \textit{card}~\qty{type} option for a function or product type
1818 implicitly enables boxing for that type.
1821 {\small See also \textit{verbose} (\S\ref{output-format})
1822 and \textit{debug} (\S\ref{output-format}).}
1824 \opsmart{box}{dont\_box}
1825 Specifies the default boxing setting to use. This can be overridden on a
1826 per-type basis using the \textit{box}~\qty{type} option described above.
1828 \opusmart{mono}{type}{non\_mono}
1829 Specifies whether the specified type should be considered monotonic when
1830 enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a
1831 monotonicity check on the type. Setting this option to \textit{true} can reduce
1832 the number of scopes tried, but it also diminishes the theoretical chance of
1833 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
1836 {\small See also \textit{card} (\S\ref{scope-of-search}),
1837 \textit{coalesce\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
1838 (\S\ref{output-format}).}
1840 \opsmart{mono}{non\_box}
1841 Specifies the default monotonicity setting to use. This can be overridden on a
1842 per-type basis using the \textit{mono}~\qty{type} option described above.
1844 \opfalse{coalesce\_type\_vars}{dont\_coalesce\_type\_vars}
1845 Specifies whether type variables with the same sort constraints should be
1846 merged. Setting this option to \textit{true} can reduce the number of scopes
1847 tried and the size of the generated Kodkod formulas, but it also diminishes the
1848 theoretical chance of finding a counterexample.
1850 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1853 \subsection{Output Format}
1854 \label{output-format}
1857 \opfalse{verbose}{quiet}
1858 Specifies whether the \textbf{nitpick} command should explain what it does. This
1859 option is useful to determine which scopes are tried or which SAT solver is
1860 used. This option is implicitly disabled for automatic runs.
1863 {\small See also \textit{auto} (\S\ref{mode-of-operation}).}
1865 \opfalse{debug}{no\_debug}
1866 Specifies whether Nitpick should display additional debugging information beyond
1867 what \textit{verbose} already displays. Enabling \textit{debug} also enables
1868 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
1869 option is implicitly disabled for automatic runs.
1872 {\small See also \textit{auto} (\S\ref{mode-of-operation}), \textit{overlord}
1873 (\S\ref{mode-of-operation}), and \textit{batch\_size} (\S\ref{optimizations}).}
1875 \optrue{show\_skolems}{hide\_skolem}
1876 Specifies whether the values of Skolem constants should be displayed as part of
1877 counterexamples. Skolem constants correspond to bound variables in the original
1878 formula and usually help us to understand why the counterexample falsifies the
1882 {\small See also \textit{skolemize} (\S\ref{optimizations}).}
1884 \opfalse{show\_datatypes}{hide\_datatypes}
1885 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
1886 be displayed as part of counterexamples. Such subsets are sometimes helpful when
1887 investigating whether a potential counterexample is genuine or spurious, but
1888 their potential for clutter is real.
1890 \opfalse{show\_consts}{hide\_consts}
1891 Specifies whether the values of constants occurring in the formula (including
1892 its axioms) should be displayed along with any counterexample. These values are
1893 sometimes helpful when investigating why a counterexample is
1894 genuine, but they can clutter the output.
1896 \opfalse{show\_all}{dont\_show\_all}
1897 Enabling this option effectively enables \textit{show\_skolems},
1898 \textit{show\_datatypes}, and \textit{show\_consts}.
1900 \opt{max\_potential}{int}{$\mathbf{1}$}
1901 Specifies the maximum number of potential counterexamples to display. Setting
1902 this option to 0 speeds up the search for a genuine counterexample. This option
1903 is implicitly set to 0 for automatic runs. If you set this option to a value
1904 greater than 1, you will need an incremental SAT solver: For efficiency, it is
1905 recommended to install the JNI version of MiniSat and set \textit{sat\_solver} =
1906 \textit{MiniSatJNI}. Also be aware that many of the counterexamples may look
1907 identical, unless the \textit{show\_all} (\S\ref{output-format}) option is
1911 {\small See also \textit{auto} (\S\ref{mode-of-operation}),
1912 \textit{check\_potential} (\S\ref{authentication}), and
1913 \textit{sat\_solver} (\S\ref{optimizations}).}
1915 \opt{max\_genuine}{int}{$\mathbf{1}$}
1916 Specifies the maximum number of genuine counterexamples to display. If you set
1917 this option to a value greater than 1, you will need an incremental SAT solver:
1918 For efficiency, it is recommended to install the JNI version of MiniSat and set
1919 \textit{sat\_solver} = \textit{MiniSatJNI}. Also be aware that many of the
1920 counterexamples may look identical, unless the \textit{show\_all}
1921 (\S\ref{output-format}) option is enabled.
1924 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
1925 \textit{sat\_solver} (\S\ref{optimizations}).}
1927 \ops{eval}{term\_list}
1928 Specifies the list of terms whose values should be displayed along with
1929 counterexamples. This option suffers from an ``observer effect'': Nitpick might
1930 find different counterexamples for different values of this option.
1932 \opu{format}{term}{int\_seq}
1933 Specifies how to uncurry the value displayed for a variable or constant.
1934 Uncurrying sometimes increases the readability of the output for high-arity
1935 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
1936 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
1937 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
1938 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
1939 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
1940 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
1941 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
1942 arguments that are not accounted for are left alone, as if the specification had
1943 been $1,\ldots,1,n_1,\ldots,n_k$.
1946 {\small See also \textit{uncurry} (\S\ref{optimizations}).}
1948 \opt{format}{int\_seq}{$\mathbf{1}$}
1949 Specifies the default format to use. Irrespective of the default format, the
1950 extra arguments to a Skolem constant corresponding to the outer bound variables
1951 are kept separated from the remaining arguments, the \textbf{for} arguments of
1952 an inductive definitions are kept separated from the remaining arguments, and
1953 the iteration counter of an unrolled inductive definition is shown alone. The
1954 default format can be overridden on a per-variable or per-constant basis using
1955 the \textit{format}~\qty{term} option described above.
1958 %% MARK: Authentication
1959 \subsection{Authentication}
1960 \label{authentication}
1963 \opfalse{check\_potential}{trust\_potential}
1964 Specifies whether potential counterexamples should be given to Isabelle's
1965 \textit{auto} tactic to assess their validity. If a potential counterexample is
1966 shown to be genuine, Nitpick displays a message to this effect and terminates.
1969 {\small See also \textit{max\_potential} (\S\ref{output-format}) and
1970 \textit{auto\_timeout} (\S\ref{timeouts}).}
1972 \opfalse{check\_genuine}{trust\_genuine}
1973 Specifies whether genuine and likely genuine counterexamples should be given to
1974 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
1975 counterexample is shown to be spurious, the user is kindly asked to send a bug
1976 report to the author at
1977 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
1980 {\small See also \textit{max\_genuine} (\S\ref{output-format}) and
1981 \textit{auto\_timeout} (\S\ref{timeouts}).}
1983 \ops{expect}{string}
1984 Specifies the expected outcome, which must be one of the following:
1987 \item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
1988 \item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a ``likely
1989 genuine'' counterexample (i.e., a counterexample that is genuine unless
1990 it contradicts a missing axiom or a dangerous option was used inappropriately).
1991 \item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
1992 \item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
1993 \item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
1994 Kodkod ran out of memory).
1997 Nitpick emits an error if the actual outcome differs from the expected outcome.
1998 This option is useful for regression testing.
2001 \subsection{Optimizations}
2002 \label{optimizations}
2004 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
2009 \opt{sat\_solver}{string}{smart}
2010 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2011 to be faster than their Java counterparts, but they can be more difficult to
2012 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2013 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2014 you will need an incremental SAT solver, such as \textit{MiniSatJNI}
2015 (recommended) or \textit{SAT4J}.
2017 The supported solvers are listed below:
2021 \item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
2022 written in \cpp{}. To use MiniSat, set the environment variable
2023 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2024 executable. The \cpp{} sources and executables for MiniSat are available at
2025 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2026 and 2.0 beta (2007-07-21).
2028 \item[$\bullet$] \textbf{\textit{MiniSatJNI}}: The JNI (Java Native Interface)
2029 version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which
2030 you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2031 version of MiniSat, the JNI version can be used incrementally.
2033 \item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
2034 written in C. It is bundled with Kodkodi and requires no further installation or
2035 configuration steps. Alternatively, you can install a standard version of
2036 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2037 that contains the \texttt{picosat} executable. The C sources for PicoSAT are
2038 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2039 Nitpick has been tested with version 913.
2041 \item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
2042 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2043 the directory that contains the \texttt{zchaff} executable. The \cpp{} sources
2044 and executables for zChaff are available at
2045 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2046 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2048 \item[$\bullet$] \textbf{\textit{zChaffJNI}}: The JNI version of zChaff is
2049 bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on
2050 Kodkod's web site \cite{kodkod-2009}.
2052 \item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
2053 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2054 directory that contains the \texttt{rsat} executable. The \cpp{} sources for
2055 RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been
2056 tested with version 2.01.
2058 \item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
2059 written in C. To use BerkMin, set the environment variable
2060 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2061 executable. The BerkMin executables are available at
2062 \url{http://eigold.tripod.com/BerkMin.html}.
2064 \item[$\bullet$] \textbf{\textit{BerkMinAlloy}}: Variant of BerkMin that is
2065 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2066 version of BerkMin, set the environment variable
2067 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2070 \item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
2071 written in C. To use Jerusat, set the environment variable
2072 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2073 executable. The C sources for Jerusat are available at
2074 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2076 \item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
2077 written in Java that can be used incrementally. It is bundled with Kodkodi and
2078 requires no further installation or configuration steps. Do not attempt to
2079 install the official SAT4J packages, because their API is incompatible with
2082 \item[$\bullet$] \textbf{\textit{SAT4JLight}}: Variant of SAT4J that is
2083 optimized for small problems. It can also be used incrementally.
2085 \item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
2086 experimental solver written in \cpp. To use HaifaSat, set the environment
2087 variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
2088 \texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at
2089 \url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
2091 \item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
2092 \textit{smart}, Nitpick selects the first solver among MiniSat, PicoSAT, zChaff,
2093 RSat, BerkMin, BerkMinAlloy, and Jerusat that is recognized by Isabelle. If none
2094 is found, it falls back on SAT4J, which should always be available. If
2095 \textit{verbose} is enabled, Nitpick displays which SAT solver was chosen.
2100 \opt{batch\_size}{int\_or\_smart}{smart}
2101 Specifies the maximum number of Kodkod problems that should be lumped together
2102 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2103 together ensures that Kodkodi is launched less often, but it makes the verbose
2104 output less readable and is sometimes detrimental to performance. If
2105 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2106 \textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
2108 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2109 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2110 be rewritten to use (automatically generated) discriminators and destructors.
2111 This optimization can drastically reduce the size of the Boolean formulas given
2115 {\small See also \textit{debug} (\S\ref{output-format}).}
2117 \optrue{specialize}{dont\_specialize}
2118 Specifies whether functions invoked with static arguments should be specialized.
2119 This optimization can drastically reduce the search space, especially for
2120 higher-order functions.
2123 {\small See also \textit{debug} (\S\ref{output-format}) and
2124 \textit{show\_consts} (\S\ref{output-format}).}
2126 \optrue{skolemize}{dont\_skolemize}
2127 Specifies whether the formula should be skolemized. For performance reasons,
2128 (positive) $\forall$-quanti\-fiers that occur in the scope of a higher-order
2129 (positive) $\exists$-quanti\-fier are left unchanged.
2132 {\small See also \textit{debug} (\S\ref{output-format}) and
2133 \textit{show\_skolems} (\S\ref{output-format}).}
2135 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2136 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2137 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2138 predicates for which each the predicate occurs in at most one assumption of each
2139 introduction rule. Using the reflexive transitive closure is in principle
2140 equivalent to setting \textit{iter} to the cardinality of the predicate's
2141 domain, but it is usually more efficient.
2143 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2144 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2146 \optrue{uncurry}{dont\_uncurry}
2147 Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no
2148 tangible effect on efficiency, but it creates opportunities for the boxing
2152 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug}
2153 (\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).}
2155 \optrue{fast\_descrs}{full\_descrs}
2156 Specifies whether Nitpick should optimize the definite and indefinite
2157 description operators (THE and SOME). The optimized versions usually help
2158 Nitpick generate more counterexamples or at least find them faster, but only the
2159 unoptimized versions are complete when all types occurring in the formula are
2162 {\small See also \textit{debug} (\S\ref{output-format}).}
2164 \optrue{peephole\_optim}{no\_peephole\_optim}
2165 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2166 peephole optimizer. These optimizations can make a significant difference.
2167 Unless you are tracking down a bug in Nitpick or distrust the peephole
2168 optimizer, you should leave this option enabled.
2170 \opt{sym\_break}{int}{20}
2171 Specifies an upper bound on the number of relations for which Kodkod generates
2172 symmetry breaking predicates. According to the Kodkod documentation
2173 \cite{kodkod-2009-options}, ``in general, the higher this value, the more
2174 symmetries will be broken, and the faster the formula will be solved. But,
2175 setting the value too high may have the opposite effect and slow down the
2178 \opt{sharing\_depth}{int}{3}
2179 Specifies the depth to which Kodkod should check circuits for equivalence during
2180 the translation to SAT. The default of 3 is the same as in Alloy. The minimum
2181 allowed depth is 1. Increasing the sharing may result in a smaller SAT problem,
2182 but can also slow down Kodkod.
2184 \opfalse{flatten\_props}{dont\_flatten\_props}
2185 Specifies whether Kodkod should try to eliminate intermediate Boolean variables.
2186 Although this might sound like a good idea, in practice it can drastically slow
2189 \opt{max\_threads}{int}{0}
2190 Specifies the maximum number of threads to use in Kodkod. If this option is set
2191 to 0, Kodkod will compute an appropriate value based on the number of processor
2195 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2196 \textit{timeout} (\S\ref{timeouts}).}
2199 \subsection{Timeouts}
2203 \opt{timeout}{time}{$\mathbf{30}$ s}
2204 Specifies the maximum amount of time that the \textbf{nitpick} command should
2205 spend looking for a counterexample. Nitpick tries to honor this constraint as
2206 well as it can but offers no guarantees. For automatic runs,
2207 \textit{auto\_timeout} is used instead.
2210 {\small See also \textit{auto} (\S\ref{mode-of-operation})
2211 and \textit{max\_threads} (\S\ref{optimizations}).}
2213 \opt{auto\_timeout}{time}{$\mathbf{5}$ s}
2214 Specifies the maximum amount of time that Nitpick should use to find a
2215 counterexample when running automatically. Nitpick tries to honor this
2216 constraint as well as it can but offers no guarantees.
2219 {\small See also \textit{auto} (\S\ref{mode-of-operation}).}
2221 \opt{tac\_timeout}{time}{$\mathbf{500}$ ms}
2222 Specifies the maximum amount of time that the \textit{auto} tactic should use
2223 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2224 and \textit{sizechange} should use when checking whether a (co)in\-duc\-tive
2225 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2226 can but offers no guarantees.
2229 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2230 \textit{check\_potential} (\S\ref{authentication}),
2231 and \textit{check\_genuine} (\S\ref{authentication}).}
2234 \section{Attribute Reference}
2235 \label{attribute-reference}
2237 Nitpick needs to consider the definitions of all constants occurring in a
2238 formula in order to falsify it. For constants introduced using the
2239 \textbf{definition} command, the definition is simply the associated
2240 \textit{\_def} axiom. In contrast, instead of using the internal representation
2241 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2242 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2243 equational specification entered by the user.
2245 Behind the scenes, Isabelle's built-in packages and theories rely on the
2246 following attributes to affect Nitpick's behavior:
2249 \flushitem{\textit{nitpick\_def}}
2252 This attribute specifies an alternative definition of a constant. The
2253 alternative definition should be logically equivalent to the constant's actual
2254 axiomatic definition and should be of the form
2256 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2258 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2261 \flushitem{\textit{nitpick\_simp}}
2264 This attribute specifies the equations that constitute the specification of a
2265 constant. For functions defined using the \textbf{primrec}, \textbf{function},
2266 and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
2267 \textit{simps} rules. The equations must be of the form
2269 \qquad $c~t_1~\ldots\ t_n \,=\, u.$
2271 \flushitem{\textit{nitpick\_psimp}}
2274 This attribute specifies the equations that constitute the partial specification
2275 of a constant. For functions defined using the \textbf{function} package, this
2276 corresponds to the \textit{psimps} rules. The conditional equations must be of
2279 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
2281 \flushitem{\textit{nitpick\_intro}}
2284 This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
2285 For predicates defined using the \textbf{inductive} or \textbf{coinductive}
2286 command, this corresponds to the \textit{intros} rules. The introduction rules
2289 \qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
2290 \ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\
2293 where the $P_i$'s are side conditions that do not involve $c$ and $M$ is an
2294 optional monotonic operator. The order of the assumptions is irrelevant.
2298 When faced with a constant, Nitpick proceeds as follows:
2301 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2302 is not empty, Nitpick uses these rules as the specification of the constant.
2304 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2305 the constant is not empty, it uses these rules as the specification of the
2308 \item[3.] Otherwise, it looks up the definition of the constant:
2311 \item[1.] If the \textit{nitpick\_def} set associated with the constant
2312 is not empty, it uses the latest rule added to the set as the definition of the
2313 constant; otherwise it uses the actual definition axiom.
2314 \item[2.] If the definition is of the form
2316 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
2318 then Nitpick assumes that the definition was made using an inductive package and
2319 based on the introduction rules marked with \textit{nitpick\_\allowbreak
2320 ind\_\allowbreak intros} tries to determine whether the definition is
2325 As an illustration, consider the inductive definition
2328 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2329 ``\textit{odd}~1'' $\,\mid$ \\
2330 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2333 Isabelle automatically attaches the \textit{nitpick\_intro} attribute to
2334 the above rules. Nitpick then uses the \textit{lfp}-based definition in
2335 conjunction with these rules. To override this, we can specify an alternative
2336 definition as follows:
2339 \textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]: ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2342 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2343 = 1$. Alternatively, we can specify an equational specification of the constant:
2346 \textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]: ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2349 Such tweaks should be done with great care, because Nitpick will assume that the
2350 constant is completely defined by its equational specification. For example, if
2351 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
2352 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2353 (\S\ref{output-format}) option is extremely useful to understand what is going
2354 on when experimenting with \textit{nitpick\_} attributes.
2356 \section{Standard ML Interface}
2357 \label{standard-ml-interface}
2359 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2360 and debugging. Among the most interesting functions exported by Nitpick are
2361 those that let you invoke the tool programmatically and those that let you
2362 register and unregister custom coinductive datatypes.
2364 \subsection{Invocation of Nitpick}
2365 \label{invocation-of-nitpick}
2367 The \textit{Nitpick} structure offers the following functions for invoking your
2368 favorite counterexample generator:
2371 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2372 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2373 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2374 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2375 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2378 The return value is a new proof state paired with an outcome string
2379 (``genuine'', ``likely\_genuine'', ``potential'', ``none'', or ``unknown''). The
2380 \textit{params} type is a large record that lets you set Nitpick's options. The
2381 current default options can be retrieved by calling the following function
2382 defined in the \textit{NitpickIsar} structure:
2385 $\textbf{val}\,~\textit{default\_params} :\,
2386 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2389 The second argument lets you override option values before they are parsed and
2390 put into a \textit{params} record. Here is an example:
2393 $\textbf{val}\,~\textit{params} = \textit{NitpickIsar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2394 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2395 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2396 & \textit{subgoal}\end{aligned}$
2399 \subsection{Registration of Coinductive Datatypes}
2400 \label{registration-of-coinductive-datatypes}
2404 If you have defined a custom coinductive datatype, you can tell Nitpick about
2405 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2406 uses for lazy lists. The interface for registering and unregistering coinductive
2407 datatypes consists of the following pair of functions defined in the
2408 \textit{Nitpick} structure:
2411 $\textbf{val}\,~\textit{register\_codatatype} :\,
2412 \textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2413 $\textbf{val}\,~\textit{unregister\_codatatype} :\,
2414 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2417 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2418 not been, you could have told Nitpick about it by adding the following line
2419 to your theory file:
2422 $\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
2423 & \textit{Nitpick.register\_codatatype} \\[-2pt]
2424 & \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
2425 & \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
2428 The \textit{register\_codatatype} function takes a coinductive type, its case
2429 function, and the list of its constructors. The case function must take its
2430 arguments in the order that the constructors are listed. If no case function
2431 with the correct signature is available, simply pass the empty string.
2433 On the other hand, if your goal is to cripple Nitpick, add the following line to
2434 your theory file and try to check a few conjectures about lazy lists:
2437 $\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
2438 \kern1pt'a~\textit{list}\textrm{''}\}\ \,{*}\}$
2441 \section{Known Bugs and Limitations}
2442 \label{known-bugs-and-limitations}
2444 Here are the known bugs and limitations in Nitpick at the time of writing:
2447 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2448 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2449 Nitpick to generate spurious counterexamples for theorems that refer to values
2450 for which the function is not defined. For example:
2453 \textbf{primrec} \textit{prec} \textbf{where} \\
2454 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2455 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2456 \textbf{nitpick} \\[2\smallskipamount]
2457 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2459 \\[2\smallskipamount]
2460 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2461 \textbf{by}~(\textit{auto simp}: \textit{prec\_def})
2464 Such theorems are considered bad style because they rely on the internal
2465 representation of functions synthesized by Isabelle, which is an implementation
2468 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2469 \textbf{guess} command in a structured proof.
2471 \item[$\bullet$] The \textit{nitpick\_} attributes and the
2472 \textit{Nitpick.register\_} functions can cause havoc if used improperly.
2474 \item[$\bullet$] Local definitions are not supported and result in an error.
2476 \item[$\bullet$] All constants and types whose names start with
2477 \textit{Nitpick}{.} or \textit{NitpickDefs}{.} are reserved for internal use.
2481 \bibliography{../manual}{}
2482 \bibliographystyle{abbrv}