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.
111 Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
112 machine called \texttt{java}. The examples presented in this manual can be found
113 in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.
115 Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
116 Nitpick also provides an automatic mode that can be enabled using the
117 ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof General. In this
118 mode, Nitpick is run on every newly entered theorem, much like Auto Quickcheck.
119 The collective time limit for Auto Nitpick and Auto Quickcheck can be set using
120 the ``Auto Counterexample Time Limit'' option.
123 \setbox\boxA=\hbox{\texttt{nospam}}
125 The known bugs and limitations at the time of writing are listed in
126 \S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
127 or this manual should be directed to
128 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
129 in.\allowbreak tum.\allowbreak de}.
131 \vskip2.5\smallskipamount
133 \textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
134 suggesting several textual improvements.
135 % and Perry James for reporting a typo.
137 \section{First Steps}
140 This section introduces Nitpick by presenting small examples. If possible, you
141 should try out the examples on your workstation. Your theory file should start
145 \textbf{theory}~\textit{Scratch} \\
146 \textbf{imports}~\textit{Main} \\
150 The results presented here were obtained using the JNI version of MiniSat and
151 with multithreading disabled to reduce nondeterminism. This was done by adding
155 \textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSatJNI}, \,\textit{max\_threads}~= 1]
158 after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
159 Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
160 be installed, as explained in \S\ref{optimizations}. If you have already
161 configured SAT solvers in Isabelle (e.g., for Refute), these will also be
162 available to Nitpick.
164 \subsection{Propositional Logic}
165 \label{propositional-logic}
167 Let's start with a trivial example from propositional logic:
170 \textbf{lemma}~``$P \longleftrightarrow Q$'' \\
174 You should get the following output:
178 Nitpick found a counterexample: \\[2\smallskipamount]
179 \hbox{}\qquad Free variables: \nopagebreak \\
180 \hbox{}\qquad\qquad $P = \textit{True}$ \\
181 \hbox{}\qquad\qquad $Q = \textit{False}$
184 Nitpick can also be invoked on individual subgoals, as in the example below:
187 \textbf{apply}~\textit{auto} \\[2\smallskipamount]
188 {\slshape goal (2 subgoals): \\
189 \ 1. $P\,\Longrightarrow\, Q$ \\
190 \ 2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
191 \textbf{nitpick}~1 \\[2\smallskipamount]
192 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
193 \hbox{}\qquad Free variables: \nopagebreak \\
194 \hbox{}\qquad\qquad $P = \textit{True}$ \\
195 \hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
196 \textbf{nitpick}~2 \\[2\smallskipamount]
197 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
198 \hbox{}\qquad Free variables: \nopagebreak \\
199 \hbox{}\qquad\qquad $P = \textit{False}$ \\
200 \hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
204 \subsection{Type Variables}
205 \label{type-variables}
207 If you are left unimpressed by the previous example, don't worry. The next
208 one is more mind- and computer-boggling:
211 \textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
213 \pagebreak[2] %% TYPESETTING
215 The putative lemma involves the definite description operator, {THE}, presented
216 in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
217 operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
218 lemma is merely asserting the indefinite description operator axiom with {THE}
219 substituted for {SOME}.
221 The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
222 containing type variables, Nitpick enumerates the possible domains for each type
223 variable, up to a given cardinality (8 by default), looking for a finite
227 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
229 Trying 8 scopes: \nopagebreak \\
230 \hbox{}\qquad \textit{card}~$'a$~= 1; \\
231 \hbox{}\qquad \textit{card}~$'a$~= 2; \\
232 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
233 \hbox{}\qquad \textit{card}~$'a$~= 8. \\[2\smallskipamount]
234 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
235 \hbox{}\qquad Free variables: \nopagebreak \\
236 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
237 \hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
241 Nitpick found a counterexample in which $'a$ has cardinality 3. (For
242 cardinalities 1 and 2, the formula holds.) In the counterexample, the three
243 values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
245 The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
246 \textit{verbose} is enabled. You can specify \textit{verbose} each time you
247 invoke \textbf{nitpick}, or you can set it globally using the command
250 \textbf{nitpick\_params} [\textit{verbose}]
253 This command also displays the current default values for all of the options
254 supported by Nitpick. The options are listed in \S\ref{option-reference}.
256 \subsection{Constants}
259 By just looking at Nitpick's output, it might not be clear why the
260 counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
261 this time telling it to show the values of the constants that occur in the
265 \textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
266 \textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
268 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
269 \hbox{}\qquad Free variables: \nopagebreak \\
270 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
271 \hbox{}\qquad\qquad $x = a_3$ \\
272 \hbox{}\qquad Constant: \nopagebreak \\
273 \hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
276 We can see more clearly now. Since the predicate $P$ isn't true for a unique
277 value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
278 $a_1$. Since $P~a_1$ is false, the entire formula is falsified.
280 As an optimization, Nitpick's preprocessor introduced the special constant
281 ``\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
282 $\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
283 satisfying $P~y$. We disable this optimization by passing the
284 \textit{full\_descrs} option:
287 \textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
289 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
290 \hbox{}\qquad Free variables: \nopagebreak \\
291 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
292 \hbox{}\qquad\qquad $x = a_3$ \\
293 \hbox{}\qquad Constant: \nopagebreak \\
294 \hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
297 As the result of another optimization, Nitpick directly assigned a value to the
298 subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
299 disable this second optimization by using the command
302 \textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
303 \textit{show\_consts}]
306 we finally get \textit{The}:
309 \slshape Constant: \nopagebreak \\
310 \hbox{}\qquad $\mathit{The} = \undef{}
311 (\!\begin{aligned}[t]%
312 & \{\} := a_3,\> \{a_3\} := a_3,\> \{a_2\} := a_2, \\[-2pt] %% TYPESETTING
313 & \{a_2, a_3\} := a_1,\> \{a_1\} := a_1,\> \{a_1, a_3\} := a_3, \\[-2pt]
314 & \{a_1, a_2\} := a_3,\> \{a_1, a_2, a_3\} := a_3)\end{aligned}$
317 Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
318 just like before.\footnote{The \undef{} symbol's presence is explained as
319 follows: In higher-order logic, any function can be built from the undefined
320 function using repeated applications of the function update operator $f(x :=
321 y)$, just like any list can be built from the empty list using $x \mathbin{\#}
324 Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
325 unique $x$ such that'') at the front of our putative lemma's assumption:
328 \textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
331 The fix appears to work:
334 \textbf{nitpick} \\[2\smallskipamount]
335 \slshape Nitpick found no counterexample.
338 We can further increase our confidence in the formula by exhausting all
339 cardinalities up to 50:
342 \textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
343 can be entered as \texttt{-} (hyphen) or
344 \texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
345 \slshape Nitpick found no counterexample.
348 Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
351 \textbf{sledgehammer} \\[2\smallskipamount]
352 {\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
353 $\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
354 Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
355 \textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
356 {\slshape No subgoals!}% \\[2\smallskipamount]
360 This must be our lucky day.
362 \subsection{Skolemization}
363 \label{skolemization}
365 Are all invertible functions onto? Let's find out:
368 \textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
369 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
370 \textbf{nitpick} \\[2\smallskipamount]
372 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
373 \hbox{}\qquad Free variable: \nopagebreak \\
374 \hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
375 \hbox{}\qquad Skolem constants: \nopagebreak \\
376 \hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
377 \hbox{}\qquad\qquad $y = a_2$
380 Although $f$ is the only free variable occurring in the formula, Nitpick also
381 displays values for the bound variables $g$ and $y$. These values are available
382 to Nitpick because it performs skolemization as a preprocessing step.
384 In the previous example, skolemization only affected the outermost quantifiers.
385 This is not always the case, as illustrated below:
388 \textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
389 \textbf{nitpick} \\[2\smallskipamount]
391 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
392 \hbox{}\qquad Skolem constant: \nopagebreak \\
393 \hbox{}\qquad\qquad $\lambda x.\; f =
394 \undef{}(\!\begin{aligned}[t]
395 & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
396 & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
399 The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
400 $x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
401 function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
402 maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
404 The source of the Skolem constants is sometimes more obscure:
407 \textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
408 \textbf{nitpick} \\[2\smallskipamount]
410 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
411 \hbox{}\qquad Free variable: \nopagebreak \\
412 \hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
413 \hbox{}\qquad Skolem constants: \nopagebreak \\
414 \hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
415 \hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
418 What happened here is that Nitpick expanded the \textit{sym} constant to its
422 $\mathit{sym}~r \,\equiv\,
423 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
426 As their names suggest, the Skolem constants $\mathit{sym}.x$ and
427 $\mathit{sym}.y$ are simply the bound variables $x$ and $y$
428 from \textit{sym}'s definition.
430 Although skolemization is a useful optimization, you can disable it by invoking
431 Nitpick with \textit{dont\_skolemize}. See \S\ref{optimizations} for details.
433 \subsection{Natural Numbers and Integers}
434 \label{natural-numbers-and-integers}
436 Because of the axiom of infinity, the type \textit{nat} does not admit any
437 finite models. To deal with this, Nitpick considers prefixes $\{0,\, 1,\,
438 \ldots,\, K - 1\}$ of \textit{nat} (where $K = \textit{card}~\textit{nat}$) and
439 maps all other numbers to the undefined value ($\unk$). The type \textit{int} is
440 handled in a similar way: If $K = \textit{card}~\textit{int}$, the subset of
441 \textit{int} known to Nitpick is $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor
442 K/2 \rfloor\}$. Undefined values lead to a three-valued logic.
444 Here is an example involving \textit{int}:
447 \textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
448 \textbf{nitpick} \\[2\smallskipamount]
449 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
450 \hbox{}\qquad Free variables: \nopagebreak \\
451 \hbox{}\qquad\qquad $i = 0$ \\
452 \hbox{}\qquad\qquad $j = 1$ \\
453 \hbox{}\qquad\qquad $m = 1$ \\
454 \hbox{}\qquad\qquad $n = 0$
457 With infinite types, we don't always have the luxury of a genuine counterexample
458 and must often content ourselves with a potential one. The tedious task of
459 finding out whether the potential counterexample is in fact genuine can be
460 outsourced to \textit{auto} by passing the option \textit{check\_potential}. For
464 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
465 \textbf{nitpick} [\textit{card~nat}~= 100,\, \textit{check\_potential}] \\[2\smallskipamount]
466 \slshape Nitpick found a potential counterexample: \\[2\smallskipamount]
467 \hbox{}\qquad Free variable: \nopagebreak \\
468 \hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
469 Confirmation by ``\textit{auto}'': The above counterexample is genuine.
472 You might wonder why the counterexample is first reported as potential. The root
473 of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
474 \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
475 that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
476 \textit{False}; but otherwise, it does not know anything about values of $n \ge
477 \textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
478 \textit{True}. Since the assumption can never be satisfied, the putative lemma
479 can never be falsified.
481 Incidentally, if you distrust the so-called genuine counterexamples, you can
482 enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
483 aware that \textit{auto} will often fail to prove that the counterexample is
486 Some conjectures involving elementary number theory make Nitpick look like a
487 giant with feet of clay:
490 \textbf{lemma} ``$P~\textit{Suc}$'' \\
491 \textbf{nitpick} [\textit{card} = 1--6] \\[2\smallskipamount]
493 Nitpick found no counterexample.
496 For any cardinality $k$, \textit{Suc} is the partial function $\{0 \mapsto 1,\,
497 1 \mapsto 2,\, \ldots,\, k - 1 \mapsto \unk\}$, which evaluates to $\unk$ when
498 it is passed as argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$.
499 The next example is similar:
502 \textbf{lemma} ``$P~(\textit{op}~{+}\Colon
503 \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
504 \textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
505 {\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
506 \hbox{}\qquad Free variable: \nopagebreak \\
507 \hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
508 \textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
509 {\slshape Nitpick found no counterexample.}
512 The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
513 $\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
516 Because numbers are infinite and are approximated using a three-valued logic,
517 there is usually no need to systematically enumerate domain sizes. If Nitpick
518 cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
519 unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
520 example above is an exception to this principle.) Nitpick nonetheless enumerates
521 all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
522 cardinalities are fast to handle and give rise to simpler counterexamples. This
523 is explained in more detail in \S\ref{scope-monotonicity}.
525 \subsection{Inductive Datatypes}
526 \label{inductive-datatypes}
528 Like natural numbers and integers, inductive datatypes with recursive
529 constructors admit no finite models and must be approximated by a subterm-closed
530 subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
531 Nitpick looks for all counterexamples that can be built using at most 10
534 Let's see with an example involving \textit{hd} (which returns the first element
535 of a list) and $@$ (which concatenates two lists):
538 \textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
539 \textbf{nitpick} \\[2\smallskipamount]
540 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
541 \hbox{}\qquad Free variables: \nopagebreak \\
542 \hbox{}\qquad\qquad $\textit{xs} = []$ \\
543 \hbox{}\qquad\qquad $\textit{y} = a_3$
546 To see why the counterexample is genuine, we enable \textit{show\_consts}
547 and \textit{show\_\allowbreak datatypes}:
550 {\slshape Datatype:} \\
551 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_3, a_3],\, [a_3],\, \unr\}$ \\
552 {\slshape Constants:} \\
553 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_3, a_3],\> [a_3, a_3] := \unk,\> [a_3] := \unk)$ \\
554 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_3, a_3] := a_3,\> [a_3] := a_3)$
557 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
560 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
561 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_3,
562 a_3]$ to $[a_3]$ would normally give $[a_3, a_3, a_3]$, but this value is not
563 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
564 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
565 appending $[a_3, a_3]$ to itself gives $\unk$.
567 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
568 considers the following subsets:
570 \kern-.5\smallskipamount %% TYPESETTING
574 $\{[],\, [a_1],\, [a_2]\}$; \\
575 $\{[],\, [a_1],\, [a_3]\}$; \\
576 $\{[],\, [a_2],\, [a_3]\}$; \\
577 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\
578 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\
579 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\
580 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\
581 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\
582 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\
583 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\
584 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\
585 $\{[],\, [a_3],\, [a_3, a_3]\}$.
589 \kern-2\smallskipamount %% TYPESETTING
591 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
592 are listed and only those. As an example of a non-subterm-closed subset,
593 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_3]\}$, and observe
594 that $[a_1, a_3]$ (i.e., $a_1 \mathbin{\#} [a_3]$) has $[a_3] \notin
595 \mathcal{S}$ as a subterm.
597 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
600 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
601 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
603 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
604 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
605 \hbox{}\qquad Free variables: \nopagebreak \\
606 \hbox{}\qquad\qquad $\textit{xs} = [a_2]$ \\
607 \hbox{}\qquad\qquad $\textit{ys} = [a_3]$ \\
608 \hbox{}\qquad Datatypes: \\
609 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
610 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_3],\, [a_2],\, \unr\}$
613 Because datatypes are approximated using a three-valued logic, there is usually
614 no need to systematically enumerate cardinalities: If Nitpick cannot find a
615 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
616 unlikely that one could be found for smaller cardinalities.
618 \subsection{Typedefs, Records, Rationals, and Reals}
619 \label{typedefs-records-rationals-and-reals}
621 Nitpick generally treats types declared using \textbf{typedef} as datatypes
622 whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
626 \textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
627 \textbf{by}~\textit{blast} \\[2\smallskipamount]
628 \textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
629 \textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
630 \textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
631 \textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
632 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
633 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
634 \hbox{}\qquad Free variables: \nopagebreak \\
635 \hbox{}\qquad\qquad $P = \{\Abs{1},\, \Abs{0}\}$ \\
636 \hbox{}\qquad\qquad $x = \Abs{2}$ \\
637 \hbox{}\qquad Datatypes: \\
638 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
639 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{2},\, \Abs{1},\, \Abs{0},\, \unr\}$
643 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
646 Records, which are implemented as \textbf{typedef}s behind the scenes, are
647 handled in much the same way:
650 \textbf{record} \textit{point} = \\
651 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
652 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
653 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
654 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
655 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
656 \hbox{}\qquad Free variables: \nopagebreak \\
657 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
658 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
659 \hbox{}\qquad Datatypes: \\
660 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
661 \hbox{}\qquad\qquad $\textit{point} = \{\lparr\textit{Xcoord} = 1,\>
662 \textit{Ycoord} = 1\rparr,\> \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr,\, \unr\}$\kern-1pt %% QUIET
665 Finally, Nitpick provides rudimentary support for rationals and reals using a
669 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
670 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
671 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
672 \hbox{}\qquad Free variables: \nopagebreak \\
673 \hbox{}\qquad\qquad $x = 1/2$ \\
674 \hbox{}\qquad\qquad $y = -1/2$ \\
675 \hbox{}\qquad Datatypes: \\
676 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
677 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
678 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
681 \subsection{Inductive and Coinductive Predicates}
682 \label{inductive-and-coinductive-predicates}
684 Inductively defined predicates (and sets) are particularly problematic for
685 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
686 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
687 the problem is that they are defined using a least fixed point construction.
689 Nitpick's philosophy is that not all inductive predicates are equal. Consider
690 the \textit{even} predicate below:
693 \textbf{inductive}~\textit{even}~\textbf{where} \\
694 ``\textit{even}~0'' $\,\mid$ \\
695 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
698 This predicate enjoys the desirable property of being well-founded, which means
699 that the introduction rules don't give rise to infinite chains of the form
702 $\cdots\,\Longrightarrow\, \textit{even}~k''
703 \,\Longrightarrow\, \textit{even}~k'
704 \,\Longrightarrow\, \textit{even}~k.$
707 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
711 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
712 \,\Longrightarrow\, \textit{even}~(k - 2)
713 \,\Longrightarrow\, \textit{even}~k.$
716 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
717 fixed point computation.%
718 \footnote{If an inductive predicate is
719 well-founded, then it has exactly one fixed point, which is simultaneously the
720 least and the greatest fixed point. In these circumstances, the computation of
721 the least fixed point amounts to the computation of an arbitrary fixed point,
722 which can be performed using a straightforward recursive equation.}
723 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
724 just as Isabelle's \textbf{function} package usually discharges termination
725 proof obligations automatically.
727 Let's try an example:
730 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
731 \textbf{nitpick}~[\textit{card nat}~= 100,\, \textit{verbose}] \\[2\smallskipamount]
732 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
733 Nitpick can compute it efficiently. \\[2\smallskipamount]
735 \hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
736 Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
737 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
738 Nitpick could not find a better counterexample. \\[2\smallskipamount]
742 No genuine counterexample is possible because Nitpick cannot rule out the
743 existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
744 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
745 existential quantifier:
748 \textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
749 \textbf{nitpick}~[\textit{card nat}~= 100] \\[2\smallskipamount]
750 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
751 \hbox{}\qquad Empty assignment
754 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
755 we use the following definition instead?
758 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
759 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
760 ``$\textit{even}'~2$'' $\,\mid$ \\
761 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
764 This definition is not well-founded: From $\textit{even}'~0$ and
765 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
766 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
768 Let's check a property involving $\textit{even}'$. To make up for the
769 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
770 \textit{nat}'s cardinality to a mere 10:
773 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
774 \lnot\;\textit{even}'~n$'' \\
775 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
777 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
778 Nitpick might need to unroll it. \\[2\smallskipamount]
780 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
781 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
782 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
783 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
784 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
785 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
786 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
787 \hbox{}\qquad Constant: \nopagebreak \\
788 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
789 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
790 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
791 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
795 Nitpick's output is very instructive. First, it tells us that the predicate is
796 unrolled, meaning that it is computed iteratively from the empty set. Then it
797 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
800 The output also shows how each iteration contributes to $\textit{even}'$. The
801 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
802 predicate depends on an iteration counter. Iteration 0 provides the basis
803 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
804 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
805 iterations would not contribute any new elements.
807 Some values are marked with superscripted question
808 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
809 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
810 \textit{True} or $\unk$, never \textit{False}.
812 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
813 iterations. However, these numbers are bounded by the cardinality of the
814 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
815 ever needed to compute the value of a \textit{nat} predicate. You can specify
816 the number of iterations using the \textit{iter} option, as explained in
817 \S\ref{scope-of-search}.
819 In the next formula, $\textit{even}'$ occurs both positively and negatively:
822 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
823 \textbf{nitpick} [\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
824 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
825 \hbox{}\qquad Free variable: \nopagebreak \\
826 \hbox{}\qquad\qquad $n = 1$ \\
827 \hbox{}\qquad Constants: \nopagebreak \\
828 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
829 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
830 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
833 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
834 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
835 fixed point (not necessarily the least one). It is used to falsify
836 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
837 $\textit{even}'~(n - 2)$.
839 Coinductive predicates are handled dually. For example:
842 \textbf{coinductive} \textit{nats} \textbf{where} \\
843 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
844 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
845 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
846 \slshape Nitpick found a counterexample:
847 \\[2\smallskipamount]
848 \hbox{}\qquad Constants: \nopagebreak \\
849 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
850 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
851 & \unr\})\end{aligned}$ \\
852 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
855 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
856 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
857 inductive predicates for which each the predicate occurs in at most one
858 assumption of each introduction rule. For example:
861 \textbf{inductive} \textit{odd} \textbf{where} \\
862 ``$\textit{odd}~1$'' $\,\mid$ \\
863 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
864 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
865 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
866 \slshape Nitpick found a counterexample:
867 \\[2\smallskipamount]
868 \hbox{}\qquad Free variable: \nopagebreak \\
869 \hbox{}\qquad\qquad $n = 1$ \\
870 \hbox{}\qquad Constants: \nopagebreak \\
871 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
872 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
873 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
875 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
876 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
878 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
879 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
880 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
884 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
885 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
886 elements from known ones. The set $\textit{odd}$ consists of all the values
887 reachable through the reflexive transitive closure of
888 $\textit{odd}_{\textrm{step}}$ starting with any element from
889 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
890 transitive closure to encode linear predicates is normally either more thorough
891 or more efficient than unrolling (depending on the value of \textit{iter}), but
892 for those cases where it isn't you can disable it by passing the
893 \textit{dont\_star\_linear\_preds} option.
895 \subsection{Coinductive Datatypes}
896 \label{coinductive-datatypes}
898 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
899 datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
900 list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
901 these lazy lists seamlessly and provides a hook, described in
902 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
905 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
906 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
907 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
908 1, 2, 3, \ldots]$ can be defined as lazy lists using the
909 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
910 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
911 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
913 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
914 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
918 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
919 \textbf{nitpick} \\[2\smallskipamount]
920 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
921 \hbox{}\qquad Free variables: \nopagebreak \\
922 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
923 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
926 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
927 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
928 infinite list $[a_1, a_1, a_1, \ldots]$.
930 The next example is more interesting:
933 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
934 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
935 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
936 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
937 some scopes. \\[2\smallskipamount]
939 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 1,
940 and \textit{bisim\_depth}~= 0. \\
941 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
942 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 8,
943 and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
944 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
945 \textit{card}~``\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
947 \\[2\smallskipamount]
948 \hbox{}\qquad Free variables: \nopagebreak \\
949 \hbox{}\qquad\qquad $\textit{a} = a_2$ \\
950 \hbox{}\qquad\qquad $\textit{b} = a_1$ \\
951 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
952 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_1~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega)$ \\[2\smallskipamount]
956 The lazy list $\textit{xs}$ is simply $[a_2, a_2, a_2, \ldots]$, whereas
957 $\textit{ys}$ is $[a_1, a_2, a_2, a_2, \ldots]$, i.e., a lasso-shaped list with
958 $[a_1]$ as its stem and $[a_2]$ as its cycle. In general, the list segment
959 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
960 the segment leading to the binder is the stem.
962 A salient property of coinductive datatypes is that two objects are considered
963 equal if and only if they lead to the same observations. For example, the lazy
964 lists $\textrm{THE}~\omega.\; \omega =
965 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
966 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
967 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
968 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
969 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
970 concept of equality for coinductive datatypes is called bisimulation and is
971 defined coinductively.
973 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
974 the Kodkod problem to ensure that distinct objects lead to different
975 observations. This precaution is somewhat expensive and often unnecessary, so it
976 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
977 bisimilarity check is then performed \textsl{after} the counterexample has been
978 found to ensure correctness. If this after-the-fact check fails, the
979 counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
980 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
981 check for the previous example saves approximately 150~milli\-seconds; the speed
982 gains can be more significant for larger scopes.
984 The next formula illustrates the need for bisimilarity (either as a Kodkod
985 predicate or as an after-the-fact check) to prevent spurious counterexamples:
988 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
989 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
990 \textbf{nitpick} [\textit{bisim\_depth} = $-1$,\, \textit{show\_datatypes}] \\[2\smallskipamount]
991 \slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
992 \hbox{}\qquad Free variables: \nopagebreak \\
993 \hbox{}\qquad\qquad $a = a_2$ \\
994 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
995 \textit{LCons}~a_2~\omega$ \\
996 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
997 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
998 \hbox{}\qquad\qquad $'a~\textit{llist} =
999 \{\!\begin{aligned}[t]
1000 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega, \\[-2pt]
1001 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega,\> \unr\}\end{aligned}$
1002 \\[2\smallskipamount]
1003 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1004 that the counterexample is genuine. \\[2\smallskipamount]
1005 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1006 \slshape Nitpick found no counterexample.
1009 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1010 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1012 A compromise between leaving out the bisimilarity predicate from the Kodkod
1013 problem and performing the after-the-fact check is to specify a lower
1014 nonnegative \textit{bisim\_depth} value than the default one provided by
1015 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1016 be distinguished from each other by their prefixes of length $K$. Be aware that
1017 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1018 finding any counterexamples.
1023 Nitpick normally maps function and product types directly to the corresponding
1024 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1025 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1026 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1027 off to treat these types in the same way as plain datatypes, by approximating
1028 them by a subset of a given cardinality. This technique is called ``boxing'' and
1029 is particularly useful for functions passed as arguments to other functions, for
1030 high-arity functions, and for large tuples. Under the hood, boxing involves
1031 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1032 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1034 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1035 using de Bruijn's notation:
1038 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1041 The $\textit{lift}~t~k$ function increments all variables with indices greater
1042 than or equal to $k$ by one:
1045 \textbf{primrec} \textit{lift} \textbf{where} \\
1046 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1047 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1048 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1051 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1052 term $t$ has a loose variable with index $k$ or more:
1055 \textbf{primrec}~\textit{loose} \textbf{where} \\
1056 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1057 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1058 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1061 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1065 \textbf{primrec}~\textit{subst} \textbf{where} \\
1066 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1067 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1068 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1069 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1072 A substitution is a function that maps variable indices to terms. Observe that
1073 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1074 away, because the recursive call for the \textit{Lam} case involves an altered
1075 version. Also notice the \textit{lift} call, which increments the variable
1076 indices when moving under a \textit{Lam}.
1078 A reasonable property to expect of substitution is that it should leave closed
1079 terms unchanged. Alas, even this simple property does not hold:
1082 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1083 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1085 Trying 8 scopes: \nopagebreak \\
1086 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1087 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1088 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1089 \hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
1090 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1091 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1092 \hbox{}\qquad Free variables: \nopagebreak \\
1093 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1094 & 0 := \textit{Var}~0,\>
1095 1 := \textit{Var}~0,\>
1096 2 := \textit{Var}~0, \\[-2pt]
1097 & 3 := \textit{Var}~0,\>
1098 4 := \textit{Var}~0,\>
1099 5 := \textit{Var}~0)\end{aligned}$ \\
1100 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1101 Total time: $4679$ ms.
1104 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1105 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1106 $\lambda$-term notation, $t$~is
1107 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1108 The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
1109 replaced with $\textit{lift}~(\sigma~m)~0$.
1111 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1112 cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1113 For the formula of interest, knowing 6 values of that type was enough to find
1114 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1115 considered, a hopeless undertaking:
1118 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1119 {\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
1123 Boxing can be enabled or disabled globally or on a per-type basis using the
1124 \textit{box} option. Moreover, setting the cardinality of a function or
1125 product type implicitly enables boxing for that type. Nitpick usually performs
1126 reasonable choices about which types should be boxed, but option tweaking
1131 \subsection{Scope Monotonicity}
1132 \label{scope-monotonicity}
1134 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1135 and \textit{max}) controls which scopes are actually tested. In general, to
1136 exhaust all models below a certain cardinality bound, the number of scopes that
1137 Nitpick must consider increases exponentially with the number of type variables
1138 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1139 cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
1140 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1142 Fortunately, many formulas exhibit a property called \textsl{scope
1143 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1144 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1146 Consider the formula
1149 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1152 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1153 $'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
1154 exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
1155 that any counterexample found with a small scope would still be a counterexample
1156 in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
1157 by the larger scope. Nitpick comes to the same conclusion after a careful
1158 inspection of the formula and the relevant definitions:
1161 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1163 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1164 Nitpick might be able to skip some scopes.
1165 \\[2\smallskipamount]
1167 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1168 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1169 \textit{list}''~= 1, \\
1170 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
1171 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
1172 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1173 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1174 \textit{list}''~= 2, \\
1175 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
1176 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
1177 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1178 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
1179 \textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
1180 \textit{list}''~= 8, \\
1181 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
1182 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
1183 \\[2\smallskipamount]
1184 Nitpick found a counterexample for
1185 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1186 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1187 \textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
1188 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
1189 \\[2\smallskipamount]
1190 \hbox{}\qquad Free variables: \nopagebreak \\
1191 \hbox{}\qquad\qquad $\textit{xs} = [a_4, a_5]$ \\
1192 \hbox{}\qquad\qquad $\textit{ys} = [b_3, b_3]$ \\[2\smallskipamount]
1193 Total time: 1636 ms.
1196 In theory, it should be sufficient to test a single scope:
1199 \textbf{nitpick}~[\textit{card}~= 8]
1202 However, this is often less efficient in practice and may lead to overly complex
1205 If the monotonicity check fails but we believe that the formula is monotonic (or
1206 we don't mind missing some counterexamples), we can pass the
1207 \textit{mono} option. To convince yourself that this option is risky,
1208 simply consider this example from \S\ref{skolemization}:
1211 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1212 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1213 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1214 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1215 \textbf{nitpick} \\[2\smallskipamount]
1217 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1218 \hbox{}\qquad $\vdots$
1221 (It turns out the formula holds if and only if $\textit{card}~'a \le
1222 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1223 monotonicity checks can be disabled by passing \textit{non\_mono}
1224 (\S\ref{optimizations}).
1226 As insinuated in \S\ref{natural-numbers-and-integers} and
1227 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1228 are normally monotonic and treated as such. The same is true for record types,
1229 \textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
1230 cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
1231 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1232 consider only 8~scopes instead of $32\,768$.
1234 \section{Case Studies}
1235 \label{case-studies}
1237 As a didactic device, the previous section focused mostly on toy formulas whose
1238 validity can easily be assessed just by looking at the formula. We will now
1239 review two somewhat more realistic case studies that are within Nitpick's
1240 reach:\ a context-free grammar modeled by mutually inductive sets and a
1241 functional implementation of AA trees. The results presented in this
1242 section were produced with the following settings:
1245 \textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
1248 \subsection{A Context-Free Grammar}
1249 \label{a-context-free-grammar}
1251 Our first case study is taken from section 7.4 in the Isabelle tutorial
1252 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1253 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1256 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1257 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1258 $A$ & $::=$ & $aS \mid bAA$ \\
1259 $B$ & $::=$ & $bS \mid aBB$
1263 The intuition behind the grammar is that $A$ generates all string with one more
1264 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1266 The alphabet consists exclusively of $a$'s and $b$'s:
1269 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1272 Strings over the alphabet are represented by \textit{alphabet list}s.
1273 Nonterminals in the grammar become sets of strings. The production rules
1274 presented above can be expressed as a mutually inductive definition:
1277 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1278 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1279 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1280 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1281 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1282 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1283 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1286 The conversion of the grammar into the inductive definition was done manually by
1287 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1290 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1291 d'\^etre}. A good approach is to state desirable properties of the specification
1292 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1293 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1294 counterexamples will point to bugs in the specification. For our grammar
1295 example, we will proceed in two steps, separating the soundness and the
1296 completeness of the set $S$. First, soundness:
1299 \textbf{theorem}~\textit{S\_sound}: \\
1300 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1301 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1302 \textbf{nitpick} \\[2\smallskipamount]
1303 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1304 \hbox{}\qquad Free variable: \nopagebreak \\
1305 \hbox{}\qquad\qquad $w = [b]$
1308 It would seem that $[b] \in S$. How could this be? An inspection of the
1309 introduction rules reveals that the only rule with a right-hand side of the form
1310 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1314 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1317 On closer inspection, we can see that this rule is wrong. To match the
1318 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1322 \textbf{nitpick} \\[2\smallskipamount]
1323 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1324 \hbox{}\qquad Free variable: \nopagebreak \\
1325 \hbox{}\qquad\qquad $w = [a, a, b]$
1328 Some detective work is necessary to find out what went wrong here. To get $[a,
1329 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1333 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1336 Now, this formula must be wrong: The same assumption occurs twice, and the
1337 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1338 the assumptions should have been a $w$.
1340 With the correction made, we don't get any counterexample from Nitpick. Let's
1341 move on and check completeness:
1344 \textbf{theorem}~\textit{S\_complete}: \\
1345 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1346 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1347 \longrightarrow w \in S$'' \\
1348 \textbf{nitpick} \\[2\smallskipamount]
1349 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1350 \hbox{}\qquad Free variable: \nopagebreak \\
1351 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1354 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1355 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1356 the introduction rules we have are probably correct. Perhaps we simply lack an
1357 introduction rule. Comparing the grammar with the inductive definition, our
1358 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1359 without which the grammar cannot generate two or more $b$'s in a row. So we add
1363 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1366 With this last change, we don't get any counterexamples from Nitpick for either
1367 soundness or completeness. We can even generalize our result to cover $A$ and
1371 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1372 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1373 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1374 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1375 \textbf{nitpick} \\[2\smallskipamount]
1376 \slshape Nitpick found no counterexample.
1379 \subsection{AA Trees}
1382 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1383 similar performance to red-black trees, but with a simpler implementation
1384 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1385 a total order $<$. We start by defining the datatype and some basic extractor
1389 \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]
1390 \textbf{primrec} \textit{data} \textbf{where} \\
1391 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1392 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1393 \textbf{primrec} \textit{dataset} \textbf{where} \\
1394 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1395 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1396 \textbf{primrec} \textit{level} \textbf{where} \\
1397 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1398 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1399 \textbf{primrec} \textit{left} \textbf{where} \\
1400 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1401 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1402 \textbf{primrec} \textit{right} \textbf{where} \\
1403 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1404 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1407 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1408 as follows \cite{wikipedia-2009-aa-trees}:
1410 \kern.2\parskip %% TYPESETTING
1413 Each node has a level field, and the following invariants must remain true for
1414 the tree to be valid:
1418 \kern-.4\parskip %% TYPESETTING
1423 \item[1.] The level of a leaf node is one.
1424 \item[2.] The level of a left child is strictly less than that of its parent.
1425 \item[3.] The level of a right child is less than or equal to that of its parent.
1426 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1427 \item[5.] Every node of level greater than one must have two children.
1432 \kern.4\parskip %% TYPESETTING
1434 The \textit{wf} predicate formalizes this description:
1437 \textbf{primrec} \textit{wf} \textbf{where} \\
1438 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1439 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1440 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1441 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1442 \phantom{``$($}$\textrm{else}$ \\
1443 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1444 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1445 \mathrel{\land} \textit{level}~u \le k$ \\
1446 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1449 Rebalancing the tree upon insertion and removal of elements is performed by two
1450 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1453 \textbf{primrec} \textit{skew} \textbf{where} \\
1454 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1455 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1456 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1457 \textit{level}~t~\textrm{then}$ \\
1458 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1459 (\textit{right}~t)~u)$ \\
1460 \phantom{``(}$\textrm{else}$ \\
1461 \phantom{``(\quad}$N~x~k~t~u)$''
1465 \textbf{primrec} \textit{split} \textbf{where} \\
1466 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1467 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1468 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1469 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1470 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1471 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1472 \phantom{``(}$\textrm{else}$ \\
1473 \phantom{``(\quad}$N~x~k~t~u)$''
1476 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1477 of elements stored in the tree:
1480 \textbf{theorem}~\textit{dataset\_skew\_split}:\\
1481 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1482 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1483 \textbf{nitpick} \\[2\smallskipamount]
1484 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1487 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1488 should not alter the tree:
1491 \textbf{theorem}~\textit{wf\_skew\_split}:\\
1492 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1493 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1494 \textbf{nitpick} \\[2\smallskipamount]
1495 {\slshape Nitpick found no counterexample.}
1498 Insertion is implemented recursively. It preserves the sort order:
1501 \textbf{primrec}~\textit{insort} \textbf{where} \\
1502 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1503 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1504 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1505 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1508 Notice that we deliberately commented out the application of \textit{skew} and
1509 \textit{split}. Let's see if this causes any problems:
1512 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1513 \textbf{nitpick} \\[2\smallskipamount]
1514 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1515 \hbox{}\qquad Free variables: \nopagebreak \\
1516 \hbox{}\qquad\qquad $t = N~a_3~1~\Lambda~\Lambda$ \\
1517 \hbox{}\qquad\qquad $x = a_4$ \\[2\smallskipamount]
1518 Hint: Maybe you forgot a type constraint?
1521 It's hard to see why this is a counterexample. The hint is of no help here. To
1522 improve readability, we will restrict the theorem to \textit{nat}, so that we
1523 don't need to look up the value of the $\textit{op}~{<}$ constant to find out
1524 which element is smaller than the other. In addition, we will tell Nitpick to
1525 display the value of $\textit{insort}~t~x$ using the \textit{eval} option. This
1529 \textbf{theorem} \textit{wf\_insort\_nat}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1530 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1531 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1532 \hbox{}\qquad Free variables: \nopagebreak \\
1533 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1534 \hbox{}\qquad\qquad $x = 0$ \\
1535 \hbox{}\qquad Evaluated term: \\
1536 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1539 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1540 where both have a level of 1. This violates the second AA tree invariant, which
1541 states that a left child's level must be less than its parent's. This shouldn't
1542 come as a surprise, considering that we commented out the tree rebalancing code.
1543 Reintroducing the code seems to solve the problem:
1546 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1547 \textbf{nitpick} \\[2\smallskipamount]
1548 {\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
1551 Insertion should transform the set of elements represented by the tree in the
1555 \textbf{theorem} \textit{dataset\_insort}:\kern.4em
1556 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1557 \textbf{nitpick} \\[2\smallskipamount]
1558 {\slshape Nitpick ran out of time after checking 5 of 8 scopes.}
1561 We could continue like this and sketch a complete theory of AA trees without
1562 performing a single proof. Once the definitions and main theorems are in place
1563 and have been thoroughly tested using Nitpick, we could start working on the
1564 proofs. Developing theories this way usually saves time, because faulty theorems
1565 and definitions are discovered much earlier in the process.
1567 \section{Option Reference}
1568 \label{option-reference}
1570 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1571 \def\qty#1{$\left<\textit{#1}\right>$}
1572 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1573 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1574 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1575 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1576 \def\ops#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1577 \def\opt#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1578 \def\opu#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1579 \def\opusmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1581 Nitpick's behavior can be influenced by various options, which can be specified
1582 in brackets after the \textbf{nitpick} command. Default values can be set
1583 using \textbf{nitpick\_\allowbreak params}. For example:
1586 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1589 The options are categorized as follows:\ mode of operation
1590 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1591 format (\S\ref{output-format}), automatic counterexample checks
1592 (\S\ref{authentication}), optimizations
1593 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1595 You can instruct Nitpick to run automatically on newly entered theorems by
1596 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1597 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
1598 and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1599 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1600 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1601 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1602 \textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
1603 Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
1606 The number of options can be overwhelming at first glance. Do not let that worry
1607 you: Nitpick's defaults have been chosen so that it almost always does the right
1608 thing, and the most important options have been covered in context in
1609 \S\ref{first-steps}.
1611 The descriptions below refer to the following syntactic quantities:
1614 \item[$\bullet$] \qtybf{string}: A string.
1615 \item[$\bullet$] \qtybf{bool}: \textit{true} or \textit{false}.
1616 \item[$\bullet$] \qtybf{bool\_or\_smart}: \textit{true}, \textit{false}, or \textit{smart}.
1617 \item[$\bullet$] \qtybf{int}: An integer. Negative integers are prefixed with a hyphen.
1618 \item[$\bullet$] \qtybf{int\_or\_smart}: An integer or \textit{smart}.
1619 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1620 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1622 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1623 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1624 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1625 \item[$\bullet$] \qtybf{const}: The name of a HOL constant.
1626 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1627 \item[$\bullet$] \qtybf{term\_list}: A space-separated list of HOL terms (e.g.,
1628 ``$f~x$''~``$g~y$'').
1629 \item[$\bullet$] \qtybf{type}: A HOL type.
1632 Default values are indicated in square brackets. Boolean options have a negated
1633 counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
1634 Boolean options, ``= \textit{true}'' may be omitted.
1636 \subsection{Mode of Operation}
1637 \label{mode-of-operation}
1640 \optrue{blocking}{non\_blocking}
1641 Specifies whether the \textbf{nitpick} command should operate synchronously.
1642 The asynchronous (non-blocking) mode lets the user start proving the putative
1643 theorem while Nitpick looks for a counterexample, but it can also be more
1644 confusing. For technical reasons, automatic runs currently always block.
1646 \optrue{falsify}{satisfy}
1647 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1648 satisfying examples (models). This manual assumes throughout that
1649 \textit{falsify} is enabled.
1651 \opsmart{user\_axioms}{no\_user\_axioms}
1652 Specifies whether the user-defined axioms (specified using
1653 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1654 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1655 the constants that occur in the formula to falsify. The option is implicitly set
1656 to \textit{true} for automatic runs.
1658 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1659 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1660 these conditions are tagged as ``likely genuine.'' The \textit{debug}
1661 (\S\ref{output-format}) option can be used to find out which axioms were
1665 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1666 (\S\ref{output-format}).}
1668 \optrue{assms}{no\_assms}
1669 Specifies whether the relevant assumptions in structured proof should be
1670 considered. The option is implicitly enabled for automatic runs.
1673 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1675 \opfalse{overlord}{no\_overlord}
1676 Specifies whether Nitpick should put its temporary files in
1677 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1678 debugging Nitpick but also unsafe if several instances of the tool are run
1682 {\small See also \textit{debug} (\S\ref{output-format}).}
1685 \subsection{Scope of Search}
1686 \label{scope-of-search}
1689 \opu{card}{type}{int\_seq}
1690 Specifies the sequence of cardinalities to use for a given type. For
1691 \textit{nat} and \textit{int}, the cardinality fully specifies the subset used
1692 to approximate the type. For example:
1694 $$\hbox{\begin{tabular}{@{}rll@{}}%
1695 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
1696 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
1697 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
1702 $$\hbox{\begin{tabular}{@{}rll@{}}%
1703 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
1704 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
1707 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1708 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1709 Although function and product types are normally mapped directly to the
1710 corresponding Kodkod concepts, setting
1711 the cardinality of such types is also allowed and implicitly enables ``boxing''
1712 for them, as explained in the description of the \textit{box}~\qty{type}
1713 and \textit{box} (\S\ref{scope-of-search}) options.
1716 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1718 \opt{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
1719 Specifies the default sequence of cardinalities to use. This can be overridden
1720 on a per-type basis using the \textit{card}~\qty{type} option described above.
1722 \opu{max}{const}{int\_seq}
1723 Specifies the sequence of maximum multiplicities to use for a given
1724 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1725 number of distinct values that it can construct. Nonsensical values (e.g.,
1726 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1727 datatypes equipped with several constructors.
1730 Specifies the default sequence of maximum multiplicities to use for
1731 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
1732 basis using the \textit{max}~\qty{const} option described above.
1734 \opusmart{wf}{const}{non\_wf}
1735 Specifies whether the specified (co)in\-duc\-tively defined predicate is
1736 well-founded. The option can take the following values:
1739 \item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
1740 predicate as if it were well-founded. Since this is generally not sound when the
1741 predicate is not well-founded, the counterexamples are tagged as ``likely
1744 \item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
1745 as if it were not well-founded. The predicate is then unrolled as prescribed by
1746 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
1749 \item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
1750 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
1751 \textit{sizechange} tactics. If this succeeds (or the predicate occurs with an
1752 appropriate polarity in the formula to falsify), use an efficient fixed point
1753 equation as specification of the predicate; otherwise, unroll the predicates
1754 according to the \textit{iter}~\qty{const} and \textit{iter} options.
1758 {\small See also \textit{iter} (\S\ref{scope-of-search}),
1759 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
1760 (\S\ref{timeouts}).}
1762 \opsmart{wf}{non\_wf}
1763 Specifies the default wellfoundedness setting to use. This can be overridden on
1764 a per-predicate basis using the \textit{wf}~\qty{const} option above.
1766 \opu{iter}{const}{int\_seq}
1767 Specifies the sequence of iteration counts to use when unrolling a given
1768 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
1769 predicates that occur negatively and coinductive predicates that occur
1770 positively in the formula to falsify and that cannot be proved to be
1771 well-founded, but this behavior is influenced by the \textit{wf} option. The
1772 iteration counts are automatically bounded by the cardinality of the predicate's
1775 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
1776 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
1778 \opt{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
1779 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
1780 predicates. This can be overridden on a per-predicate basis using the
1781 \textit{iter} \qty{const} option above.
1783 \opt{bisim\_depth}{int\_seq}{$\mathbf{7}$}
1784 Specifies the sequence of iteration counts to use when unrolling the
1785 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
1786 of $-1$ means that no predicate is generated, in which case Nitpick performs an
1787 after-the-fact check to see if the known coinductive datatype values are
1788 bidissimilar. If two values are found to be bisimilar, the counterexample is
1789 tagged as ``likely genuine.'' The iteration counts are automatically bounded by
1790 the sum of the cardinalities of the coinductive datatypes occurring in the
1793 \opusmart{box}{type}{dont\_box}
1794 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
1795 product type in an isomorphic datatype internally. Boxing is an effective mean
1796 to reduce the search space and speed up Nitpick, because the isomorphic datatype
1797 is approximated by a subset of the possible function or pair values;
1798 like other drastic optimizations, it can also prevent the discovery of
1799 counterexamples. The option can take the following values:
1802 \item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
1804 \item[$\bullet$] \textbf{\textit{false}}: Never box the type.
1805 \item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
1806 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
1807 higher-order functions are good candidates for boxing.
1810 Setting the \textit{card}~\qty{type} option for a function or product type
1811 implicitly enables boxing for that type.
1814 {\small See also \textit{verbose} (\S\ref{output-format})
1815 and \textit{debug} (\S\ref{output-format}).}
1817 \opsmart{box}{dont\_box}
1818 Specifies the default boxing setting to use. This can be overridden on a
1819 per-type basis using the \textit{box}~\qty{type} option described above.
1821 \opusmart{mono}{type}{non\_mono}
1822 Specifies whether the specified type should be considered monotonic when
1823 enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a
1824 monotonicity check on the type. Setting this option to \textit{true} can reduce
1825 the number of scopes tried, but it also diminishes the theoretical chance of
1826 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
1829 {\small See also \textit{card} (\S\ref{scope-of-search}),
1830 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
1831 (\S\ref{output-format}).}
1833 \opsmart{mono}{non\_box}
1834 Specifies the default monotonicity setting to use. This can be overridden on a
1835 per-type basis using the \textit{mono}~\qty{type} option described above.
1837 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
1838 Specifies whether type variables with the same sort constraints should be
1839 merged. Setting this option to \textit{true} can reduce the number of scopes
1840 tried and the size of the generated Kodkod formulas, but it also diminishes the
1841 theoretical chance of finding a counterexample.
1843 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1846 \subsection{Output Format}
1847 \label{output-format}
1850 \opfalse{verbose}{quiet}
1851 Specifies whether the \textbf{nitpick} command should explain what it does. This
1852 option is useful to determine which scopes are tried or which SAT solver is
1853 used. This option is implicitly disabled for automatic runs.
1855 \opfalse{debug}{no\_debug}
1856 Specifies whether Nitpick should display additional debugging information beyond
1857 what \textit{verbose} already displays. Enabling \textit{debug} also enables
1858 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
1859 option is implicitly disabled for automatic runs.
1862 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
1863 \textit{batch\_size} (\S\ref{optimizations}).}
1865 \optrue{show\_skolems}{hide\_skolem}
1866 Specifies whether the values of Skolem constants should be displayed as part of
1867 counterexamples. Skolem constants correspond to bound variables in the original
1868 formula and usually help us to understand why the counterexample falsifies the
1872 {\small See also \textit{skolemize} (\S\ref{optimizations}).}
1874 \opfalse{show\_datatypes}{hide\_datatypes}
1875 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
1876 be displayed as part of counterexamples. Such subsets are sometimes helpful when
1877 investigating whether a potential counterexample is genuine or spurious, but
1878 their potential for clutter is real.
1880 \opfalse{show\_consts}{hide\_consts}
1881 Specifies whether the values of constants occurring in the formula (including
1882 its axioms) should be displayed along with any counterexample. These values are
1883 sometimes helpful when investigating why a counterexample is
1884 genuine, but they can clutter the output.
1886 \opfalse{show\_all}{dont\_show\_all}
1887 Enabling this option effectively enables \textit{show\_skolems},
1888 \textit{show\_datatypes}, and \textit{show\_consts}.
1890 \opt{max\_potential}{int}{$\mathbf{1}$}
1891 Specifies the maximum number of potential counterexamples to display. Setting
1892 this option to 0 speeds up the search for a genuine counterexample. This option
1893 is implicitly set to 0 for automatic runs. If you set this option to a value
1894 greater than 1, you will need an incremental SAT solver: For efficiency, it is
1895 recommended to install the JNI version of MiniSat and set \textit{sat\_solver} =
1896 \textit{MiniSatJNI}. Also be aware that many of the counterexamples may look
1897 identical, unless the \textit{show\_all} (\S\ref{output-format}) option is
1901 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
1902 \textit{sat\_solver} (\S\ref{optimizations}).}
1904 \opt{max\_genuine}{int}{$\mathbf{1}$}
1905 Specifies the maximum number of genuine counterexamples to display. If you set
1906 this option to a value greater than 1, you will need an incremental SAT solver:
1907 For efficiency, it is recommended to install the JNI version of MiniSat and set
1908 \textit{sat\_solver} = \textit{MiniSatJNI}. Also be aware that many of the
1909 counterexamples may look identical, unless the \textit{show\_all}
1910 (\S\ref{output-format}) option is enabled.
1913 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
1914 \textit{sat\_solver} (\S\ref{optimizations}).}
1916 \ops{eval}{term\_list}
1917 Specifies the list of terms whose values should be displayed along with
1918 counterexamples. This option suffers from an ``observer effect'': Nitpick might
1919 find different counterexamples for different values of this option.
1921 \opu{format}{term}{int\_seq}
1922 Specifies how to uncurry the value displayed for a variable or constant.
1923 Uncurrying sometimes increases the readability of the output for high-arity
1924 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
1925 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
1926 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
1927 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
1928 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
1929 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
1930 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
1931 arguments that are not accounted for are left alone, as if the specification had
1932 been $1,\ldots,1,n_1,\ldots,n_k$.
1935 {\small See also \textit{uncurry} (\S\ref{optimizations}).}
1937 \opt{format}{int\_seq}{$\mathbf{1}$}
1938 Specifies the default format to use. Irrespective of the default format, the
1939 extra arguments to a Skolem constant corresponding to the outer bound variables
1940 are kept separated from the remaining arguments, the \textbf{for} arguments of
1941 an inductive definitions are kept separated from the remaining arguments, and
1942 the iteration counter of an unrolled inductive definition is shown alone. The
1943 default format can be overridden on a per-variable or per-constant basis using
1944 the \textit{format}~\qty{term} option described above.
1947 %% MARK: Authentication
1948 \subsection{Authentication}
1949 \label{authentication}
1952 \opfalse{check\_potential}{trust\_potential}
1953 Specifies whether potential counterexamples should be given to Isabelle's
1954 \textit{auto} tactic to assess their validity. If a potential counterexample is
1955 shown to be genuine, Nitpick displays a message to this effect and terminates.
1958 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
1960 \opfalse{check\_genuine}{trust\_genuine}
1961 Specifies whether genuine and likely genuine counterexamples should be given to
1962 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
1963 counterexample is shown to be spurious, the user is kindly asked to send a bug
1964 report to the author at
1965 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
1968 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
1970 \ops{expect}{string}
1971 Specifies the expected outcome, which must be one of the following:
1974 \item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
1975 \item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a ``likely
1976 genuine'' counterexample (i.e., a counterexample that is genuine unless
1977 it contradicts a missing axiom or a dangerous option was used inappropriately).
1978 \item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
1979 \item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
1980 \item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
1981 Kodkod ran out of memory).
1984 Nitpick emits an error if the actual outcome differs from the expected outcome.
1985 This option is useful for regression testing.
1988 \subsection{Optimizations}
1989 \label{optimizations}
1991 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
1996 \opt{sat\_solver}{string}{smart}
1997 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
1998 to be faster than their Java counterparts, but they can be more difficult to
1999 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2000 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2001 you will need an incremental SAT solver, such as \textit{MiniSatJNI}
2002 (recommended) or \textit{SAT4J}.
2004 The supported solvers are listed below:
2008 \item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
2009 written in \cpp{}. To use MiniSat, set the environment variable
2010 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2011 executable. The \cpp{} sources and executables for MiniSat are available at
2012 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2013 and 2.0 beta (2007-07-21).
2015 \item[$\bullet$] \textbf{\textit{MiniSatJNI}}: The JNI (Java Native Interface)
2016 version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which
2017 you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2018 version of MiniSat, the JNI version can be used incrementally.
2020 \item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
2021 written in C. It is bundled with Kodkodi and requires no further installation or
2022 configuration steps. Alternatively, you can install a standard version of
2023 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2024 that contains the \texttt{picosat} executable. The C sources for PicoSAT are
2025 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2026 Nitpick has been tested with version 913.
2028 \item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
2029 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2030 the directory that contains the \texttt{zchaff} executable. The \cpp{} sources
2031 and executables for zChaff are available at
2032 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2033 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2035 \item[$\bullet$] \textbf{\textit{zChaffJNI}}: The JNI version of zChaff is
2036 bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on
2037 Kodkod's web site \cite{kodkod-2009}.
2039 \item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
2040 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2041 directory that contains the \texttt{rsat} executable. The \cpp{} sources for
2042 RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been
2043 tested with version 2.01.
2045 \item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
2046 written in C. To use BerkMin, set the environment variable
2047 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2048 executable. The BerkMin executables are available at
2049 \url{http://eigold.tripod.com/BerkMin.html}.
2051 \item[$\bullet$] \textbf{\textit{BerkMinAlloy}}: Variant of BerkMin that is
2052 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2053 version of BerkMin, set the environment variable
2054 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2057 \item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
2058 written in C. To use Jerusat, set the environment variable
2059 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2060 executable. The C sources for Jerusat are available at
2061 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2063 \item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
2064 written in Java that can be used incrementally. It is bundled with Kodkodi and
2065 requires no further installation or configuration steps. Do not attempt to
2066 install the official SAT4J packages, because their API is incompatible with
2069 \item[$\bullet$] \textbf{\textit{SAT4JLight}}: Variant of SAT4J that is
2070 optimized for small problems. It can also be used incrementally.
2072 \item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
2073 experimental solver written in \cpp. To use HaifaSat, set the environment
2074 variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
2075 \texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at
2076 \url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
2078 \item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
2079 \textit{smart}, Nitpick selects the first solver among MiniSatJNI, MiniSat,
2080 PicoSAT, zChaffJNI, zChaff, RSat, BerkMin, BerkMinAlloy, and Jerusat that is
2081 recognized by Isabelle. If none is found, it falls back on SAT4J, which should
2082 always be available. If \textit{verbose} is enabled, Nitpick displays which SAT
2087 \opt{batch\_size}{int\_or\_smart}{smart}
2088 Specifies the maximum number of Kodkod problems that should be lumped together
2089 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2090 together ensures that Kodkodi is launched less often, but it makes the verbose
2091 output less readable and is sometimes detrimental to performance. If
2092 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2093 \textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
2095 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2096 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2097 be rewritten to use (automatically generated) discriminators and destructors.
2098 This optimization can drastically reduce the size of the Boolean formulas given
2102 {\small See also \textit{debug} (\S\ref{output-format}).}
2104 \optrue{specialize}{dont\_specialize}
2105 Specifies whether functions invoked with static arguments should be specialized.
2106 This optimization can drastically reduce the search space, especially for
2107 higher-order functions.
2110 {\small See also \textit{debug} (\S\ref{output-format}) and
2111 \textit{show\_consts} (\S\ref{output-format}).}
2113 \optrue{skolemize}{dont\_skolemize}
2114 Specifies whether the formula should be skolemized. For performance reasons,
2115 (positive) $\forall$-quanti\-fiers that occur in the scope of a higher-order
2116 (positive) $\exists$-quanti\-fier are left unchanged.
2119 {\small See also \textit{debug} (\S\ref{output-format}) and
2120 \textit{show\_skolems} (\S\ref{output-format}).}
2122 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2123 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2124 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2125 predicates for which each the predicate occurs in at most one assumption of each
2126 introduction rule. Using the reflexive transitive closure is in principle
2127 equivalent to setting \textit{iter} to the cardinality of the predicate's
2128 domain, but it is usually more efficient.
2130 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2131 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2133 \optrue{uncurry}{dont\_uncurry}
2134 Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no
2135 tangible effect on efficiency, but it creates opportunities for the boxing
2139 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug}
2140 (\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).}
2142 \optrue{fast\_descrs}{full\_descrs}
2143 Specifies whether Nitpick should optimize the definite and indefinite
2144 description operators (THE and SOME). The optimized versions usually help
2145 Nitpick generate more counterexamples or at least find them faster, but only the
2146 unoptimized versions are complete when all types occurring in the formula are
2149 {\small See also \textit{debug} (\S\ref{output-format}).}
2151 \optrue{peephole\_optim}{no\_peephole\_optim}
2152 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2153 peephole optimizer. These optimizations can make a significant difference.
2154 Unless you are tracking down a bug in Nitpick or distrust the peephole
2155 optimizer, you should leave this option enabled.
2157 \opt{sym\_break}{int}{20}
2158 Specifies an upper bound on the number of relations for which Kodkod generates
2159 symmetry breaking predicates. According to the Kodkod documentation
2160 \cite{kodkod-2009-options}, ``in general, the higher this value, the more
2161 symmetries will be broken, and the faster the formula will be solved. But,
2162 setting the value too high may have the opposite effect and slow down the
2165 \opt{sharing\_depth}{int}{3}
2166 Specifies the depth to which Kodkod should check circuits for equivalence during
2167 the translation to SAT. The default of 3 is the same as in Alloy. The minimum
2168 allowed depth is 1. Increasing the sharing may result in a smaller SAT problem,
2169 but can also slow down Kodkod.
2171 \opfalse{flatten\_props}{dont\_flatten\_props}
2172 Specifies whether Kodkod should try to eliminate intermediate Boolean variables.
2173 Although this might sound like a good idea, in practice it can drastically slow
2176 \opt{max\_threads}{int}{0}
2177 Specifies the maximum number of threads to use in Kodkod. If this option is set
2178 to 0, Kodkod will compute an appropriate value based on the number of processor
2182 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2183 \textit{timeout} (\S\ref{timeouts}).}
2186 \subsection{Timeouts}
2190 \opt{timeout}{time}{$\mathbf{30}$ s}
2191 Specifies the maximum amount of time that the \textbf{nitpick} command should
2192 spend looking for a counterexample. Nitpick tries to honor this constraint as
2193 well as it can but offers no guarantees. For automatic runs,
2194 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2195 a time slot whose length is specified by the ``Auto Counterexample Time
2196 Limit'' option in Proof General.
2199 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2201 \opt{tac\_timeout}{time}{$\mathbf{500}$\,ms}
2202 Specifies the maximum amount of time that the \textit{auto} tactic should use
2203 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2204 and \textit{sizechange} should use when checking whether a (co)in\-duc\-tive
2205 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2206 can but offers no guarantees.
2209 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2210 \textit{check\_potential} (\S\ref{authentication}),
2211 and \textit{check\_genuine} (\S\ref{authentication}).}
2214 \section{Attribute Reference}
2215 \label{attribute-reference}
2217 Nitpick needs to consider the definitions of all constants occurring in a
2218 formula in order to falsify it. For constants introduced using the
2219 \textbf{definition} command, the definition is simply the associated
2220 \textit{\_def} axiom. In contrast, instead of using the internal representation
2221 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2222 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2223 equational specification entered by the user.
2225 Behind the scenes, Isabelle's built-in packages and theories rely on the
2226 following attributes to affect Nitpick's behavior:
2229 \flushitem{\textit{nitpick\_def}}
2232 This attribute specifies an alternative definition of a constant. The
2233 alternative definition should be logically equivalent to the constant's actual
2234 axiomatic definition and should be of the form
2236 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2238 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2241 \flushitem{\textit{nitpick\_simp}}
2244 This attribute specifies the equations that constitute the specification of a
2245 constant. For functions defined using the \textbf{primrec}, \textbf{function},
2246 and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
2247 \textit{simps} rules. The equations must be of the form
2249 \qquad $c~t_1~\ldots\ t_n \,=\, u.$
2251 \flushitem{\textit{nitpick\_psimp}}
2254 This attribute specifies the equations that constitute the partial specification
2255 of a constant. For functions defined using the \textbf{function} package, this
2256 corresponds to the \textit{psimps} rules. The conditional equations must be of
2259 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
2261 \flushitem{\textit{nitpick\_intro}}
2264 This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
2265 For predicates defined using the \textbf{inductive} or \textbf{coinductive}
2266 command, this corresponds to the \textit{intros} rules. The introduction rules
2269 \qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
2270 \ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\
2273 where the $P_i$'s are side conditions that do not involve $c$ and $M$ is an
2274 optional monotonic operator. The order of the assumptions is irrelevant.
2278 When faced with a constant, Nitpick proceeds as follows:
2281 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2282 is not empty, Nitpick uses these rules as the specification of the constant.
2284 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2285 the constant is not empty, it uses these rules as the specification of the
2288 \item[3.] Otherwise, it looks up the definition of the constant:
2291 \item[1.] If the \textit{nitpick\_def} set associated with the constant
2292 is not empty, it uses the latest rule added to the set as the definition of the
2293 constant; otherwise it uses the actual definition axiom.
2294 \item[2.] If the definition is of the form
2296 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
2298 then Nitpick assumes that the definition was made using an inductive package and
2299 based on the introduction rules marked with \textit{nitpick\_\allowbreak
2300 ind\_\allowbreak intros} tries to determine whether the definition is
2305 As an illustration, consider the inductive definition
2308 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2309 ``\textit{odd}~1'' $\,\mid$ \\
2310 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2313 Isabelle automatically attaches the \textit{nitpick\_intro} attribute to
2314 the above rules. Nitpick then uses the \textit{lfp}-based definition in
2315 conjunction with these rules. To override this, we can specify an alternative
2316 definition as follows:
2319 \textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]: ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2322 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2323 = 1$. Alternatively, we can specify an equational specification of the constant:
2326 \textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]: ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2329 Such tweaks should be done with great care, because Nitpick will assume that the
2330 constant is completely defined by its equational specification. For example, if
2331 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
2332 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2333 (\S\ref{output-format}) option is extremely useful to understand what is going
2334 on when experimenting with \textit{nitpick\_} attributes.
2336 \section{Standard ML Interface}
2337 \label{standard-ml-interface}
2339 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2340 and debugging. Among the most interesting functions exported by Nitpick are
2341 those that let you invoke the tool programmatically and those that let you
2342 register and unregister custom coinductive datatypes.
2344 \subsection{Invocation of Nitpick}
2345 \label{invocation-of-nitpick}
2347 The \textit{Nitpick} structure offers the following functions for invoking your
2348 favorite counterexample generator:
2351 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2352 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2353 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2354 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2355 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2358 The return value is a new proof state paired with an outcome string
2359 (``genuine'', ``likely\_genuine'', ``potential'', ``none'', or ``unknown''). The
2360 \textit{params} type is a large record that lets you set Nitpick's options. The
2361 current default options can be retrieved by calling the following function
2362 defined in the \textit{Nitpick\_Isar} structure:
2365 $\textbf{val}\,~\textit{default\_params} :\,
2366 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2369 The second argument lets you override option values before they are parsed and
2370 put into a \textit{params} record. Here is an example:
2373 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2374 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2375 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2376 & \textit{subgoal}\end{aligned}$
2381 \subsection{Registration of Coinductive Datatypes}
2382 \label{registration-of-coinductive-datatypes}
2384 If you have defined a custom coinductive datatype, you can tell Nitpick about
2385 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2386 uses for lazy lists. The interface for registering and unregistering coinductive
2387 datatypes consists of the following pair of functions defined in the
2388 \textit{Nitpick} structure:
2391 $\textbf{val}\,~\textit{register\_codatatype} :\,
2392 \textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2393 $\textbf{val}\,~\textit{unregister\_codatatype} :\,
2394 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2397 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2398 not been, you could have told Nitpick about it by adding the following line
2399 to your theory file:
2402 $\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
2403 & \textit{Nitpick.register\_codatatype} \\[-2pt]
2404 & \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
2405 & \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
2408 The \textit{register\_codatatype} function takes a coinductive type, its case
2409 function, and the list of its constructors. The case function must take its
2410 arguments in the order that the constructors are listed. If no case function
2411 with the correct signature is available, simply pass the empty string.
2413 On the other hand, if your goal is to cripple Nitpick, add the following line to
2414 your theory file and try to check a few conjectures about lazy lists:
2417 $\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
2418 \kern1pt'a~\textit{list}\textrm{''}\}\ \,{*}\}$
2421 \section{Known Bugs and Limitations}
2422 \label{known-bugs-and-limitations}
2424 Here are the known bugs and limitations in Nitpick at the time of writing:
2427 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2428 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2429 Nitpick to generate spurious counterexamples for theorems that refer to values
2430 for which the function is not defined. For example:
2433 \textbf{primrec} \textit{prec} \textbf{where} \\
2434 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2435 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2436 \textbf{nitpick} \\[2\smallskipamount]
2437 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2439 \\[2\smallskipamount]
2440 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2441 \textbf{by}~(\textit{auto simp}: \textit{prec\_def})
2444 Such theorems are considered bad style because they rely on the internal
2445 representation of functions synthesized by Isabelle, which is an implementation
2448 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2449 which can become invalid if you change the definition of an inductive predicate
2450 that is registered in the cache. To clear the cache,
2451 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2452 501$\,\textit{ms}$).
2454 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2455 \textbf{guess} command in a structured proof.
2457 \item[$\bullet$] The \textit{nitpick\_} attributes and the
2458 \textit{Nitpick.register\_} functions can cause havoc if used improperly.
2460 \item[$\bullet$] Local definitions are not supported and result in an error.
2462 \item[$\bullet$] All constants and types whose names start with
2463 \textit{Nitpick}{.} are reserved for internal use.
2467 \bibliography{../manual}{}
2468 \bibliographystyle{abbrv}