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23 \def\Colon{\mathord{:\mkern-1.5mu:}}
24 %\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}}
25 %\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}}
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27 \def\rparr{\mathclose{\mid\mkern-4mu)}}
30 \def\undef{(\lambda x.\; \unk)}
31 %\def\unr{\textit{others}}
33 \def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
34 \def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
36 \hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
37 counter-example counter-examples data-type data-types co-data-type
38 co-data-types in-duc-tive co-in-duc-tive}
44 \selectlanguage{english}
46 \title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
47 Picking Nits \\[\smallskipamount]
48 \Large A User's Guide to Nitpick for Isabelle/HOL}
50 Jasmin Christian Blanchette \\
51 {\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\
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81 \section{Introduction}
84 Nitpick \cite{blanchette-nipkow-2010} is a counterexample generator for
85 Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
86 combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
87 quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
88 first-order relational model finder developed by the Software Design Group at
89 MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
90 borrows many ideas and code fragments, but it benefits from Kodkod's
91 optimizations and a new encoding scheme. The name Nitpick is shamelessly
92 appropriated from a now retired Alloy precursor.
94 Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
95 theorem and wait a few seconds. Nonetheless, there are situations where knowing
96 how it works under the hood and how it reacts to various options helps
97 increase the test coverage. This manual also explains how to install the tool on
98 your workstation. Should the motivation fail you, think of the many hours of
99 hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.
101 Another common use of Nitpick is to find out whether the axioms of a locale are
102 satisfiable, while the locale is being developed. To check this, it suffices to
106 \textbf{lemma}~``$\textit{False}$'' \\
107 \textbf{nitpick}~[\textit{show\_all}]
110 after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
111 must find a model for the axioms. If it finds no model, we have an indication
112 that the axioms might be unsatisfiable.
114 You can also invoke Nitpick from the ``Commands'' submenu of the
115 ``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c C-a
116 C-n. This is equivalent to entering the \textbf{nitpick} command with no
117 arguments in the theory text.
119 Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
120 machine called \texttt{java}. The examples presented in this manual can be found
121 in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.
123 Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
124 Nitpick also provides an automatic mode that can be enabled using the
125 ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof General. In this
126 mode, Nitpick is run on every newly entered theorem, much like Auto Quickcheck.
127 The collective time limit for Auto Nitpick and Auto Quickcheck can be set using
128 the ``Auto Counterexample Time Limit'' option.
131 \setbox\boxA=\hbox{\texttt{nospam}}
133 The known bugs and limitations at the time of writing are listed in
134 \S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
135 or this manual should be directed to
136 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
137 in.\allowbreak tum.\allowbreak de}.
139 \vskip2.5\smallskipamount
141 \textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
142 suggesting several textual improvements.
143 % and Perry James for reporting a typo.
145 %\section{Installation}
146 %\label{installation}
150 % * Nitpick is part of Isabelle/HOL
151 % * but it relies on an external tool called Kodkodi (Kodkod wrapper)
153 % * if you use a prebuilt Isabelle package, Kodkodi is automatically there
154 % * if you work from sources, the latest Kodkodi can be obtained from ...
155 % download it, install it in some directory of your choice (e.g.,
156 % $ISABELLE_HOME/contrib/kodkodi), and add the absolute path to Kodkodi
157 % in your .isabelle/etc/components file
159 % * If you're not sure, just try the example in the next section
161 \section{First Steps}
164 This section introduces Nitpick by presenting small examples. If possible, you
165 should try out the examples on your workstation. Your theory file should start
169 \textbf{theory}~\textit{Scratch} \\
170 \textbf{imports}~\textit{Main~Quotient\_Product~RealDef} \\
174 The results presented here were obtained using the JNI (Java Native Interface)
175 version of MiniSat and with multithreading disabled to reduce nondeterminism.
176 This was done by adding the line
179 \textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSat\_JNI}, \,\textit{max\_threads}~= 1]
182 after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
183 Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
184 be installed, as explained in \S\ref{optimizations}. If you have already
185 configured SAT solvers in Isabelle (e.g., for Refute), these will also be
186 available to Nitpick.
188 \subsection{Propositional Logic}
189 \label{propositional-logic}
191 Let's start with a trivial example from propositional logic:
194 \textbf{lemma}~``$P \longleftrightarrow Q$'' \\
198 You should get the following output:
202 Nitpick found a counterexample: \\[2\smallskipamount]
203 \hbox{}\qquad Free variables: \nopagebreak \\
204 \hbox{}\qquad\qquad $P = \textit{True}$ \\
205 \hbox{}\qquad\qquad $Q = \textit{False}$
208 %FIXME: If you get the output:...
209 %Then do such-and-such.
211 Nitpick can also be invoked on individual subgoals, as in the example below:
214 \textbf{apply}~\textit{auto} \\[2\smallskipamount]
215 {\slshape goal (2 subgoals): \\
216 \phantom{0}1. $P\,\Longrightarrow\, Q$ \\
217 \phantom{0}2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
218 \textbf{nitpick}~1 \\[2\smallskipamount]
219 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
220 \hbox{}\qquad Free variables: \nopagebreak \\
221 \hbox{}\qquad\qquad $P = \textit{True}$ \\
222 \hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
223 \textbf{nitpick}~2 \\[2\smallskipamount]
224 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
225 \hbox{}\qquad Free variables: \nopagebreak \\
226 \hbox{}\qquad\qquad $P = \textit{False}$ \\
227 \hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
231 \subsection{Type Variables}
232 \label{type-variables}
234 If you are left unimpressed by the previous example, don't worry. The next
235 one is more mind- and computer-boggling:
238 \textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
240 \pagebreak[2] %% TYPESETTING
242 The putative lemma involves the definite description operator, {THE}, presented
243 in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
244 operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
245 lemma is merely asserting the indefinite description operator axiom with {THE}
246 substituted for {SOME}.
248 The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
249 containing type variables, Nitpick enumerates the possible domains for each type
250 variable, up to a given cardinality (10 by default), looking for a finite
254 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
256 Trying 10 scopes: \nopagebreak \\
257 \hbox{}\qquad \textit{card}~$'a$~= 1; \\
258 \hbox{}\qquad \textit{card}~$'a$~= 2; \\
259 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
260 \hbox{}\qquad \textit{card}~$'a$~= 10. \\[2\smallskipamount]
261 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
262 \hbox{}\qquad Free variables: \nopagebreak \\
263 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
264 \hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
268 Nitpick found a counterexample in which $'a$ has cardinality 3. (For
269 cardinalities 1 and 2, the formula holds.) In the counterexample, the three
270 values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
272 The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
273 \textit{verbose} is enabled. You can specify \textit{verbose} each time you
274 invoke \textbf{nitpick}, or you can set it globally using the command
277 \textbf{nitpick\_params} [\textit{verbose}]
280 This command also displays the current default values for all of the options
281 supported by Nitpick. The options are listed in \S\ref{option-reference}.
283 \subsection{Constants}
286 By just looking at Nitpick's output, it might not be clear why the
287 counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
288 this time telling it to show the values of the constants that occur in the
292 \textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
293 \textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
295 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
296 \hbox{}\qquad Free variables: \nopagebreak \\
297 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
298 \hbox{}\qquad\qquad $x = a_3$ \\
299 \hbox{}\qquad Constant: \nopagebreak \\
300 \hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
303 We can see more clearly now. Since the predicate $P$ isn't true for a unique
304 value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
305 $a_1$. Since $P~a_1$ is false, the entire formula is falsified.
307 As an optimization, Nitpick's preprocessor introduced the special constant
308 ``\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
309 $\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
310 satisfying $P~y$. We disable this optimization by passing the
311 \textit{full\_descrs} option:
314 \textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
316 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
317 \hbox{}\qquad Free variables: \nopagebreak \\
318 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
319 \hbox{}\qquad\qquad $x = a_3$ \\
320 \hbox{}\qquad Constant: \nopagebreak \\
321 \hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
324 As the result of another optimization, Nitpick directly assigned a value to the
325 subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
326 disable this second optimization by using the command
329 \textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
330 \textit{show\_consts}]
333 we finally get \textit{The}:
336 \slshape Constant: \nopagebreak \\
337 \hbox{}\qquad $\mathit{The} = \undef{}
338 (\!\begin{aligned}[t]%
339 & \{a_1, a_2, a_3\} := a_3,\> \{a_1, a_2\} := a_3,\> \{a_1, a_3\} := a_3, \\[-2pt] %% TYPESETTING
340 & \{a_1\} := a_1,\> \{a_2, a_3\} := a_1,\> \{a_2\} := a_2, \\[-2pt]
341 & \{a_3\} := a_3,\> \{\} := a_3)\end{aligned}$
344 Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
345 just like before.\footnote{The Isabelle/HOL notation $f(x :=
346 y)$ denotes the function that maps $x$ to $y$ and that otherwise behaves like
349 Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
350 unique $x$ such that'') at the front of our putative lemma's assumption:
353 \textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
356 The fix appears to work:
359 \textbf{nitpick} \\[2\smallskipamount]
360 \slshape Nitpick found no counterexample.
363 We can further increase our confidence in the formula by exhausting all
364 cardinalities up to 50:
367 \textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
368 can be entered as \texttt{-} (hyphen) or
369 \texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
370 \slshape Nitpick found no counterexample.
373 Let's see if Sledgehammer can find a proof:
376 \textbf{sledgehammer} \\[2\smallskipamount]
377 {\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
378 $\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
379 Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
380 \textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
381 {\slshape No subgoals!}% \\[2\smallskipamount]
385 This must be our lucky day.
387 \subsection{Skolemization}
388 \label{skolemization}
390 Are all invertible functions onto? Let's find out:
393 \textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
394 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
395 \textbf{nitpick} \\[2\smallskipamount]
397 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
398 \hbox{}\qquad Free variable: \nopagebreak \\
399 \hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
400 \hbox{}\qquad Skolem constants: \nopagebreak \\
401 \hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
402 \hbox{}\qquad\qquad $y = a_2$
405 Although $f$ is the only free variable occurring in the formula, Nitpick also
406 displays values for the bound variables $g$ and $y$. These values are available
407 to Nitpick because it performs skolemization as a preprocessing step.
409 In the previous example, skolemization only affected the outermost quantifiers.
410 This is not always the case, as illustrated below:
413 \textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
414 \textbf{nitpick} \\[2\smallskipamount]
416 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
417 \hbox{}\qquad Skolem constant: \nopagebreak \\
418 \hbox{}\qquad\qquad $\lambda x.\; f =
419 \undef{}(\!\begin{aligned}[t]
420 & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
421 & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
424 The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
425 $x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
426 function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
427 maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
429 The source of the Skolem constants is sometimes more obscure:
432 \textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
433 \textbf{nitpick} \\[2\smallskipamount]
435 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
436 \hbox{}\qquad Free variable: \nopagebreak \\
437 \hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
438 \hbox{}\qquad Skolem constants: \nopagebreak \\
439 \hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
440 \hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
443 What happened here is that Nitpick expanded the \textit{sym} constant to its
447 $\mathit{sym}~r \,\equiv\,
448 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
451 As their names suggest, the Skolem constants $\mathit{sym}.x$ and
452 $\mathit{sym}.y$ are simply the bound variables $x$ and $y$
453 from \textit{sym}'s definition.
455 \subsection{Natural Numbers and Integers}
456 \label{natural-numbers-and-integers}
458 Because of the axiom of infinity, the type \textit{nat} does not admit any
459 finite models. To deal with this, Nitpick's approach is to consider finite
460 subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined
461 value (displayed as `$\unk$'). The type \textit{int} is handled similarly.
462 Internally, undefined values lead to a three-valued logic.
464 Here is an example involving \textit{int\/}:
467 \textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
468 \textbf{nitpick} \\[2\smallskipamount]
469 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
470 \hbox{}\qquad Free variables: \nopagebreak \\
471 \hbox{}\qquad\qquad $i = 0$ \\
472 \hbox{}\qquad\qquad $j = 1$ \\
473 \hbox{}\qquad\qquad $m = 1$ \\
474 \hbox{}\qquad\qquad $n = 0$
477 Internally, Nitpick uses either a unary or a binary representation of numbers.
478 The unary representation is more efficient but only suitable for numbers very
479 close to zero. By default, Nitpick attempts to choose the more appropriate
480 encoding by inspecting the formula at hand. This behavior can be overridden by
481 passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For
482 binary notation, the number of bits to use can be specified using
483 the \textit{bits} option. For example:
486 \textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$]
489 With infinite types, we don't always have the luxury of a genuine counterexample
490 and must often content ourselves with a potential one. The tedious task of
491 finding out whether the potential counterexample is in fact genuine can be
492 outsourced to \textit{auto} by passing \textit{check\_potential}. For example:
495 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
496 \textbf{nitpick} [\textit{card~nat}~= 50, \textit{check\_potential}] \\[2\smallskipamount]
497 \slshape Warning: The conjecture either trivially holds for the given scopes or lies outside Nitpick's supported
498 fragment. Only potential counterexamples may be found. \\[2\smallskipamount]
499 Nitpick found a potential counterexample: \\[2\smallskipamount]
500 \hbox{}\qquad Free variable: \nopagebreak \\
501 \hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
502 Confirmation by ``\textit{auto}'': The above counterexample is genuine.
505 You might wonder why the counterexample is first reported as potential. The root
506 of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
507 \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
508 that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
509 \textit{False}; but otherwise, it does not know anything about values of $n \ge
510 \textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
511 \textit{True}. Since the assumption can never be satisfied, the putative lemma
512 can never be falsified.
514 Incidentally, if you distrust the so-called genuine counterexamples, you can
515 enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
516 aware that \textit{auto} will usually fail to prove that the counterexample is
519 Some conjectures involving elementary number theory make Nitpick look like a
520 giant with feet of clay:
523 \textbf{lemma} ``$P~\textit{Suc}$'' \\
524 \textbf{nitpick} \\[2\smallskipamount]
526 Nitpick found no counterexample.
529 On any finite set $N$, \textit{Suc} is a partial function; for example, if $N =
530 \{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\,
531 \ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when passed as
532 argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$. The next
536 \textbf{lemma} ``$P~(\textit{op}~{+}\Colon
537 \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
538 \textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
539 {\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
540 \hbox{}\qquad Free variable: \nopagebreak \\
541 \hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
542 \textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
543 {\slshape Nitpick found no counterexample.}
546 The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
547 $\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
550 Because numbers are infinite and are approximated using a three-valued logic,
551 there is usually no need to systematically enumerate domain sizes. If Nitpick
552 cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
553 unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
554 example above is an exception to this principle.) Nitpick nonetheless enumerates
555 all cardinalities from 1 to 10 for \textit{nat}, mainly because smaller
556 cardinalities are fast to handle and give rise to simpler counterexamples. This
557 is explained in more detail in \S\ref{scope-monotonicity}.
559 \subsection{Inductive Datatypes}
560 \label{inductive-datatypes}
562 Like natural numbers and integers, inductive datatypes with recursive
563 constructors admit no finite models and must be approximated by a subterm-closed
564 subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
565 Nitpick looks for all counterexamples that can be built using at most 10
568 Let's see with an example involving \textit{hd} (which returns the first element
569 of a list) and $@$ (which concatenates two lists):
572 \textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
573 \textbf{nitpick} \\[2\smallskipamount]
574 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
575 \hbox{}\qquad Free variables: \nopagebreak \\
576 \hbox{}\qquad\qquad $\textit{xs} = []$ \\
577 \hbox{}\qquad\qquad $\textit{y} = a_1$
580 To see why the counterexample is genuine, we enable \textit{show\_consts}
581 and \textit{show\_\allowbreak datatypes}:
584 {\slshape Datatype:} \\
585 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\
586 {\slshape Constants:} \\
587 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_1, a_1])$ \\
588 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$
591 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
594 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
595 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_1,
596 a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not
597 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
598 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
599 appending $[a_1, a_1]$ to itself gives $\unk$.
601 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
602 considers the following subsets:
604 \kern-.5\smallskipamount %% TYPESETTING
608 $\{[],\, [a_1],\, [a_2]\}$; \\
609 $\{[],\, [a_1],\, [a_3]\}$; \\
610 $\{[],\, [a_2],\, [a_3]\}$; \\
611 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\
612 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\
613 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\
614 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\
615 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\
616 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\
617 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\
618 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\
619 $\{[],\, [a_3],\, [a_3, a_3]\}$.
623 \kern-2\smallskipamount %% TYPESETTING
625 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
626 are listed and only those. As an example of a non-subterm-closed subset,
627 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe
628 that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin
629 \mathcal{S}$ as a subterm.
631 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
634 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
635 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
637 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
638 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
639 \hbox{}\qquad Free variables: \nopagebreak \\
640 \hbox{}\qquad\qquad $\textit{xs} = [a_1]$ \\
641 \hbox{}\qquad\qquad $\textit{ys} = [a_2]$ \\
642 \hbox{}\qquad Datatypes: \\
643 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
644 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$
647 Because datatypes are approximated using a three-valued logic, there is usually
648 no need to systematically enumerate cardinalities: If Nitpick cannot find a
649 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
650 unlikely that one could be found for smaller cardinalities.
652 \subsection{Typedefs, Quotient Types, Records, Rationals, and Reals}
653 \label{typedefs-quotient-types-records-rationals-and-reals}
655 Nitpick generally treats types declared using \textbf{typedef} as datatypes
656 whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
660 \textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
661 \textbf{by}~\textit{blast} \\[2\smallskipamount]
662 \textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
663 \textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
664 \textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
665 \textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
666 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
667 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
668 \hbox{}\qquad Free variables: \nopagebreak \\
669 \hbox{}\qquad\qquad $P = \{\Abs{0},\, \Abs{1}\}$ \\
670 \hbox{}\qquad\qquad $x = \Abs{2}$ \\
671 \hbox{}\qquad Datatypes: \\
672 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
673 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$
676 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
678 Quotient types are handled in much the same way. The following fragment defines
679 the integer type \textit{my\_int} by encoding the integer $x$ by a pair of
680 natural numbers $(m, n)$ such that $x + n = m$:
683 \textbf{fun} \textit{my\_int\_rel} \textbf{where} \\
684 ``$\textit{my\_int\_rel}~(x,\, y)~(u,\, v) = (x + v = u + y)$'' \\[2\smallskipamount]
686 \textbf{quotient\_type}~\textit{my\_int} = ``$\textit{nat} \times \textit{nat\/}$''$\;{/}\;$\textit{my\_int\_rel} \\
687 \textbf{by}~(\textit{auto simp add\/}:\ \textit{equivp\_def expand\_fun\_eq}) \\[2\smallskipamount]
689 \textbf{definition}~\textit{add\_raw}~\textbf{where} \\
690 ``$\textit{add\_raw} \,\equiv\, \lambda(x,\, y)~(u,\, v).\; (x + (u\Colon\textit{nat}), y + (v\Colon\textit{nat}))$'' \\[2\smallskipamount]
692 \textbf{quotient\_definition} ``$\textit{add\/}\Colon\textit{my\_int} \Rightarrow \textit{my\_int} \Rightarrow \textit{my\_int\/}$'' \textbf{is} \textit{add\_raw} \\[2\smallskipamount]
694 \textbf{lemma} ``$\textit{add}~x~y = \textit{add}~x~x$'' \\
695 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
696 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
697 \hbox{}\qquad Free variables: \nopagebreak \\
698 \hbox{}\qquad\qquad $x = \Abs{(0,\, 0)}$ \\
699 \hbox{}\qquad\qquad $y = \Abs{(1,\, 0)}$ \\
700 \hbox{}\qquad Datatypes: \\
701 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, \unr\}$ \\
702 \hbox{}\qquad\qquad $\textit{nat} \times \textit{nat}~[\textsl{boxed\/}] = \{(0,\, 0),\> (1,\, 0),\> \unr\}$ \\
703 \hbox{}\qquad\qquad $\textit{my\_int} = \{\Abs{(0,\, 0)},\> \Abs{(1,\, 0)},\> \unr\}$
706 In the counterexample, $\Abs{(0,\, 0)}$ and $\Abs{(1,\, 0)}$ represent the
707 integers $0$ and $1$, respectively. Other representants would have been
708 possible---e.g., $\Abs{(5,\, 5)}$ and $\Abs{(12,\, 11)}$. If we are going to
709 use \textit{my\_int} extensively, it pays off to install a term postprocessor
710 that converts the pair notation to the standard mathematical notation:
713 $\textbf{ML}~\,\{{*} \\
715 %& ({*}~\,\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \\[-2pt]
716 %& \phantom{(*}~\,{\rightarrow}\;\textit{term}~\,{*}) \\[-2pt]
717 & \textbf{fun}\,~\textit{my\_int\_postproc}~\_~\_~\_~T~(\textit{Const}~\_~\$~(\textit{Const}~\_~\$~\textit{t1}~\$~\textit{t2\/})) = {} \\[-2pt]
718 & \phantom{fun}\,~\textit{HOLogic.mk\_number}~T~(\textit{snd}~(\textit{HOLogic.dest\_number~t1}) \\[-2pt]
719 & \phantom{fun\,~\textit{HOLogic.mk\_number}~T~(}{-}~\textit{snd}~(\textit{HOLogic.dest\_number~t2\/})) \\[-2pt]
720 & \phantom{fun}\!{\mid}\,~\textit{my\_int\_postproc}~\_~\_~\_~\_~t = t \\[-2pt]
721 {*}\}\end{aligned}$ \\[2\smallskipamount]
722 $\textbf{setup}~\,\{{*} \\
724 & \textit{Nitpick\_Model.register\_term\_postprocessor\_global\/}~\!\begin{aligned}[t]
725 & @\{\textrm{typ}~\textit{my\_int}\} \\[-2pt]
726 & \textit{my\_int\_postproc}\end{aligned} \\[-2pt]
730 Records are also handled as datatypes with a single constructor:
733 \textbf{record} \textit{point} = \\
734 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
735 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
736 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
737 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
738 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
739 \hbox{}\qquad Free variables: \nopagebreak \\
740 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
741 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
742 \hbox{}\qquad Datatypes: \\
743 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
744 \hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
745 & \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
746 & \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
749 Finally, Nitpick provides rudimentary support for rationals and reals using a
753 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
754 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
755 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
756 \hbox{}\qquad Free variables: \nopagebreak \\
757 \hbox{}\qquad\qquad $x = 1/2$ \\
758 \hbox{}\qquad\qquad $y = -1/2$ \\
759 \hbox{}\qquad Datatypes: \\
760 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
761 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
762 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
765 \subsection{Inductive and Coinductive Predicates}
766 \label{inductive-and-coinductive-predicates}
768 Inductively defined predicates (and sets) are particularly problematic for
769 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
770 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
771 the problem is that they are defined using a least fixed-point construction.
773 Nitpick's philosophy is that not all inductive predicates are equal. Consider
774 the \textit{even} predicate below:
777 \textbf{inductive}~\textit{even}~\textbf{where} \\
778 ``\textit{even}~0'' $\,\mid$ \\
779 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
782 This predicate enjoys the desirable property of being well-founded, which means
783 that the introduction rules don't give rise to infinite chains of the form
786 $\cdots\,\Longrightarrow\, \textit{even}~k''
787 \,\Longrightarrow\, \textit{even}~k'
788 \,\Longrightarrow\, \textit{even}~k.$
791 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
795 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
796 \,\Longrightarrow\, \textit{even}~(k - 2)
797 \,\Longrightarrow\, \textit{even}~k.$
800 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
801 fixed-point computation.%
802 \footnote{If an inductive predicate is
803 well-founded, then it has exactly one fixed point, which is simultaneously the
804 least and the greatest fixed point. In these circumstances, the computation of
805 the least fixed point amounts to the computation of an arbitrary fixed point,
806 which can be performed using a straightforward recursive equation.}
807 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
808 just as Isabelle's \textbf{function} package usually discharges termination
809 proof obligations automatically.
811 Let's try an example:
814 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
815 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
816 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
817 Nitpick can compute it efficiently. \\[2\smallskipamount]
819 \hbox{}\qquad \textit{card nat}~= 50. \\[2\smallskipamount]
820 Nitpick found a potential counterexample for \textit{card nat}~= 50: \\[2\smallskipamount]
821 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
822 Nitpick could not find a better counterexample. It checked 0 of 1 scope. \\[2\smallskipamount]
826 No genuine counterexample is possible because Nitpick cannot rule out the
827 existence of a natural number $n \ge 50$ such that both $\textit{even}~n$ and
828 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
829 existential quantifier:
832 \textbf{lemma} ``$\exists n \mathbin{\le} 49.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
833 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}] \\[2\smallskipamount]
834 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
835 \hbox{}\qquad Empty assignment
838 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
839 we use the following definition instead?
842 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
843 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
844 ``$\textit{even}'~2$'' $\,\mid$ \\
845 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
848 This definition is not well-founded: From $\textit{even}'~0$ and
849 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
850 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
852 Let's check a property involving $\textit{even}'$. To make up for the
853 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
854 \textit{nat}'s cardinality to a mere 10:
857 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
858 \lnot\;\textit{even}'~n$'' \\
859 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
861 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
862 Nitpick might need to unroll it. \\[2\smallskipamount]
864 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
865 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
866 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
867 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
868 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
869 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
870 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
871 \hbox{}\qquad Constant: \nopagebreak \\
872 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
873 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
874 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
875 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
879 Nitpick's output is very instructive. First, it tells us that the predicate is
880 unrolled, meaning that it is computed iteratively from the empty set. Then it
881 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
884 The output also shows how each iteration contributes to $\textit{even}'$. The
885 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
886 predicate depends on an iteration counter. Iteration 0 provides the basis
887 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
888 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
889 iterations would not contribute any new elements.
891 Some values are marked with superscripted question
892 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
893 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
894 \textit{True} or $\unk$, never \textit{False}.
896 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, 20, 24, and 28
897 iterations. However, these numbers are bounded by the cardinality of the
898 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
899 ever needed to compute the value of a \textit{nat} predicate. You can specify
900 the number of iterations using the \textit{iter} option, as explained in
901 \S\ref{scope-of-search}.
903 In the next formula, $\textit{even}'$ occurs both positively and negatively:
906 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
907 \textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
908 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
909 \hbox{}\qquad Free variable: \nopagebreak \\
910 \hbox{}\qquad\qquad $n = 1$ \\
911 \hbox{}\qquad Constants: \nopagebreak \\
912 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
913 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
914 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
917 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
918 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
919 fixed point (not necessarily the least one). It is used to falsify
920 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
921 $\textit{even}'~(n - 2)$.
923 Coinductive predicates are handled dually. For example:
926 \textbf{coinductive} \textit{nats} \textbf{where} \\
927 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
928 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
929 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
930 \slshape Nitpick found a counterexample:
931 \\[2\smallskipamount]
932 \hbox{}\qquad Constants: \nopagebreak \\
933 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
934 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
935 & \unr\})\end{aligned}$ \\
936 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
939 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
940 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
941 inductive predicates for which each the predicate occurs in at most one
942 assumption of each introduction rule. For example:
945 \textbf{inductive} \textit{odd} \textbf{where} \\
946 ``$\textit{odd}~1$'' $\,\mid$ \\
947 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
948 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
949 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
950 \slshape Nitpick found a counterexample:
951 \\[2\smallskipamount]
952 \hbox{}\qquad Free variable: \nopagebreak \\
953 \hbox{}\qquad\qquad $n = 1$ \\
954 \hbox{}\qquad Constants: \nopagebreak \\
955 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
956 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
957 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
959 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
960 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
962 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
963 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
964 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
968 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
969 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
970 elements from known ones. The set $\textit{odd}$ consists of all the values
971 reachable through the reflexive transitive closure of
972 $\textit{odd}_{\textrm{step}}$ starting with any element from
973 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
974 transitive closure to encode linear predicates is normally either more thorough
975 or more efficient than unrolling (depending on the value of \textit{iter}), but
976 for those cases where it isn't you can disable it by passing the
977 \textit{dont\_star\_linear\_preds} option.
979 \subsection{Coinductive Datatypes}
980 \label{coinductive-datatypes}
982 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
983 datatypes, the \textit{Coinductive\_List} theory from Andreas Lochbihler's
984 \textit{Coinductive} AFP entry \cite{lochbihler-2010} provides a coinductive
985 ``lazy list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick
986 supports these lazy lists seamlessly and provides a hook, described in
987 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
990 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
991 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
992 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
993 1, 2, 3, \ldots]$ can be defined as lazy lists using the
994 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
995 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
996 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
998 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
999 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
1003 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
1004 \textbf{nitpick} \\[2\smallskipamount]
1005 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
1006 \hbox{}\qquad Free variables: \nopagebreak \\
1007 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
1008 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
1011 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
1012 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
1013 infinite list $[a_1, a_1, a_1, \ldots]$.
1015 The next example is more interesting:
1018 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
1019 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1020 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1021 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
1022 some scopes. \\[2\smallskipamount]
1023 Trying 10 scopes: \\
1024 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 1,
1025 and \textit{bisim\_depth}~= 0. \\
1026 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1027 \hbox{}\qquad \textit{card} $'a$~= 10, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 10,
1028 and \textit{bisim\_depth}~= 9. \\[2\smallskipamount]
1029 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
1030 \textit{card}~``\kern1pt$'a~\textit{list\/}$''~= 2, and \textit{bisim\_\allowbreak
1032 \\[2\smallskipamount]
1033 \hbox{}\qquad Free variables: \nopagebreak \\
1034 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
1035 \hbox{}\qquad\qquad $\textit{b} = a_2$ \\
1036 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1037 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
1038 Total time: 1027 ms.
1041 The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
1042 $\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
1043 $[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment
1044 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
1045 the segment leading to the binder is the stem.
1047 A salient property of coinductive datatypes is that two objects are considered
1048 equal if and only if they lead to the same observations. For example, the lazy
1049 lists $\textrm{THE}~\omega.\; \omega =
1050 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
1051 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
1052 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
1053 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
1054 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
1055 concept of equality for coinductive datatypes is called bisimulation and is
1056 defined coinductively.
1058 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
1059 the Kodkod problem to ensure that distinct objects lead to different
1060 observations. This precaution is somewhat expensive and often unnecessary, so it
1061 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
1062 bisimilarity check is then performed \textsl{after} the counterexample has been
1063 found to ensure correctness. If this after-the-fact check fails, the
1064 counterexample is tagged as ``quasi genuine'' and Nitpick recommends to try
1065 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
1066 check for the previous example saves approximately 150~milli\-seconds; the speed
1067 gains can be more significant for larger scopes.
1069 The next formula illustrates the need for bisimilarity (either as a Kodkod
1070 predicate or as an after-the-fact check) to prevent spurious counterexamples:
1073 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
1074 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1075 \textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
1076 \slshape Nitpick found a quasi genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
1077 \hbox{}\qquad Free variables: \nopagebreak \\
1078 \hbox{}\qquad\qquad $a = a_1$ \\
1079 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
1080 \textit{LCons}~a_1~\omega$ \\
1081 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1082 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
1083 \hbox{}\qquad\qquad $'a~\textit{llist} =
1084 \{\!\begin{aligned}[t]
1085 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
1086 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
1087 \\[2\smallskipamount]
1088 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1089 that the counterexample is genuine. \\[2\smallskipamount]
1090 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1091 \slshape Nitpick found no counterexample.
1094 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1095 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1097 A compromise between leaving out the bisimilarity predicate from the Kodkod
1098 problem and performing the after-the-fact check is to specify a lower
1099 nonnegative \textit{bisim\_depth} value than the default one provided by
1100 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1101 be distinguished from each other by their prefixes of length $K$. Be aware that
1102 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1103 finding any counterexamples.
1108 Nitpick normally maps function and product types directly to the corresponding
1109 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1110 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1111 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1112 off to treat these types in the same way as plain datatypes, by approximating
1113 them by a subset of a given cardinality. This technique is called ``boxing'' and
1114 is particularly useful for functions passed as arguments to other functions, for
1115 high-arity functions, and for large tuples. Under the hood, boxing involves
1116 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1117 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1119 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1120 using de Bruijn's notation:
1123 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1126 The $\textit{lift}~t~k$ function increments all variables with indices greater
1127 than or equal to $k$ by one:
1130 \textbf{primrec} \textit{lift} \textbf{where} \\
1131 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1132 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1133 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1136 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1137 term $t$ has a loose variable with index $k$ or more:
1140 \textbf{primrec}~\textit{loose} \textbf{where} \\
1141 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1142 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1143 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1146 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1150 \textbf{primrec}~\textit{subst} \textbf{where} \\
1151 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1152 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1153 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1154 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1157 A substitution is a function that maps variable indices to terms. Observe that
1158 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1159 away, because the recursive call for the \textit{Lam} case involves an altered
1160 version. Also notice the \textit{lift} call, which increments the variable
1161 indices when moving under a \textit{Lam}.
1163 A reasonable property to expect of substitution is that it should leave closed
1164 terms unchanged. Alas, even this simple property does not hold:
1167 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1168 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1170 Trying 10 scopes: \nopagebreak \\
1171 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1172 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1173 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1174 \hbox{}\qquad \textit{card~nat}~= 10, \textit{card tm}~= 10, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 10. \\[2\smallskipamount]
1175 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1176 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1177 \hbox{}\qquad Free variables: \nopagebreak \\
1178 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1179 & 0 := \textit{Var}~0,\>
1180 1 := \textit{Var}~0,\>
1181 2 := \textit{Var}~0, \\[-2pt]
1182 & 3 := \textit{Var}~0,\>
1183 4 := \textit{Var}~0,\>
1184 5 := \textit{Var}~0)\end{aligned}$ \\
1185 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1186 Total time: $3560$ ms.
1189 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1190 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1191 $\lambda$-term notation, $t$~is
1192 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1193 The bug is in \textit{subst\/}: The $\textit{lift}~(\sigma~m)~1$ call should be
1194 replaced with $\textit{lift}~(\sigma~m)~0$.
1196 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1197 cardinalities from 1 to 10 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1198 For the formula of interest, knowing 6 values of that type was enough to find
1199 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1200 considered, a hopeless undertaking:
1203 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1204 {\slshape Nitpick ran out of time after checking 3 of 10 scopes.}
1208 Boxing can be enabled or disabled globally or on a per-type basis using the
1209 \textit{box} option. Nitpick usually performs reasonable choices about which
1210 types should be boxed, but option tweaking sometimes helps. A related optimization,
1211 ``finalization,'' attempts to wrap functions that constant at all but finitely
1212 many points (e.g., finite sets); see the documentation for the \textit{finalize}
1213 option in \S\ref{scope-of-search} for details.
1217 \subsection{Scope Monotonicity}
1218 \label{scope-monotonicity}
1220 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1221 and \textit{max}) controls which scopes are actually tested. In general, to
1222 exhaust all models below a certain cardinality bound, the number of scopes that
1223 Nitpick must consider increases exponentially with the number of type variables
1224 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1225 cardinality specification of 1--10, no fewer than $10^4 = 10\,000$ scopes must be
1226 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1228 Fortunately, many formulas exhibit a property called \textsl{scope
1229 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1230 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1232 Consider the formula
1235 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1238 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1239 $'b~\textit{list}$. A priori, Nitpick would need to consider $1\,000$ scopes to
1240 exhaust the specification \textit{card}~= 1--10 (10 cardinalies for $'a$
1241 $\times$ 10 cardinalities for $'b$ $\times$ 10 cardinalities for the datatypes).
1242 However, our intuition tells us that any counterexample found with a small scope
1243 would still be a counterexample in a larger scope---by simply ignoring the fresh
1244 $'a$ and $'b$ values provided by the larger scope. Nitpick comes to the same
1245 conclusion after a careful inspection of the formula and the relevant
1249 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1251 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1252 Nitpick might be able to skip some scopes.
1253 \\[2\smallskipamount]
1254 Trying 10 scopes: \\
1255 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1256 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1257 \textit{list\/}''~= 1, \\
1258 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 1, and
1259 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 1. \\
1260 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1261 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1262 \textit{list\/}''~= 2, \\
1263 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 2, and
1264 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 2. \\
1265 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1266 \hbox{}\qquad \textit{card} $'a$~= 10, \textit{card} $'b$~= 10,
1267 \textit{card} \textit{nat}~= 10, \textit{card} ``$('a \times {'}b)$
1268 \textit{list\/}''~= 10, \\
1269 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 10, and
1270 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 10.
1271 \\[2\smallskipamount]
1272 Nitpick found a counterexample for
1273 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1274 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1275 \textit{list\/}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 5, and
1276 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 5:
1277 \\[2\smallskipamount]
1278 \hbox{}\qquad Free variables: \nopagebreak \\
1279 \hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
1280 \hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
1281 Total time: 1636 ms.
1284 In theory, it should be sufficient to test a single scope:
1287 \textbf{nitpick}~[\textit{card}~= 10]
1290 However, this is often less efficient in practice and may lead to overly complex
1293 If the monotonicity check fails but we believe that the formula is monotonic (or
1294 we don't mind missing some counterexamples), we can pass the
1295 \textit{mono} option. To convince yourself that this option is risky,
1296 simply consider this example from \S\ref{skolemization}:
1299 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1300 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1301 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1302 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1303 \textbf{nitpick} \\[2\smallskipamount]
1305 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1306 \hbox{}\qquad $\vdots$
1309 (It turns out the formula holds if and only if $\textit{card}~'a \le
1310 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1311 monotonicity checks can be disabled by passing \textit{non\_mono}
1312 (\S\ref{optimizations}).
1314 As insinuated in \S\ref{natural-numbers-and-integers} and
1315 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1316 are normally monotonic and treated as such. The same is true for record types,
1317 \textit{rat}, and \textit{real}. Thus, given the
1318 cardinality specification 1--10, a formula involving \textit{nat}, \textit{int},
1319 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1320 consider only 10~scopes instead of $10\,000$. On the other hand,
1321 \textbf{typedef}s and quotient types are generally nonmonotonic.
1323 \subsection{Inductive Properties}
1324 \label{inductive-properties}
1326 Inductive properties are a particular pain to prove, because the failure to
1327 establish an induction step can mean several things:
1330 \item The property is invalid.
1331 \item The property is valid but is too weak to support the induction step.
1332 \item The property is valid and strong enough; it's just that we haven't found
1336 Depending on which scenario applies, we would take the appropriate course of
1340 \item Repair the statement of the property so that it becomes valid.
1341 \item Generalize the property and/or prove auxiliary properties.
1342 \item Work harder on a proof.
1345 How can we distinguish between the three scenarios? Nitpick's normal mode of
1346 operation can often detect scenario 1, and Isabelle's automatic tactics help with
1347 scenario 3. Using appropriate techniques, it is also often possible to use
1348 Nitpick to identify scenario 2. Consider the following transition system,
1349 in which natural numbers represent states:
1352 \textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
1353 ``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
1354 ``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
1355 ``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
1358 We will try to prove that only even numbers are reachable:
1361 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
1364 Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
1365 so let's attempt a proof by induction:
1368 \textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
1369 \textbf{apply}~\textit{auto}
1372 This leaves us in the following proof state:
1375 {\slshape goal (2 subgoals): \\
1376 \phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
1377 \phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
1381 If we run Nitpick on the first subgoal, it still won't find any
1382 counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
1383 is helpless. However, notice the $n \in \textit{reach}$ assumption, which
1384 strengthens the induction hypothesis but is not immediately usable in the proof.
1385 If we remove it and invoke Nitpick, this time we get a counterexample:
1388 \textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
1389 \textbf{nitpick} \\[2\smallskipamount]
1390 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1391 \hbox{}\qquad Skolem constant: \nopagebreak \\
1392 \hbox{}\qquad\qquad $n = 0$
1395 Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
1396 to strength the lemma:
1399 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
1402 Unfortunately, the proof by induction still gets stuck, except that Nitpick now
1403 finds the counterexample $n = 2$. We generalize the lemma further to
1406 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
1409 and this time \textit{arith} can finish off the subgoals.
1411 A similar technique can be employed for structural induction. The
1412 following mini formalization of full binary trees will serve as illustration:
1415 \textbf{datatype} $\kern1pt'a$~\textit{bin\_tree} = $\textit{Leaf}~{\kern1pt'a}$ $\mid$ $\textit{Branch}$ ``\kern1pt$'a$ \textit{bin\_tree}'' ``\kern1pt$'a$ \textit{bin\_tree}'' \\[2\smallskipamount]
1416 \textbf{primrec}~\textit{labels}~\textbf{where} \\
1417 ``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
1418 ``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
1419 \textbf{primrec}~\textit{swap}~\textbf{where} \\
1420 ``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
1421 \phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
1422 ``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
1425 The \textit{labels} function returns the set of labels occurring on leaves of a
1426 tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
1427 labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
1428 obtained by swapping $a$ and $b$:
1431 \textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
1434 Nitpick can't find any counterexample, so we proceed with induction
1435 (this time favoring a more structured style):
1438 \textbf{proof}~(\textit{induct}~$t$) \\
1439 \hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
1441 \hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
1444 Nitpick can't find any counterexample at this point either, but it makes the
1445 following suggestion:
1449 Hint: To check that the induction hypothesis is general enough, try this command:
1450 \textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
1453 If we follow the hint, we get a ``nonstandard'' counterexample for the step:
1456 \slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
1457 \hbox{}\qquad Free variables: \nopagebreak \\
1458 \hbox{}\qquad\qquad $a = a_1$ \\
1459 \hbox{}\qquad\qquad $b = a_2$ \\
1460 \hbox{}\qquad\qquad $t = \xi_1$ \\
1461 \hbox{}\qquad\qquad $u = \xi_2$ \\
1462 \hbox{}\qquad Datatype: \nopagebreak \\
1463 \hbox{}\qquad\qquad $\alpha~\textit{btree} = \{\xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \textit{Branch}~\xi_1~\xi_2\}$ \\
1464 \hbox{}\qquad {\slshape Constants:} \nopagebreak \\
1465 \hbox{}\qquad\qquad $\textit{labels} = \undef
1466 (\!\begin{aligned}[t]%
1467 & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
1468 & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
1469 \hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \undef
1470 (\!\begin{aligned}[t]%
1471 & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
1472 & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount]
1473 The existence of a nonstandard model suggests that the induction hypothesis is not general enough or may even
1474 be wrong. See the Nitpick manual's ``Inductive Properties'' section for details (``\textit{isabelle doc nitpick}'').
1477 Reading the Nitpick manual is a most excellent idea.
1478 But what's going on? The \textit{non\_std} option told the tool to look for
1479 nonstandard models of binary trees, which means that new ``nonstandard'' trees
1480 $\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees
1481 generated by the \textit{Leaf} and \textit{Branch} constructors.%
1482 \footnote{Notice the similarity between allowing nonstandard trees here and
1483 allowing unreachable states in the preceding example (by removing the ``$n \in
1484 \textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
1485 set of objects over which the induction is performed while doing the step
1486 in order to test the induction hypothesis's strength.}
1487 Unlike standard trees, these new trees contain cycles. We will see later that
1488 every property of acyclic trees that can be proved without using induction also
1489 holds for cyclic trees. Hence,
1492 \textsl{If the induction
1493 hypothesis is strong enough, the induction step will hold even for nonstandard
1494 objects, and Nitpick won't find any nonstandard counterexample.}
1497 But here the tool find some nonstandard trees $t = \xi_1$
1498 and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in
1499 \textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$.
1500 Because neither tree contains both $a$ and $b$, the induction hypothesis tells
1501 us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
1502 and as a result we know nothing about the labels of the tree
1503 $\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
1504 $\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
1505 labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
1506 \textit{labels}$ $(\textit{swap}~u~a~b)$.
1508 The solution is to ensure that we always know what the labels of the subtrees
1509 are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
1510 $t$ in the statement of the lemma:
1513 \textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
1514 \phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
1515 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
1516 \phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
1517 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
1520 This time, Nitpick won't find any nonstandard counterexample, and we can perform
1521 the induction step using \textit{auto}.
1523 \section{Case Studies}
1524 \label{case-studies}
1526 As a didactic device, the previous section focused mostly on toy formulas whose
1527 validity can easily be assessed just by looking at the formula. We will now
1528 review two somewhat more realistic case studies that are within Nitpick's
1529 reach:\ a context-free grammar modeled by mutually inductive sets and a
1530 functional implementation of AA trees. The results presented in this
1531 section were produced with the following settings:
1534 \textbf{nitpick\_params} [\textit{max\_potential}~= 0]
1537 \subsection{A Context-Free Grammar}
1538 \label{a-context-free-grammar}
1540 Our first case study is taken from section 7.4 in the Isabelle tutorial
1541 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1542 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1545 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1546 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1547 $A$ & $::=$ & $aS \mid bAA$ \\
1548 $B$ & $::=$ & $bS \mid aBB$
1552 The intuition behind the grammar is that $A$ generates all string with one more
1553 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1555 The alphabet consists exclusively of $a$'s and $b$'s:
1558 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1561 Strings over the alphabet are represented by \textit{alphabet list}s.
1562 Nonterminals in the grammar become sets of strings. The production rules
1563 presented above can be expressed as a mutually inductive definition:
1566 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1567 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1568 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1569 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1570 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1571 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1572 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1575 The conversion of the grammar into the inductive definition was done manually by
1576 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1579 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1580 d'\^etre}. A good approach is to state desirable properties of the specification
1581 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1582 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1583 counterexamples will point to bugs in the specification. For our grammar
1584 example, we will proceed in two steps, separating the soundness and the
1585 completeness of the set $S$. First, soundness:
1588 \textbf{theorem}~\textit{S\_sound\/}: \\
1589 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1590 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1591 \textbf{nitpick} \\[2\smallskipamount]
1592 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1593 \hbox{}\qquad Free variable: \nopagebreak \\
1594 \hbox{}\qquad\qquad $w = [b]$
1597 It would seem that $[b] \in S$. How could this be? An inspection of the
1598 introduction rules reveals that the only rule with a right-hand side of the form
1599 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1603 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1606 On closer inspection, we can see that this rule is wrong. To match the
1607 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1611 \textbf{nitpick} \\[2\smallskipamount]
1612 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1613 \hbox{}\qquad Free variable: \nopagebreak \\
1614 \hbox{}\qquad\qquad $w = [a, a, b]$
1617 Some detective work is necessary to find out what went wrong here. To get $[a,
1618 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1622 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1625 Now, this formula must be wrong: The same assumption occurs twice, and the
1626 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1627 the assumptions should have been a $w$.
1629 With the correction made, we don't get any counterexample from Nitpick. Let's
1630 move on and check completeness:
1633 \textbf{theorem}~\textit{S\_complete}: \\
1634 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1635 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1636 \longrightarrow w \in S$'' \\
1637 \textbf{nitpick} \\[2\smallskipamount]
1638 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1639 \hbox{}\qquad Free variable: \nopagebreak \\
1640 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1643 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1644 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1645 the introduction rules we have are probably correct. Perhaps we simply lack an
1646 introduction rule. Comparing the grammar with the inductive definition, our
1647 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1648 without which the grammar cannot generate two or more $b$'s in a row. So we add
1652 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1655 With this last change, we don't get any counterexamples from Nitpick for either
1656 soundness or completeness. We can even generalize our result to cover $A$ and
1660 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1661 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1662 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1663 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1664 \textbf{nitpick} \\[2\smallskipamount]
1665 \slshape Nitpick found no counterexample.
1668 \subsection{AA Trees}
1671 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1672 similar performance to red-black trees, but with a simpler implementation
1673 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1674 a total order $<$. We start by defining the datatype and some basic extractor
1678 \textbf{datatype} $'a$~\textit{aa\_tree} = \\
1679 \hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}'' \\[2\smallskipamount]
1680 \textbf{primrec} \textit{data} \textbf{where} \\
1681 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1682 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1683 \textbf{primrec} \textit{dataset} \textbf{where} \\
1684 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1685 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1686 \textbf{primrec} \textit{level} \textbf{where} \\
1687 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1688 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1689 \textbf{primrec} \textit{left} \textbf{where} \\
1690 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1691 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1692 \textbf{primrec} \textit{right} \textbf{where} \\
1693 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1694 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1697 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1698 as follows \cite{wikipedia-2009-aa-trees}:
1700 \kern.2\parskip %% TYPESETTING
1703 Each node has a level field, and the following invariants must remain true for
1704 the tree to be valid:
1708 \kern-.4\parskip %% TYPESETTING
1713 \item[1.] The level of a leaf node is one.
1714 \item[2.] The level of a left child is strictly less than that of its parent.
1715 \item[3.] The level of a right child is less than or equal to that of its parent.
1716 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1717 \item[5.] Every node of level greater than one must have two children.
1722 \kern.4\parskip %% TYPESETTING
1724 The \textit{wf} predicate formalizes this description:
1727 \textbf{primrec} \textit{wf} \textbf{where} \\
1728 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1729 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1730 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1731 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1732 \phantom{``$($}$\textrm{else}$ \\
1733 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1734 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1735 \mathrel{\land} \textit{level}~u \le k$ \\
1736 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1739 Rebalancing the tree upon insertion and removal of elements is performed by two
1740 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1743 \textbf{primrec} \textit{skew} \textbf{where} \\
1744 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1745 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1746 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1747 \textit{level}~t~\textrm{then}$ \\
1748 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1749 (\textit{right}~t)~u)$ \\
1750 \phantom{``(}$\textrm{else}$ \\
1751 \phantom{``(\quad}$N~x~k~t~u)$''
1755 \textbf{primrec} \textit{split} \textbf{where} \\
1756 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1757 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1758 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1759 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1760 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1761 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1762 \phantom{``(}$\textrm{else}$ \\
1763 \phantom{``(\quad}$N~x~k~t~u)$''
1766 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1767 of elements stored in the tree:
1770 \textbf{theorem}~\textit{dataset\_skew\_split\/}:\\
1771 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1772 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1773 \textbf{nitpick} \\[2\smallskipamount]
1774 {\slshape Nitpick ran out of time after checking 9 of 10 scopes.}
1777 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1778 should not alter the tree:
1781 \textbf{theorem}~\textit{wf\_skew\_split\/}:\\
1782 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1783 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1784 \textbf{nitpick} \\[2\smallskipamount]
1785 {\slshape Nitpick found no counterexample.}
1788 Insertion is implemented recursively. It preserves the sort order:
1791 \textbf{primrec}~\textit{insort} \textbf{where} \\
1792 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1793 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1794 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1795 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1798 Notice that we deliberately commented out the application of \textit{skew} and
1799 \textit{split}. Let's see if this causes any problems:
1802 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1803 \textbf{nitpick} \\[2\smallskipamount]
1804 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1805 \hbox{}\qquad Free variables: \nopagebreak \\
1806 \hbox{}\qquad\qquad $t = N~a_1~1~\Lambda~\Lambda$ \\
1807 \hbox{}\qquad\qquad $x = a_2$
1810 It's hard to see why this is a counterexample. To improve readability, we will
1811 restrict the theorem to \textit{nat}, so that we don't need to look up the value
1812 of the $\textit{op}~{<}$ constant to find out which element is smaller than the
1813 other. In addition, we will tell Nitpick to display the value of
1814 $\textit{insort}~t~x$ using the \textit{eval} option. This gives
1817 \textbf{theorem} \textit{wf\_insort\_nat\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1818 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1819 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1820 \hbox{}\qquad Free variables: \nopagebreak \\
1821 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1822 \hbox{}\qquad\qquad $x = 0$ \\
1823 \hbox{}\qquad Evaluated term: \\
1824 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1827 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1828 where both have a level of 1. This violates the second AA tree invariant, which
1829 states that a left child's level must be less than its parent's. This shouldn't
1830 come as a surprise, considering that we commented out the tree rebalancing code.
1831 Reintroducing the code seems to solve the problem:
1834 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1835 \textbf{nitpick} \\[2\smallskipamount]
1836 {\slshape Nitpick ran out of time after checking 8 of 10 scopes.}
1839 Insertion should transform the set of elements represented by the tree in the
1843 \textbf{theorem} \textit{dataset\_insort\/}:\kern.4em
1844 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1845 \textbf{nitpick} \\[2\smallskipamount]
1846 {\slshape Nitpick ran out of time after checking 7 of 10 scopes.}
1849 We could continue like this and sketch a complete theory of AA trees. Once the
1850 definitions and main theorems are in place and have been thoroughly tested using
1851 Nitpick, we could start working on the proofs. Developing theories this way
1852 usually saves time, because faulty theorems and definitions are discovered much
1853 earlier in the process.
1855 \section{Option Reference}
1856 \label{option-reference}
1858 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1859 \def\qty#1{$\left<\textit{#1}\right>$}
1860 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1861 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1862 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1863 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1864 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1865 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1866 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1867 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1868 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1870 Nitpick's behavior can be influenced by various options, which can be specified
1871 in brackets after the \textbf{nitpick} command. Default values can be set
1872 using \textbf{nitpick\_\allowbreak params}. For example:
1875 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1878 The options are categorized as follows:\ mode of operation
1879 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1880 format (\S\ref{output-format}), automatic counterexample checks
1881 (\S\ref{authentication}), optimizations
1882 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1884 You can instruct Nitpick to run automatically on newly entered theorems by
1885 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1886 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
1887 and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1888 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1889 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1890 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1891 \textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
1892 Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
1895 The number of options can be overwhelming at first glance. Do not let that worry
1896 you: Nitpick's defaults have been chosen so that it almost always does the right
1897 thing, and the most important options have been covered in context in
1898 \S\ref{first-steps}.
1900 The descriptions below refer to the following syntactic quantities:
1903 \item[$\bullet$] \qtybf{string}: A string.
1904 \item[$\bullet$] \qtybf{string\_list\/}: A space-separated list of strings
1905 (e.g., ``\textit{ichi ni san}'').
1906 \item[$\bullet$] \qtybf{bool\/}: \textit{true} or \textit{false}.
1907 \item[$\bullet$] \qtybf{bool\_or\_smart\/}: \textit{true}, \textit{false}, or \textit{smart}.
1908 \item[$\bullet$] \qtybf{int\/}: An integer. Negative integers are prefixed with a hyphen.
1909 \item[$\bullet$] \qtybf{int\_or\_smart\/}: An integer or \textit{smart}.
1910 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1911 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1913 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1914 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1915 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1916 \item[$\bullet$] \qtybf{const\/}: The name of a HOL constant.
1917 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1918 \item[$\bullet$] \qtybf{term\_list\/}: A space-separated list of HOL terms (e.g.,
1919 ``$f~x$''~``$g~y$'').
1920 \item[$\bullet$] \qtybf{type}: A HOL type.
1923 Default values are indicated in square brackets. Boolean options have a negated
1924 counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
1925 Boolean options, ``= \textit{true}'' may be omitted.
1927 \subsection{Mode of Operation}
1928 \label{mode-of-operation}
1931 \optrue{blocking}{non\_blocking}
1932 Specifies whether the \textbf{nitpick} command should operate synchronously.
1933 The asynchronous (non-blocking) mode lets the user start proving the putative
1934 theorem while Nitpick looks for a counterexample, but it can also be more
1935 confusing. For technical reasons, automatic runs currently always block.
1937 \optrue{falsify}{satisfy}
1938 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1939 satisfying examples (models). This manual assumes throughout that
1940 \textit{falsify} is enabled.
1942 \opsmart{user\_axioms}{no\_user\_axioms}
1943 Specifies whether the user-defined axioms (specified using
1944 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1945 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1946 the constants that occur in the formula to falsify. The option is implicitly set
1947 to \textit{true} for automatic runs.
1949 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1950 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1951 these conditions are tagged as ``quasi genuine.'' The \textit{debug}
1952 (\S\ref{output-format}) option can be used to find out which axioms were
1956 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1957 (\S\ref{output-format}).}
1959 \optrue{assms}{no\_assms}
1960 Specifies whether the relevant assumptions in structured proofs should be
1961 considered. The option is implicitly enabled for automatic runs.
1964 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1966 \opfalse{overlord}{no\_overlord}
1967 Specifies whether Nitpick should put its temporary files in
1968 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1969 debugging Nitpick but also unsafe if several instances of the tool are run
1970 simultaneously. The files are identified by the extensions
1971 \texttt{.kki}, \texttt{.cnf}, \texttt{.out}, and
1972 \texttt{.err}; you may safely remove them after Nitpick has run.
1975 {\small See also \textit{debug} (\S\ref{output-format}).}
1978 \subsection{Scope of Search}
1979 \label{scope-of-search}
1982 \oparg{card}{type}{int\_seq}
1983 Specifies the sequence of cardinalities to use for a given type.
1984 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1985 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1988 {\small See also \textit{box} (\S\ref{scope-of-search}) and \textit{mono}
1989 (\S\ref{scope-of-search}).}
1991 \opdefault{card}{int\_seq}{$\mathbf{1}$--$\mathbf{10}$}
1992 Specifies the default sequence of cardinalities to use. This can be overridden
1993 on a per-type basis using the \textit{card}~\qty{type} option described above.
1995 \oparg{max}{const}{int\_seq}
1996 Specifies the sequence of maximum multiplicities to use for a given
1997 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1998 number of distinct values that it can construct. Nonsensical values (e.g.,
1999 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
2000 datatypes equipped with several constructors.
2002 \opnodefault{max}{int\_seq}
2003 Specifies the default sequence of maximum multiplicities to use for
2004 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
2005 basis using the \textit{max}~\qty{const} option described above.
2007 \opsmart{binary\_ints}{unary\_ints}
2008 Specifies whether natural numbers and integers should be encoded using a unary
2009 or binary notation. In unary mode, the cardinality fully specifies the subset
2010 used to approximate the type. For example:
2012 $$\hbox{\begin{tabular}{@{}rll@{}}%
2013 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
2014 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
2015 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
2020 $$\hbox{\begin{tabular}{@{}rll@{}}%
2021 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
2022 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
2025 In binary mode, the cardinality specifies the number of distinct values that can
2026 be constructed. Each of these value is represented by a bit pattern whose length
2027 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
2028 Nitpick attempts to choose the more appropriate encoding by inspecting the
2029 formula at hand, preferring the binary notation for problems involving
2030 multiplicative operators or large constants.
2032 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
2033 problems that refer to the types \textit{rat} or \textit{real} or the constants
2034 \textit{Suc}, \textit{gcd}, or \textit{lcm}.
2036 {\small See also \textit{bits} (\S\ref{scope-of-search}) and
2037 \textit{show\_datatypes} (\S\ref{output-format}).}
2039 \opdefault{bits}{int\_seq}{$\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{8},\mathbf{10},\mathbf{12},\mathbf{14},\mathbf{16}$}
2040 Specifies the number of bits to use to represent natural numbers and integers in
2041 binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
2043 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
2045 \opargboolorsmart{wf}{const}{non\_wf}
2046 Specifies whether the specified (co)in\-duc\-tively defined predicate is
2047 well-founded. The option can take the following values:
2050 \item[$\bullet$] \textbf{\textit{true}:} Tentatively treat the (co)in\-duc\-tive
2051 predicate as if it were well-founded. Since this is generally not sound when the
2052 predicate is not well-founded, the counterexamples are tagged as ``quasi
2055 \item[$\bullet$] \textbf{\textit{false}:} Treat the (co)in\-duc\-tive predicate
2056 as if it were not well-founded. The predicate is then unrolled as prescribed by
2057 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
2060 \item[$\bullet$] \textbf{\textit{smart}:} Try to prove that the inductive
2061 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
2062 \textit{size\_change} tactics. If this succeeds (or the predicate occurs with an
2063 appropriate polarity in the formula to falsify), use an efficient fixed-point
2064 equation as specification of the predicate; otherwise, unroll the predicates
2065 according to the \textit{iter}~\qty{const} and \textit{iter} options.
2069 {\small See also \textit{iter} (\S\ref{scope-of-search}),
2070 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
2071 (\S\ref{timeouts}).}
2073 \opsmart{wf}{non\_wf}
2074 Specifies the default wellfoundedness setting to use. This can be overridden on
2075 a per-predicate basis using the \textit{wf}~\qty{const} option above.
2077 \oparg{iter}{const}{int\_seq}
2078 Specifies the sequence of iteration counts to use when unrolling a given
2079 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
2080 predicates that occur negatively and coinductive predicates that occur
2081 positively in the formula to falsify and that cannot be proved to be
2082 well-founded, but this behavior is influenced by the \textit{wf} option. The
2083 iteration counts are automatically bounded by the cardinality of the predicate's
2086 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
2087 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
2089 \opdefault{iter}{int\_seq}{$\mathbf{0{,}1{,}2{,}4{,}8{,}12{,}16{,}20{,}24{,}28}$}
2090 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
2091 predicates. This can be overridden on a per-predicate basis using the
2092 \textit{iter} \qty{const} option above.
2094 \opdefault{bisim\_depth}{int\_seq}{$\mathbf{9}$}
2095 Specifies the sequence of iteration counts to use when unrolling the
2096 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
2097 of $-1$ means that no predicate is generated, in which case Nitpick performs an
2098 after-the-fact check to see if the known coinductive datatype values are
2099 bidissimilar. If two values are found to be bisimilar, the counterexample is
2100 tagged as ``quasi genuine.'' The iteration counts are automatically bounded by
2101 the sum of the cardinalities of the coinductive datatypes occurring in the
2104 \opargboolorsmart{box}{type}{dont\_box}
2105 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
2106 product type in an isomorphic datatype internally. Boxing is an effective mean
2107 to reduce the search space and speed up Nitpick, because the isomorphic datatype
2108 is approximated by a subset of the possible function or pair values.
2109 Like other drastic optimizations, it can also prevent the discovery of
2110 counterexamples. The option can take the following values:
2113 \item[$\bullet$] \textbf{\textit{true}:} Box the specified type whenever
2115 \item[$\bullet$] \textbf{\textit{false}:} Never box the type.
2116 \item[$\bullet$] \textbf{\textit{smart}:} Box the type only in contexts where it
2117 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
2118 higher-order functions are good candidates for boxing.
2122 {\small See also \textit{finitize} (\S\ref{scope-of-search}), \textit{verbose}
2123 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}).}
2125 \opsmart{box}{dont\_box}
2126 Specifies the default boxing setting to use. This can be overridden on a
2127 per-type basis using the \textit{box}~\qty{type} option described above.
2129 \opargboolorsmart{finitize}{type}{dont\_finitize}
2130 Specifies whether Nitpick should attempt to finitize a given type, which can be
2131 a function type or an infinite ``shallow datatype'' (an infinite datatype whose
2132 constructors don't appear in the problem).
2134 For function types, Nitpick performs a monotonicity analysis to detect functions
2135 that are constant at all but finitely many points (e.g., finite sets) and treats
2136 such occurrences specially, thereby increasing the precision. The option can
2137 then take the following values:
2140 \item[$\bullet$] \textbf{\textit{false}:} Don't attempt to finitize the type.
2141 \item[$\bullet$] \textbf{\textit{true}} or \textbf{\textit{smart}:} Finitize the
2142 type wherever possible.
2145 The semantics of the option is somewhat different for infinite shallow
2149 \item[$\bullet$] \textbf{\textit{true}:} Finitize the datatype. Since this is
2150 unsound, counterexamples generated under these conditions are tagged as ``quasi
2152 \item[$\bullet$] \textbf{\textit{false}:} Don't attempt to finitize the datatype.
2153 \item[$\bullet$] \textbf{\textit{smart}:} Perform a monotonicity analysis to
2154 detect whether the datatype can be safely finitized before finitizing it.
2157 Like other drastic optimizations, finitization can sometimes prevent the
2158 discovery of counterexamples.
2161 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{mono}
2162 (\S\ref{scope-of-search}), \textit{verbose} (\S\ref{output-format}), and
2163 \textit{debug} (\S\ref{output-format}).}
2165 \opsmart{finitize}{dont\_finitize}
2166 Specifies the default finitization setting to use. This can be overridden on a
2167 per-type basis using the \textit{finitize}~\qty{type} option described above.
2169 \opargboolorsmart{mono}{type}{non\_mono}
2170 Specifies whether the given type should be considered monotonic when enumerating
2171 scopes and finitizing types. If the option is set to \textit{smart}, Nitpick
2172 performs a monotonicity check on the type. Setting this option to \textit{true}
2173 can reduce the number of scopes tried, but it can also diminish the chance of
2174 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
2177 {\small See also \textit{card} (\S\ref{scope-of-search}),
2178 \textit{finitize} (\S\ref{scope-of-search}),
2179 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
2180 (\S\ref{output-format}).}
2182 \opsmart{mono}{non\_mono}
2183 Specifies the default monotonicity setting to use. This can be overridden on a
2184 per-type basis using the \textit{mono}~\qty{type} option described above.
2186 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
2187 Specifies whether type variables with the same sort constraints should be
2188 merged. Setting this option to \textit{true} can reduce the number of scopes
2189 tried and the size of the generated Kodkod formulas, but it also diminishes the
2190 theoretical chance of finding a counterexample.
2192 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
2194 \opargbool{std}{type}{non\_std}
2195 Specifies whether the given (recursive) datatype should be given standard
2196 models. Nonstandard models are unsound but can help debug structural induction
2197 proofs, as explained in \S\ref{inductive-properties}.
2199 \optrue{std}{non\_std}
2200 Specifies the default standardness to use. This can be overridden on a per-type
2201 basis using the \textit{std}~\qty{type} option described above.
2204 \subsection{Output Format}
2205 \label{output-format}
2208 \opfalse{verbose}{quiet}
2209 Specifies whether the \textbf{nitpick} command should explain what it does. This
2210 option is useful to determine which scopes are tried or which SAT solver is
2211 used. This option is implicitly disabled for automatic runs.
2213 \opfalse{debug}{no\_debug}
2214 Specifies whether Nitpick should display additional debugging information beyond
2215 what \textit{verbose} already displays. Enabling \textit{debug} also enables
2216 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
2217 option is implicitly disabled for automatic runs.
2220 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
2221 \textit{batch\_size} (\S\ref{optimizations}).}
2223 \opfalse{show\_datatypes}{hide\_datatypes}
2224 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
2225 be displayed as part of counterexamples. Such subsets are sometimes helpful when
2226 investigating whether a potential counterexample is genuine or spurious, but
2227 their potential for clutter is real.
2229 \opfalse{show\_consts}{hide\_consts}
2230 Specifies whether the values of constants occurring in the formula (including
2231 its axioms) should be displayed along with any counterexample. These values are
2232 sometimes helpful when investigating why a counterexample is
2233 genuine, but they can clutter the output.
2235 \opnodefault{show\_all}{bool}
2236 Abbreviation for \textit{show\_datatypes} and \textit{show\_consts}.
2238 \opdefault{max\_potential}{int}{$\mathbf{1}$}
2239 Specifies the maximum number of potential counterexamples to display. Setting
2240 this option to 0 speeds up the search for a genuine counterexample. This option
2241 is implicitly set to 0 for automatic runs. If you set this option to a value
2242 greater than 1, you will need an incremental SAT solver, such as
2243 \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that many of
2244 the counterexamples may be identical.
2247 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
2248 \textit{sat\_solver} (\S\ref{optimizations}).}
2250 \opdefault{max\_genuine}{int}{$\mathbf{1}$}
2251 Specifies the maximum number of genuine counterexamples to display. If you set
2252 this option to a value greater than 1, you will need an incremental SAT solver,
2253 such as \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that
2254 many of the counterexamples may be identical.
2257 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
2258 \textit{sat\_solver} (\S\ref{optimizations}).}
2260 \opnodefault{eval}{term\_list}
2261 Specifies the list of terms whose values should be displayed along with
2262 counterexamples. This option suffers from an ``observer effect'': Nitpick might
2263 find different counterexamples for different values of this option.
2265 \oparg{atoms}{type}{string\_list}
2266 Specifies the names to use to refer to the atoms of the given type. By default,
2267 Nitpick generates names of the form $a_1, \ldots, a_n$, where $a$ is the first
2268 letter of the type's name.
2270 \opnodefault{atoms}{string\_list}
2271 Specifies the default names to use to refer to atoms of any type. For example,
2272 to call the three atoms of type ${'}a$ \textit{ichi}, \textit{ni}, and
2273 \textit{san} instead of $a_1$, $a_2$, $a_3$, specify the option
2274 ``\textit{atoms}~${'}a$ = \textit{ichi~ni~san}''. The default names can be
2275 overridden on a per-type basis using the \textit{atoms}~\qty{type} option
2278 \oparg{format}{term}{int\_seq}
2279 Specifies how to uncurry the value displayed for a variable or constant.
2280 Uncurrying sometimes increases the readability of the output for high-arity
2281 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
2282 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
2283 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
2284 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
2285 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
2286 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
2287 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
2288 arguments that are not accounted for are left alone, as if the specification had
2289 been $1,\ldots,1,n_1,\ldots,n_k$.
2291 \opdefault{format}{int\_seq}{$\mathbf{1}$}
2292 Specifies the default format to use. Irrespective of the default format, the
2293 extra arguments to a Skolem constant corresponding to the outer bound variables
2294 are kept separated from the remaining arguments, the \textbf{for} arguments of
2295 an inductive definitions are kept separated from the remaining arguments, and
2296 the iteration counter of an unrolled inductive definition is shown alone. The
2297 default format can be overridden on a per-variable or per-constant basis using
2298 the \textit{format}~\qty{term} option described above.
2301 \subsection{Authentication}
2302 \label{authentication}
2305 \opfalse{check\_potential}{trust\_potential}
2306 Specifies whether potential counterexamples should be given to Isabelle's
2307 \textit{auto} tactic to assess their validity. If a potential counterexample is
2308 shown to be genuine, Nitpick displays a message to this effect and terminates.
2311 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
2313 \opfalse{check\_genuine}{trust\_genuine}
2314 Specifies whether genuine and quasi genuine counterexamples should be given to
2315 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
2316 counterexample is shown to be spurious, the user is kindly asked to send a bug
2317 report to the author at
2318 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
2321 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
2323 \opnodefault{expect}{string}
2324 Specifies the expected outcome, which must be one of the following:
2327 \item[$\bullet$] \textbf{\textit{genuine}:} Nitpick found a genuine counterexample.
2328 \item[$\bullet$] \textbf{\textit{quasi\_genuine}:} Nitpick found a ``quasi
2329 genuine'' counterexample (i.e., a counterexample that is genuine unless
2330 it contradicts a missing axiom or a dangerous option was used inappropriately).
2331 \item[$\bullet$] \textbf{\textit{potential}:} Nitpick found a potential counterexample.
2332 \item[$\bullet$] \textbf{\textit{none}:} Nitpick found no counterexample.
2333 \item[$\bullet$] \textbf{\textit{unknown}:} Nitpick encountered some problem (e.g.,
2334 Kodkod ran out of memory).
2337 Nitpick emits an error if the actual outcome differs from the expected outcome.
2338 This option is useful for regression testing.
2341 \subsection{Optimizations}
2342 \label{optimizations}
2344 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
2349 \opdefault{sat\_solver}{string}{smart}
2350 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2351 to be faster than their Java counterparts, but they can be more difficult to
2352 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2353 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2354 you will need an incremental SAT solver, such as \textit{MiniSat\_JNI}
2355 (recommended) or \textit{SAT4J}.
2357 The supported solvers are listed below:
2361 \item[$\bullet$] \textbf{\textit{MiniSat}:} MiniSat is an efficient solver
2362 written in \cpp{}. To use MiniSat, set the environment variable
2363 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2365 \footnote{Important note for Cygwin users: The path must be specified using
2366 native Windows syntax. Make sure to escape backslashes properly.%
2367 \label{cygwin-paths}}
2368 The \cpp{} sources and executables for MiniSat are available at
2369 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2370 and 2.0 beta (2007-07-21).
2372 \item[$\bullet$] \textbf{\textit{MiniSat\_JNI}:} The JNI (Java Native Interface)
2373 version of MiniSat is bundled with Kodkodi and is precompiled for the major
2374 platforms. It is also available from \texttt{native\-solver.\allowbreak tgz},
2375 which you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2376 version of MiniSat, the JNI version can be used incrementally.
2378 \item[$\bullet$] \textbf{\textit{CryptoMiniSat}:} CryptoMiniSat is the winner of
2379 the 2010 SAT Race. To use CryptoMiniSat, set the environment variable
2380 \texttt{CRYPTO\-MINISAT\_}\discretionary{}{}{}\texttt{HOME} to the directory that contains the \texttt{crypto\-minisat}
2382 \footref{cygwin-paths}
2383 The \cpp{} sources and executables for Crypto\-Mini\-Sat are available at
2384 \url{http://planete.inrialpes.fr/~soos/}\allowbreak\url{CryptoMiniSat2/index.php}.
2385 Nitpick has been tested with version 2.51.
2387 \item[$\bullet$] \textbf{\textit{PicoSAT}:} PicoSAT is an efficient solver
2388 written in C. You can install a standard version of
2389 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2390 that contains the \texttt{picosat} executable.%
2391 \footref{cygwin-paths}
2392 The C sources for PicoSAT are
2393 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2394 Nitpick has been tested with version 913.
2396 \item[$\bullet$] \textbf{\textit{zChaff}:} zChaff is an efficient solver written
2397 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2398 the directory that contains the \texttt{zchaff} executable.%
2399 \footref{cygwin-paths}
2400 The \cpp{} sources and executables for zChaff are available at
2401 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2402 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2404 \item[$\bullet$] \textbf{\textit{zChaff\_JNI}:} The JNI version of zChaff is
2405 bundled with Kodkodi and is precompiled for the major
2406 platforms. It is also available from \texttt{native\-solver.\allowbreak tgz},
2407 which you will find on Kodkod's web site \cite{kodkod-2009}.
2409 \item[$\bullet$] \textbf{\textit{RSat}:} RSat is an efficient solver written in
2410 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2411 directory that contains the \texttt{rsat} executable.%
2412 \footref{cygwin-paths}
2413 The \cpp{} sources for RSat are available at
2414 \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been tested with version
2417 \item[$\bullet$] \textbf{\textit{BerkMin}:} BerkMin561 is an efficient solver
2418 written in C. To use BerkMin, set the environment variable
2419 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2420 executable.\footref{cygwin-paths}
2421 The BerkMin executables are available at
2422 \url{http://eigold.tripod.com/BerkMin.html}.
2424 \item[$\bullet$] \textbf{\textit{BerkMin\_Alloy}:} Variant of BerkMin that is
2425 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2426 version of BerkMin, set the environment variable
2427 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2429 \footref{cygwin-paths}
2431 \item[$\bullet$] \textbf{\textit{Jerusat}:} Jerusat 1.3 is an efficient solver
2432 written in C. To use Jerusat, set the environment variable
2433 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2435 \footref{cygwin-paths}
2436 The C sources for Jerusat are available at
2437 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2439 \item[$\bullet$] \textbf{\textit{SAT4J}:} SAT4J is a reasonably efficient solver
2440 written in Java that can be used incrementally. It is bundled with Kodkodi and
2441 requires no further installation or configuration steps. Do not attempt to
2442 install the official SAT4J packages, because their API is incompatible with
2445 \item[$\bullet$] \textbf{\textit{SAT4J\_Light}:} Variant of SAT4J that is
2446 optimized for small problems. It can also be used incrementally.
2448 \item[$\bullet$] \textbf{\textit{smart}:} If \textit{sat\_solver} is set to
2449 \textit{smart}, Nitpick selects the first solver among the above that is
2450 recognized by Isabelle. If \textit{verbose} (\S\ref{output-format}) is enabled,
2451 Nitpick displays which SAT solver was chosen.
2455 \opdefault{batch\_size}{int\_or\_smart}{smart}
2456 Specifies the maximum number of Kodkod problems that should be lumped together
2457 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2458 together ensures that Kodkodi is launched less often, but it makes the verbose
2459 output less readable and is sometimes detrimental to performance. If
2460 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2461 \textit{debug} (\S\ref{output-format}) is set and 50 otherwise.
2463 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2464 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2465 be rewritten to use (automatically generated) discriminators and destructors.
2466 This optimization can drastically reduce the size of the Boolean formulas given
2470 {\small See also \textit{debug} (\S\ref{output-format}).}
2472 \optrue{specialize}{dont\_specialize}
2473 Specifies whether functions invoked with static arguments should be specialized.
2474 This optimization can drastically reduce the search space, especially for
2475 higher-order functions.
2478 {\small See also \textit{debug} (\S\ref{output-format}) and
2479 \textit{show\_consts} (\S\ref{output-format}).}
2481 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2482 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2483 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2484 predicates for which each the predicate occurs in at most one assumption of each
2485 introduction rule. Using the reflexive transitive closure is in principle
2486 equivalent to setting \textit{iter} to the cardinality of the predicate's
2487 domain, but it is usually more efficient.
2489 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2490 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2492 \optrue{fast\_descrs}{full\_descrs}
2493 Specifies whether Nitpick should optimize the definite and indefinite
2494 description operators (THE and SOME). The optimized versions usually help
2495 Nitpick generate more counterexamples or at least find them faster, but only the
2496 unoptimized versions are complete when all types occurring in the formula are
2499 {\small See also \textit{debug} (\S\ref{output-format}).}
2501 \opnodefault{whack}{term\_list}
2502 Specifies a list of atomic terms (usually constants, but also free and schematic
2503 variables) that should be taken as being $\unk$ (unknown). This can be useful to
2504 reduce the size of the Kodkod problem if you can guess in advance that a
2505 constant might not be needed to find a countermodel.
2507 {\small See also \textit{debug} (\S\ref{output-format}).}
2509 \optrue{peephole\_optim}{no\_peephole\_optim}
2510 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2511 peephole optimizer. These optimizations can make a significant difference.
2512 Unless you are tracking down a bug in Nitpick or distrust the peephole
2513 optimizer, you should leave this option enabled.
2515 \opdefault{datatype\_sym\_break}{int}{5}
2516 Specifies an upper bound on the number of datatypes for which Nitpick generates
2517 symmetry breaking predicates. Symmetry breaking can speed up the SAT solver
2518 considerably, especially for unsatisfiable problems, but too much of it can slow
2521 \opdefault{kodkod\_sym\_break}{int}{15}
2522 Specifies an upper bound on the number of relations for which Kodkod generates
2523 symmetry breaking predicates. Symmetry breaking can speed up the SAT solver
2524 considerably, especially for unsatisfiable problems, but too much of it can slow
2527 \opdefault{max\_threads}{int}{0}
2528 Specifies the maximum number of threads to use in Kodkod. If this option is set
2529 to 0, Kodkod will compute an appropriate value based on the number of processor
2533 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2534 \textit{timeout} (\S\ref{timeouts}).}
2537 \subsection{Timeouts}
2541 \opdefault{timeout}{time}{$\mathbf{30}$ s}
2542 Specifies the maximum amount of time that the \textbf{nitpick} command should
2543 spend looking for a counterexample. Nitpick tries to honor this constraint as
2544 well as it can but offers no guarantees. For automatic runs,
2545 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2546 a time slot whose length is specified by the ``Auto Counterexample Time
2547 Limit'' option in Proof General.
2550 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2552 \opdefault{tac\_timeout}{time}{$\mathbf{500}$\,ms}
2553 Specifies the maximum amount of time that the \textit{auto} tactic should use
2554 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2555 and \textit{size\_change} should use when checking whether a (co)in\-duc\-tive
2556 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2557 can but offers no guarantees.
2560 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2561 \textit{check\_potential} (\S\ref{authentication}),
2562 and \textit{check\_genuine} (\S\ref{authentication}).}
2565 \section{Attribute Reference}
2566 \label{attribute-reference}
2568 Nitpick needs to consider the definitions of all constants occurring in a
2569 formula in order to falsify it. For constants introduced using the
2570 \textbf{definition} command, the definition is simply the associated
2571 \textit{\_def} axiom. In contrast, instead of using the internal representation
2572 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2573 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2574 equational specification entered by the user.
2576 Behind the scenes, Isabelle's built-in packages and theories rely on the
2577 following attributes to affect Nitpick's behavior:
2580 \flushitem{\textit{nitpick\_def}}
2583 This attribute specifies an alternative definition of a constant. The
2584 alternative definition should be logically equivalent to the constant's actual
2585 axiomatic definition and should be of the form
2587 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2589 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2592 \flushitem{\textit{nitpick\_simp}}
2595 This attribute specifies the equations that constitute the specification of a
2596 constant. The \textbf{primrec}, \textbf{function}, and
2597 \textbf{nominal\_\allowbreak primrec} packages automatically attach this
2598 attribute to their \textit{simps} rules. The equations must be of the form
2600 \qquad $c~t_1~\ldots\ t_n \;\bigl[{=}\; u\bigr]$
2604 \qquad $c~t_1~\ldots\ t_n \,\equiv\, u.$
2606 \flushitem{\textit{nitpick\_psimp}}
2609 This attribute specifies the equations that constitute the partial specification
2610 of a constant. The \textbf{function} package automatically attaches this
2611 attribute to its \textit{psimps} rules. The conditional equations must be of the
2614 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \;\bigl[{=}\; u\bigr]$
2618 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,\equiv\, u$.
2620 \flushitem{\textit{nitpick\_choice\_spec}}
2623 This attribute specifies the (free-form) specification of a constant defined
2624 using the \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command.
2627 When faced with a constant, Nitpick proceeds as follows:
2630 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2631 is not empty, Nitpick uses these rules as the specification of the constant.
2633 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2634 the constant is not empty, it uses these rules as the specification of the
2637 \item[3.] Otherwise, if the constant was defined using the
2638 \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command and the
2639 \textit{nitpick\_choice\_spec} set associated with the constant is not empty, it
2640 uses these theorems as the specification of the constant.
2642 \item[4.] Otherwise, it looks up the definition of the constant. If the
2643 \textit{nitpick\_def} set associated with the constant is not empty, it uses the
2644 latest rule added to the set as the definition of the constant; otherwise it
2645 uses the actual definition axiom.
2648 \item[1.] If the definition is of the form
2650 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$
2654 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{gfp}~(\lambda f.\; t).$
2656 Nitpick assumes that the definition was made using a (co)inductive package
2657 based on the user-specified introduction rules registered in Isabelle's internal
2658 \textit{Spec\_Rules} table. The tool uses the introduction rules to ascertain
2659 whether the definition is well-founded and the definition to generate a
2660 fixed-point equation or an unrolled equation.
2662 \item[2.] If the definition is compact enough, the constant is \textsl{unfolded}
2663 wherever it appears; otherwise, it is defined equationally, as with
2664 the \textit{nitpick\_simp} attribute.
2668 As an illustration, consider the inductive definition
2671 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2672 ``\textit{odd}~1'' $\,\mid$ \\
2673 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2676 By default, Nitpick uses the \textit{lfp}-based definition in conjunction with
2677 the introduction rules. To override this, you can specify an alternative
2678 definition as follows:
2681 \textbf{lemma} $\mathit{odd\_alt\_def}$ [\textit{nitpick\_def}]:\kern.4em ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2684 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2685 = 1$. Alternatively, you can specify an equational specification of the constant:
2688 \textbf{lemma} $\mathit{odd\_simp}$ [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2691 Such tweaks should be done with great care, because Nitpick will assume that the
2692 constant is completely defined by its equational specification. For example, if
2693 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
2694 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2695 (\S\ref{output-format}) option is extremely useful to understand what is going
2696 on when experimenting with \textit{nitpick\_} attributes.
2698 Because of its internal three-valued logic, Nitpick tends to lose a
2699 lot of precision in the presence of partially specified constants. For example,
2702 \textbf{lemma} \textit{odd\_simp} [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd~x} = \lnot\, \textit{even}~x$''
2708 \textbf{lemma} \textit{odd\_psimps} [\textit{nitpick\_simp}]: \\
2709 ``$\textit{even~x} \,\Longrightarrow\, \textit{odd~x} = \textit{False\/}$'' \\
2710 ``$\lnot\, \textit{even~x} \,\Longrightarrow\, \textit{odd~x} = \textit{True\/}$''
2713 Because Nitpick sometimes unfolds definitions but never simplification rules,
2714 you can ensure that a constant is defined explicitly using the
2715 \textit{nitpick\_simp}. For example:
2718 \textbf{definition}~\textit{optimum} \textbf{where} [\textit{nitpick\_simp}]: \\
2719 ``$\textit{optimum}~t =
2720 (\forall u.\; \textit{consistent}~u \mathrel{\land} \textit{alphabet}~t = \textit{alphabet}~u$ \\
2721 \phantom{``$\textit{optimum}~t = (\forall u.\;$}${\mathrel{\land}}\; \textit{freq}~t = \textit{freq}~u \longrightarrow
2722 \textit{cost}~t \le \textit{cost}~u)$''
2725 In some rare occasions, you might want to provide an inductive or coinductive
2726 view on top of an existing constant $c$. The easiest way to achieve this is to
2727 define a new constant $c'$ (co)inductively. Then prove that $c$ equals $c'$
2728 and let Nitpick know about it:
2731 \textbf{lemma} \textit{c\_alt\_def} [\textit{nitpick\_def}]:\kern.4em ``$c \equiv c'$\kern2pt ''
2734 This ensures that Nitpick will substitute $c'$ for $c$ and use the (co)inductive
2737 \section{Standard ML Interface}
2738 \label{standard-ml-interface}
2740 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2741 and debugging. Among the most interesting functions exported by Nitpick are
2742 those that let you invoke the tool programmatically and those that let you
2743 register and unregister custom coinductive datatypes as well as term
2746 \subsection{Invocation of Nitpick}
2747 \label{invocation-of-nitpick}
2749 The \textit{Nitpick} structure offers the following functions for invoking your
2750 favorite counterexample generator:
2753 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2754 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2755 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2756 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2757 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2760 The return value is a new proof state paired with an outcome string
2761 (``genuine'', ``quasi\_genuine'', ``potential'', ``none'', or ``unknown''). The
2762 \textit{params} type is a large record that lets you set Nitpick's options. The
2763 current default options can be retrieved by calling the following function
2764 defined in the \textit{Nitpick\_Isar} structure:
2767 $\textbf{val}\,~\textit{default\_params} :\,
2768 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2771 The second argument lets you override option values before they are parsed and
2772 put into a \textit{params} record. Here is an example:
2775 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout\/}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2776 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2777 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2778 & \textit{subgoal}\end{aligned}$
2783 \subsection{Registration of Coinductive Datatypes}
2784 \label{registration-of-coinductive-datatypes}
2786 If you have defined a custom coinductive datatype, you can tell Nitpick about
2787 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2788 uses for lazy lists. The interface for registering and unregistering coinductive
2789 datatypes consists of the following pair of functions defined in the
2790 \textit{Nitpick\_HOL} structure:
2793 $\textbf{val}\,~\textit{register\_codatatype\_global\/} : {}$ \\
2794 $\hbox{}\quad\textit{typ} \rightarrow \textit{string} \rightarrow (\textit{string} \times \textit{typ})\;\textit{list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2795 $\textbf{val}\,~\textit{unregister\_codatatype\_global\/} :\,
2796 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2799 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2800 not been, you could have told Nitpick about it by adding the following line
2801 to your theory file:
2804 $\textbf{setup}~\,\{{*}$ \\
2805 $\hbox{}\quad\textit{Nitpick\_HOL.register\_codatatype\_global\/}~@\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}$ \\
2806 $\hbox{}\qquad\quad @\{\antiq{const\_name}~ \textit{llist\_case}\}$ \\
2807 $\hbox{}\qquad\quad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])$ \\
2811 The \textit{register\_codatatype\_global\/} function takes a coinductive type, its
2812 case function, and the list of its constructors. The case function must take its
2813 arguments in the order that the constructors are listed. If no case function
2814 with the correct signature is available, simply pass the empty string.
2816 On the other hand, if your goal is to cripple Nitpick, add the following line to
2817 your theory file and try to check a few conjectures about lazy lists:
2820 $\textbf{setup}~\,\{{*}$ \\
2821 $\hbox{}\quad\textit{Nitpick\_HOL.unregister\_codatatype\_global\/}~@\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}$ \\
2825 Inductive datatypes can be registered as coinductive datatypes, given
2826 appropriate coinductive constructors. However, doing so precludes
2827 the use of the inductive constructors---Nitpick will generate an error if they
2830 \subsection{Registration of Term Postprocessors}
2831 \label{registration-of-term-postprocessors}
2833 It is possible to change the output of any term that Nitpick considers a
2834 datatype by registering a term postprocessor. The interface for registering and
2835 unregistering postprocessors consists of the following pair of functions defined
2836 in the \textit{Nitpick\_Model} structure:
2839 $\textbf{type}\,~\textit{term\_postprocessor}\,~{=} {}$ \\
2840 $\hbox{}\quad\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \rightarrow \textit{term}$ \\
2841 $\textbf{val}\,~\textit{register\_term\_postprocessor\_global\/} : {}$ \\
2842 $\hbox{}\quad\textit{typ} \rightarrow \textit{term\_postprocessor} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2843 $\textbf{val}\,~\textit{unregister\_term\_postprocessor\_global\/} :\,
2844 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2847 \S\ref{typedefs-quotient-types-records-rationals-and-reals} and
2848 \texttt{src/HOL/Library/Multiset.thy} illustrate this feature in context.
2850 \section{Known Bugs and Limitations}
2851 \label{known-bugs-and-limitations}
2853 Here are the known bugs and limitations in Nitpick at the time of writing:
2856 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2857 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2858 Nitpick to generate spurious counterexamples for theorems that refer to values
2859 for which the function is not defined. For example:
2862 \textbf{primrec} \textit{prec} \textbf{where} \\
2863 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2864 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2865 \textbf{nitpick} \\[2\smallskipamount]
2866 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2868 \\[2\smallskipamount]
2869 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2870 \textbf{by}~(\textit{auto simp}:~\textit{prec\_def})
2873 Such theorems are generally considered bad style because they rely on the
2874 internal representation of functions synthesized by Isabelle, an implementation
2877 \item[$\bullet$] Similarly, Nitpick might find spurious counterexamples for
2878 theorems that rely on the use of the indefinite description operator internally
2879 by \textbf{specification} and \textbf{quot\_type}.
2881 \item[$\bullet$] Axioms or definitions that restrict the possible values of the
2882 \textit{undefined} constant or other partially specified built-in Isabelle
2883 constants (e.g., \textit{Abs\_} and \textit{Rep\_} constants) are in general
2884 ignored. Again, such nonconservative extensions are generally considered bad
2887 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2888 which can become invalid if you change the definition of an inductive predicate
2889 that is registered in the cache. To clear the cache,
2890 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2891 501$\,\textit{ms}$).
2893 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2894 \textbf{guess} command in a structured proof.
2896 \item[$\bullet$] The \textit{nitpick\_xxx} attributes and the
2897 \textit{Nitpick\_xxx.register\_yyy} functions can cause havoc if used
2900 \item[$\bullet$] Although this has never been observed, arbitrary theorem
2901 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
2903 \item[$\bullet$] All constants, types, free variables, and schematic variables
2904 whose names start with \textit{Nitpick}{.} are reserved for internal use.
2908 \bibliography{../manual}{}
2909 \bibliographystyle{abbrv}