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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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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 (8 by default), looking for a finite
254 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
256 Trying 8 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}~$'a$~= 8. \\[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 \cite{sledgehammer-2009} 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 8 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.register\_term\_postprocessor}~\!\begin{aligned}[t]
725 & @\{\textrm{typ}~\textit{my\_int}\}~\textit{my\_int\_postproc}\end{aligned} \\[-2pt]
729 Records are also handled as datatypes with a single constructor:
732 \textbf{record} \textit{point} = \\
733 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
734 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
735 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
736 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
737 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
738 \hbox{}\qquad Free variables: \nopagebreak \\
739 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
740 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
741 \hbox{}\qquad Datatypes: \\
742 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
743 \hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
744 & \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
745 & \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
748 Finally, Nitpick provides rudimentary support for rationals and reals using a
752 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
753 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
754 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
755 \hbox{}\qquad Free variables: \nopagebreak \\
756 \hbox{}\qquad\qquad $x = 1/2$ \\
757 \hbox{}\qquad\qquad $y = -1/2$ \\
758 \hbox{}\qquad Datatypes: \\
759 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
760 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
761 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
764 \subsection{Inductive and Coinductive Predicates}
765 \label{inductive-and-coinductive-predicates}
767 Inductively defined predicates (and sets) are particularly problematic for
768 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
769 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
770 the problem is that they are defined using a least fixed point construction.
772 Nitpick's philosophy is that not all inductive predicates are equal. Consider
773 the \textit{even} predicate below:
776 \textbf{inductive}~\textit{even}~\textbf{where} \\
777 ``\textit{even}~0'' $\,\mid$ \\
778 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
781 This predicate enjoys the desirable property of being well-founded, which means
782 that the introduction rules don't give rise to infinite chains of the form
785 $\cdots\,\Longrightarrow\, \textit{even}~k''
786 \,\Longrightarrow\, \textit{even}~k'
787 \,\Longrightarrow\, \textit{even}~k.$
790 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
794 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
795 \,\Longrightarrow\, \textit{even}~(k - 2)
796 \,\Longrightarrow\, \textit{even}~k.$
799 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
800 fixed point computation.%
801 \footnote{If an inductive predicate is
802 well-founded, then it has exactly one fixed point, which is simultaneously the
803 least and the greatest fixed point. In these circumstances, the computation of
804 the least fixed point amounts to the computation of an arbitrary fixed point,
805 which can be performed using a straightforward recursive equation.}
806 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
807 just as Isabelle's \textbf{function} package usually discharges termination
808 proof obligations automatically.
810 Let's try an example:
813 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
814 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
815 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
816 Nitpick can compute it efficiently. \\[2\smallskipamount]
818 \hbox{}\qquad \textit{card nat}~= 50. \\[2\smallskipamount]
819 Nitpick found a potential counterexample for \textit{card nat}~= 50: \\[2\smallskipamount]
820 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
821 Nitpick could not find a better counterexample. \\[2\smallskipamount]
825 No genuine counterexample is possible because Nitpick cannot rule out the
826 existence of a natural number $n \ge 50$ such that both $\textit{even}~n$ and
827 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
828 existential quantifier:
831 \textbf{lemma} ``$\exists n \mathbin{\le} 49.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
832 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}] \\[2\smallskipamount]
833 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
834 \hbox{}\qquad Empty assignment
837 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
838 we use the following definition instead?
841 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
842 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
843 ``$\textit{even}'~2$'' $\,\mid$ \\
844 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
847 This definition is not well-founded: From $\textit{even}'~0$ and
848 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
849 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
851 Let's check a property involving $\textit{even}'$. To make up for the
852 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
853 \textit{nat}'s cardinality to a mere 10:
856 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
857 \lnot\;\textit{even}'~n$'' \\
858 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
860 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
861 Nitpick might need to unroll it. \\[2\smallskipamount]
863 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
864 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
865 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
866 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
867 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
868 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
869 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
870 \hbox{}\qquad Constant: \nopagebreak \\
871 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
872 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
873 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
874 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
878 Nitpick's output is very instructive. First, it tells us that the predicate is
879 unrolled, meaning that it is computed iteratively from the empty set. Then it
880 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
883 The output also shows how each iteration contributes to $\textit{even}'$. The
884 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
885 predicate depends on an iteration counter. Iteration 0 provides the basis
886 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
887 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
888 iterations would not contribute any new elements.
890 Some values are marked with superscripted question
891 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
892 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
893 \textit{True} or $\unk$, never \textit{False}.
895 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
896 iterations. However, these numbers are bounded by the cardinality of the
897 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
898 ever needed to compute the value of a \textit{nat} predicate. You can specify
899 the number of iterations using the \textit{iter} option, as explained in
900 \S\ref{scope-of-search}.
902 In the next formula, $\textit{even}'$ occurs both positively and negatively:
905 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
906 \textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
907 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
908 \hbox{}\qquad Free variable: \nopagebreak \\
909 \hbox{}\qquad\qquad $n = 1$ \\
910 \hbox{}\qquad Constants: \nopagebreak \\
911 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
912 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
913 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
916 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
917 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
918 fixed point (not necessarily the least one). It is used to falsify
919 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
920 $\textit{even}'~(n - 2)$.
922 Coinductive predicates are handled dually. For example:
925 \textbf{coinductive} \textit{nats} \textbf{where} \\
926 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
927 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
928 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
929 \slshape Nitpick found a counterexample:
930 \\[2\smallskipamount]
931 \hbox{}\qquad Constants: \nopagebreak \\
932 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
933 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
934 & \unr\})\end{aligned}$ \\
935 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
938 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
939 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
940 inductive predicates for which each the predicate occurs in at most one
941 assumption of each introduction rule. For example:
944 \textbf{inductive} \textit{odd} \textbf{where} \\
945 ``$\textit{odd}~1$'' $\,\mid$ \\
946 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
947 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
948 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
949 \slshape Nitpick found a counterexample:
950 \\[2\smallskipamount]
951 \hbox{}\qquad Free variable: \nopagebreak \\
952 \hbox{}\qquad\qquad $n = 1$ \\
953 \hbox{}\qquad Constants: \nopagebreak \\
954 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
955 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
956 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
958 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
959 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
961 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
962 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
963 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
967 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
968 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
969 elements from known ones. The set $\textit{odd}$ consists of all the values
970 reachable through the reflexive transitive closure of
971 $\textit{odd}_{\textrm{step}}$ starting with any element from
972 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
973 transitive closure to encode linear predicates is normally either more thorough
974 or more efficient than unrolling (depending on the value of \textit{iter}), but
975 for those cases where it isn't you can disable it by passing the
976 \textit{dont\_star\_linear\_preds} option.
978 \subsection{Coinductive Datatypes}
979 \label{coinductive-datatypes}
981 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
982 datatypes, the \textit{Coinductive\_List} theory from Andreas Lochbihler's
983 \textit{Coinductive} AFP entry \cite{lochbihler-2010} provides a coinductive
984 ``lazy list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick
985 supports these lazy lists seamlessly and provides a hook, described in
986 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
989 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
990 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
991 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
992 1, 2, 3, \ldots]$ can be defined as lazy lists using the
993 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
994 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
995 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
997 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
998 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
1002 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
1003 \textbf{nitpick} \\[2\smallskipamount]
1004 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
1005 \hbox{}\qquad Free variables: \nopagebreak \\
1006 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
1007 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
1010 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
1011 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
1012 infinite list $[a_1, a_1, a_1, \ldots]$.
1014 The next example is more interesting:
1017 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
1018 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1019 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1020 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
1021 some scopes. \\[2\smallskipamount]
1023 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 1,
1024 and \textit{bisim\_depth}~= 0. \\
1025 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1026 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 8,
1027 and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
1028 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
1029 \textit{card}~``\kern1pt$'a~\textit{list\/}$''~= 2, and \textit{bisim\_\allowbreak
1031 \\[2\smallskipamount]
1032 \hbox{}\qquad Free variables: \nopagebreak \\
1033 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
1034 \hbox{}\qquad\qquad $\textit{b} = a_2$ \\
1035 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1036 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
1040 The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
1041 $\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
1042 $[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment
1043 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
1044 the segment leading to the binder is the stem.
1046 A salient property of coinductive datatypes is that two objects are considered
1047 equal if and only if they lead to the same observations. For example, the lazy
1048 lists $\textrm{THE}~\omega.\; \omega =
1049 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
1050 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
1051 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
1052 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
1053 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
1054 concept of equality for coinductive datatypes is called bisimulation and is
1055 defined coinductively.
1057 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
1058 the Kodkod problem to ensure that distinct objects lead to different
1059 observations. This precaution is somewhat expensive and often unnecessary, so it
1060 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
1061 bisimilarity check is then performed \textsl{after} the counterexample has been
1062 found to ensure correctness. If this after-the-fact check fails, the
1063 counterexample is tagged as ``quasi genuine'' and Nitpick recommends to try
1064 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
1065 check for the previous example saves approximately 150~milli\-seconds; the speed
1066 gains can be more significant for larger scopes.
1068 The next formula illustrates the need for bisimilarity (either as a Kodkod
1069 predicate or as an after-the-fact check) to prevent spurious counterexamples:
1072 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
1073 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1074 \textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
1075 \slshape Nitpick found a quasi genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
1076 \hbox{}\qquad Free variables: \nopagebreak \\
1077 \hbox{}\qquad\qquad $a = a_1$ \\
1078 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
1079 \textit{LCons}~a_1~\omega$ \\
1080 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1081 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
1082 \hbox{}\qquad\qquad $'a~\textit{llist} =
1083 \{\!\begin{aligned}[t]
1084 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
1085 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
1086 \\[2\smallskipamount]
1087 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1088 that the counterexample is genuine. \\[2\smallskipamount]
1089 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1090 \slshape Nitpick found no counterexample.
1093 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1094 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1096 A compromise between leaving out the bisimilarity predicate from the Kodkod
1097 problem and performing the after-the-fact check is to specify a lower
1098 nonnegative \textit{bisim\_depth} value than the default one provided by
1099 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1100 be distinguished from each other by their prefixes of length $K$. Be aware that
1101 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1102 finding any counterexamples.
1107 Nitpick normally maps function and product types directly to the corresponding
1108 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1109 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1110 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1111 off to treat these types in the same way as plain datatypes, by approximating
1112 them by a subset of a given cardinality. This technique is called ``boxing'' and
1113 is particularly useful for functions passed as arguments to other functions, for
1114 high-arity functions, and for large tuples. Under the hood, boxing involves
1115 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1116 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1118 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1119 using de Bruijn's notation:
1122 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1125 The $\textit{lift}~t~k$ function increments all variables with indices greater
1126 than or equal to $k$ by one:
1129 \textbf{primrec} \textit{lift} \textbf{where} \\
1130 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1131 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1132 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1135 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1136 term $t$ has a loose variable with index $k$ or more:
1139 \textbf{primrec}~\textit{loose} \textbf{where} \\
1140 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1141 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1142 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1145 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1149 \textbf{primrec}~\textit{subst} \textbf{where} \\
1150 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1151 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1152 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1153 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1156 A substitution is a function that maps variable indices to terms. Observe that
1157 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1158 away, because the recursive call for the \textit{Lam} case involves an altered
1159 version. Also notice the \textit{lift} call, which increments the variable
1160 indices when moving under a \textit{Lam}.
1162 A reasonable property to expect of substitution is that it should leave closed
1163 terms unchanged. Alas, even this simple property does not hold:
1166 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1167 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1169 Trying 8 scopes: \nopagebreak \\
1170 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1171 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1172 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1173 \hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
1174 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1175 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1176 \hbox{}\qquad Free variables: \nopagebreak \\
1177 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1178 & 0 := \textit{Var}~0,\>
1179 1 := \textit{Var}~0,\>
1180 2 := \textit{Var}~0, \\[-2pt]
1181 & 3 := \textit{Var}~0,\>
1182 4 := \textit{Var}~0,\>
1183 5 := \textit{Var}~0)\end{aligned}$ \\
1184 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1185 Total time: $4679$ ms.
1188 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1189 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1190 $\lambda$-term notation, $t$~is
1191 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1192 The bug is in \textit{subst\/}: The $\textit{lift}~(\sigma~m)~1$ call should be
1193 replaced with $\textit{lift}~(\sigma~m)~0$.
1195 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1196 cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1197 For the formula of interest, knowing 6 values of that type was enough to find
1198 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1199 considered, a hopeless undertaking:
1202 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1203 {\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
1207 Boxing can be enabled or disabled globally or on a per-type basis using the
1208 \textit{box} option. Nitpick usually performs reasonable choices about which
1209 types should be boxed, but option tweaking sometimes helps. A related optimization,
1210 ``finalization,'' attempts to wrap functions that constant at all but finitely
1211 many points (e.g., finite sets); see the documentation for the \textit{finalize}
1212 option in \S\ref{scope-of-search} for details.
1216 \subsection{Scope Monotonicity}
1217 \label{scope-monotonicity}
1219 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1220 and \textit{max}) controls which scopes are actually tested. In general, to
1221 exhaust all models below a certain cardinality bound, the number of scopes that
1222 Nitpick must consider increases exponentially with the number of type variables
1223 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1224 cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
1225 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1227 Fortunately, many formulas exhibit a property called \textsl{scope
1228 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1229 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1231 Consider the formula
1234 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1237 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1238 $'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
1239 exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
1240 that any counterexample found with a small scope would still be a counterexample
1241 in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
1242 by the larger scope. Nitpick comes to the same conclusion after a careful
1243 inspection of the formula and the relevant definitions:
1246 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1248 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1249 Nitpick might be able to skip some scopes.
1250 \\[2\smallskipamount]
1252 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1253 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1254 \textit{list\/}''~= 1, \\
1255 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 1, and
1256 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 1. \\
1257 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1258 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1259 \textit{list\/}''~= 2, \\
1260 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 2, and
1261 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 2. \\
1262 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1263 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
1264 \textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
1265 \textit{list\/}''~= 8, \\
1266 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 8, and
1267 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 8.
1268 \\[2\smallskipamount]
1269 Nitpick found a counterexample for
1270 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1271 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1272 \textit{list\/}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 5, and
1273 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 5:
1274 \\[2\smallskipamount]
1275 \hbox{}\qquad Free variables: \nopagebreak \\
1276 \hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
1277 \hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
1278 Total time: 1636 ms.
1281 In theory, it should be sufficient to test a single scope:
1284 \textbf{nitpick}~[\textit{card}~= 8]
1287 However, this is often less efficient in practice and may lead to overly complex
1290 If the monotonicity check fails but we believe that the formula is monotonic (or
1291 we don't mind missing some counterexamples), we can pass the
1292 \textit{mono} option. To convince yourself that this option is risky,
1293 simply consider this example from \S\ref{skolemization}:
1296 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1297 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1298 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1299 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1300 \textbf{nitpick} \\[2\smallskipamount]
1302 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1303 \hbox{}\qquad $\vdots$
1306 (It turns out the formula holds if and only if $\textit{card}~'a \le
1307 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1308 monotonicity checks can be disabled by passing \textit{non\_mono}
1309 (\S\ref{optimizations}).
1311 As insinuated in \S\ref{natural-numbers-and-integers} and
1312 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1313 are normally monotonic and treated as such. The same is true for record types,
1314 \textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
1315 cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
1316 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1317 consider only 8~scopes instead of $32\,768$.
1319 \subsection{Inductive Properties}
1320 \label{inductive-properties}
1322 Inductive properties are a particular pain to prove, because the failure to
1323 establish an induction step can mean several things:
1326 \item The property is invalid.
1327 \item The property is valid but is too weak to support the induction step.
1328 \item The property is valid and strong enough; it's just that we haven't found
1332 Depending on which scenario applies, we would take the appropriate course of
1336 \item Repair the statement of the property so that it becomes valid.
1337 \item Generalize the property and/or prove auxiliary properties.
1338 \item Work harder on a proof.
1341 How can we distinguish between the three scenarios? Nitpick's normal mode of
1342 operation can often detect scenario 1, and Isabelle's automatic tactics help with
1343 scenario 3. Using appropriate techniques, it is also often possible to use
1344 Nitpick to identify scenario 2. Consider the following transition system,
1345 in which natural numbers represent states:
1348 \textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
1349 ``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
1350 ``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
1351 ``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
1354 We will try to prove that only even numbers are reachable:
1357 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
1360 Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
1361 so let's attempt a proof by induction:
1364 \textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
1365 \textbf{apply}~\textit{auto}
1368 This leaves us in the following proof state:
1371 {\slshape goal (2 subgoals): \\
1372 \phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
1373 \phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
1377 If we run Nitpick on the first subgoal, it still won't find any
1378 counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
1379 is helpless. However, notice the $n \in \textit{reach}$ assumption, which
1380 strengthens the induction hypothesis but is not immediately usable in the proof.
1381 If we remove it and invoke Nitpick, this time we get a counterexample:
1384 \textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
1385 \textbf{nitpick} \\[2\smallskipamount]
1386 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1387 \hbox{}\qquad Skolem constant: \nopagebreak \\
1388 \hbox{}\qquad\qquad $n = 0$
1391 Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
1392 to strength the lemma:
1395 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
1398 Unfortunately, the proof by induction still gets stuck, except that Nitpick now
1399 finds the counterexample $n = 2$. We generalize the lemma further to
1402 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
1405 and this time \textit{arith} can finish off the subgoals.
1407 A similar technique can be employed for structural induction. The
1408 following mini formalization of full binary trees will serve as illustration:
1411 \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]
1412 \textbf{primrec}~\textit{labels}~\textbf{where} \\
1413 ``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
1414 ``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
1415 \textbf{primrec}~\textit{swap}~\textbf{where} \\
1416 ``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
1417 \phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
1418 ``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
1421 The \textit{labels} function returns the set of labels occurring on leaves of a
1422 tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
1423 labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
1424 obtained by swapping $a$ and $b$:
1427 \textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
1430 Nitpick can't find any counterexample, so we proceed with induction
1431 (this time favoring a more structured style):
1434 \textbf{proof}~(\textit{induct}~$t$) \\
1435 \hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
1437 \hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
1440 Nitpick can't find any counterexample at this point either, but it makes the
1441 following suggestion:
1445 Hint: To check that the induction hypothesis is general enough, try this command:
1446 \textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
1449 If we follow the hint, we get a ``nonstandard'' counterexample for the step:
1452 \slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
1453 \hbox{}\qquad Free variables: \nopagebreak \\
1454 \hbox{}\qquad\qquad $a = a_1$ \\
1455 \hbox{}\qquad\qquad $b = a_2$ \\
1456 \hbox{}\qquad\qquad $t = \xi_1$ \\
1457 \hbox{}\qquad\qquad $u = \xi_2$ \\
1458 \hbox{}\qquad Datatype: \nopagebreak \\
1459 \hbox{}\qquad\qquad $\alpha~\textit{btree} = \{\xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \textit{Branch}~\xi_1~\xi_2\}$ \\
1460 \hbox{}\qquad {\slshape Constants:} \nopagebreak \\
1461 \hbox{}\qquad\qquad $\textit{labels} = \undef
1462 (\!\begin{aligned}[t]%
1463 & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
1464 & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
1465 \hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \undef
1466 (\!\begin{aligned}[t]%
1467 & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
1468 & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount]
1469 The existence of a nonstandard model suggests that the induction hypothesis is not general enough or may even
1470 be wrong. See the Nitpick manual's ``Inductive Properties'' section for details (``\textit{isabelle doc nitpick}'').
1473 Reading the Nitpick manual is a most excellent idea.
1474 But what's going on? The \textit{non\_std} option told the tool to look for
1475 nonstandard models of binary trees, which means that new ``nonstandard'' trees
1476 $\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees
1477 generated by the \textit{Leaf} and \textit{Branch} constructors.%
1478 \footnote{Notice the similarity between allowing nonstandard trees here and
1479 allowing unreachable states in the preceding example (by removing the ``$n \in
1480 \textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
1481 set of objects over which the induction is performed while doing the step
1482 in order to test the induction hypothesis's strength.}
1483 Unlike standard trees, these new trees contain cycles. We will see later that
1484 every property of acyclic trees that can be proved without using induction also
1485 holds for cyclic trees. Hence,
1488 \textsl{If the induction
1489 hypothesis is strong enough, the induction step will hold even for nonstandard
1490 objects, and Nitpick won't find any nonstandard counterexample.}
1493 But here the tool find some nonstandard trees $t = \xi_1$
1494 and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in
1495 \textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$.
1496 Because neither tree contains both $a$ and $b$, the induction hypothesis tells
1497 us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
1498 and as a result we know nothing about the labels of the tree
1499 $\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
1500 $\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
1501 labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
1502 \textit{labels}$ $(\textit{swap}~u~a~b)$.
1504 The solution is to ensure that we always know what the labels of the subtrees
1505 are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
1506 $t$ in the statement of the lemma:
1509 \textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
1510 \phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
1511 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
1512 \phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
1513 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
1516 This time, Nitpick won't find any nonstandard counterexample, and we can perform
1517 the induction step using \textit{auto}.
1519 \section{Case Studies}
1520 \label{case-studies}
1522 As a didactic device, the previous section focused mostly on toy formulas whose
1523 validity can easily be assessed just by looking at the formula. We will now
1524 review two somewhat more realistic case studies that are within Nitpick's
1525 reach:\ a context-free grammar modeled by mutually inductive sets and a
1526 functional implementation of AA trees. The results presented in this
1527 section were produced with the following settings:
1530 \textbf{nitpick\_params} [\textit{max\_potential}~= 0]
1533 \subsection{A Context-Free Grammar}
1534 \label{a-context-free-grammar}
1536 Our first case study is taken from section 7.4 in the Isabelle tutorial
1537 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1538 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1541 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1542 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1543 $A$ & $::=$ & $aS \mid bAA$ \\
1544 $B$ & $::=$ & $bS \mid aBB$
1548 The intuition behind the grammar is that $A$ generates all string with one more
1549 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1551 The alphabet consists exclusively of $a$'s and $b$'s:
1554 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1557 Strings over the alphabet are represented by \textit{alphabet list}s.
1558 Nonterminals in the grammar become sets of strings. The production rules
1559 presented above can be expressed as a mutually inductive definition:
1562 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1563 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1564 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1565 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1566 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1567 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1568 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1571 The conversion of the grammar into the inductive definition was done manually by
1572 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1575 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1576 d'\^etre}. A good approach is to state desirable properties of the specification
1577 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1578 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1579 counterexamples will point to bugs in the specification. For our grammar
1580 example, we will proceed in two steps, separating the soundness and the
1581 completeness of the set $S$. First, soundness:
1584 \textbf{theorem}~\textit{S\_sound\/}: \\
1585 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1586 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1587 \textbf{nitpick} \\[2\smallskipamount]
1588 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1589 \hbox{}\qquad Free variable: \nopagebreak \\
1590 \hbox{}\qquad\qquad $w = [b]$
1593 It would seem that $[b] \in S$. How could this be? An inspection of the
1594 introduction rules reveals that the only rule with a right-hand side of the form
1595 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1599 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1602 On closer inspection, we can see that this rule is wrong. To match the
1603 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1607 \textbf{nitpick} \\[2\smallskipamount]
1608 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1609 \hbox{}\qquad Free variable: \nopagebreak \\
1610 \hbox{}\qquad\qquad $w = [a, a, b]$
1613 Some detective work is necessary to find out what went wrong here. To get $[a,
1614 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1618 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1621 Now, this formula must be wrong: The same assumption occurs twice, and the
1622 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1623 the assumptions should have been a $w$.
1625 With the correction made, we don't get any counterexample from Nitpick. Let's
1626 move on and check completeness:
1629 \textbf{theorem}~\textit{S\_complete}: \\
1630 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1631 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1632 \longrightarrow w \in S$'' \\
1633 \textbf{nitpick} \\[2\smallskipamount]
1634 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1635 \hbox{}\qquad Free variable: \nopagebreak \\
1636 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1639 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1640 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1641 the introduction rules we have are probably correct. Perhaps we simply lack an
1642 introduction rule. Comparing the grammar with the inductive definition, our
1643 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1644 without which the grammar cannot generate two or more $b$'s in a row. So we add
1648 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1651 With this last change, we don't get any counterexamples from Nitpick for either
1652 soundness or completeness. We can even generalize our result to cover $A$ and
1656 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1657 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1658 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1659 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1660 \textbf{nitpick} \\[2\smallskipamount]
1661 \slshape Nitpick ran out of time after checking 7 of 8 scopes.
1664 \subsection{AA Trees}
1667 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1668 similar performance to red-black trees, but with a simpler implementation
1669 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1670 a total order $<$. We start by defining the datatype and some basic extractor
1674 \textbf{datatype} $'a$~\textit{aa\_tree} = \\
1675 \hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}'' \\[2\smallskipamount]
1676 \textbf{primrec} \textit{data} \textbf{where} \\
1677 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1678 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1679 \textbf{primrec} \textit{dataset} \textbf{where} \\
1680 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1681 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1682 \textbf{primrec} \textit{level} \textbf{where} \\
1683 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1684 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1685 \textbf{primrec} \textit{left} \textbf{where} \\
1686 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1687 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1688 \textbf{primrec} \textit{right} \textbf{where} \\
1689 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1690 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1693 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1694 as follows \cite{wikipedia-2009-aa-trees}:
1696 \kern.2\parskip %% TYPESETTING
1699 Each node has a level field, and the following invariants must remain true for
1700 the tree to be valid:
1704 \kern-.4\parskip %% TYPESETTING
1709 \item[1.] The level of a leaf node is one.
1710 \item[2.] The level of a left child is strictly less than that of its parent.
1711 \item[3.] The level of a right child is less than or equal to that of its parent.
1712 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1713 \item[5.] Every node of level greater than one must have two children.
1718 \kern.4\parskip %% TYPESETTING
1720 The \textit{wf} predicate formalizes this description:
1723 \textbf{primrec} \textit{wf} \textbf{where} \\
1724 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1725 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1726 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1727 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1728 \phantom{``$($}$\textrm{else}$ \\
1729 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1730 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1731 \mathrel{\land} \textit{level}~u \le k$ \\
1732 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1735 Rebalancing the tree upon insertion and removal of elements is performed by two
1736 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1739 \textbf{primrec} \textit{skew} \textbf{where} \\
1740 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1741 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1742 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1743 \textit{level}~t~\textrm{then}$ \\
1744 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1745 (\textit{right}~t)~u)$ \\
1746 \phantom{``(}$\textrm{else}$ \\
1747 \phantom{``(\quad}$N~x~k~t~u)$''
1751 \textbf{primrec} \textit{split} \textbf{where} \\
1752 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1753 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1754 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1755 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1756 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1757 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1758 \phantom{``(}$\textrm{else}$ \\
1759 \phantom{``(\quad}$N~x~k~t~u)$''
1762 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1763 of elements stored in the tree:
1766 \textbf{theorem}~\textit{dataset\_skew\_split\/}:\\
1767 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1768 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1769 \textbf{nitpick} \\[2\smallskipamount]
1770 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1773 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1774 should not alter the tree:
1777 \textbf{theorem}~\textit{wf\_skew\_split\/}:\\
1778 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1779 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1780 \textbf{nitpick} \\[2\smallskipamount]
1781 {\slshape Nitpick found no counterexample.}
1784 Insertion is implemented recursively. It preserves the sort order:
1787 \textbf{primrec}~\textit{insort} \textbf{where} \\
1788 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1789 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1790 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1791 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1794 Notice that we deliberately commented out the application of \textit{skew} and
1795 \textit{split}. Let's see if this causes any problems:
1798 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1799 \textbf{nitpick} \\[2\smallskipamount]
1800 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1801 \hbox{}\qquad Free variables: \nopagebreak \\
1802 \hbox{}\qquad\qquad $t = N~a_1~1~\Lambda~\Lambda$ \\
1803 \hbox{}\qquad\qquad $x = a_2$
1806 It's hard to see why this is a counterexample. To improve readability, we will
1807 restrict the theorem to \textit{nat}, so that we don't need to look up the value
1808 of the $\textit{op}~{<}$ constant to find out which element is smaller than the
1809 other. In addition, we will tell Nitpick to display the value of
1810 $\textit{insort}~t~x$ using the \textit{eval} option. This gives
1813 \textbf{theorem} \textit{wf\_insort\_nat\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1814 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1815 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1816 \hbox{}\qquad Free variables: \nopagebreak \\
1817 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1818 \hbox{}\qquad\qquad $x = 0$ \\
1819 \hbox{}\qquad Evaluated term: \\
1820 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1823 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1824 where both have a level of 1. This violates the second AA tree invariant, which
1825 states that a left child's level must be less than its parent's. This shouldn't
1826 come as a surprise, considering that we commented out the tree rebalancing code.
1827 Reintroducing the code seems to solve the problem:
1830 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1831 \textbf{nitpick} \\[2\smallskipamount]
1832 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1835 Insertion should transform the set of elements represented by the tree in the
1839 \textbf{theorem} \textit{dataset\_insort\/}:\kern.4em
1840 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1841 \textbf{nitpick} \\[2\smallskipamount]
1842 {\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
1845 We could continue like this and sketch a complete theory of AA trees. Once the
1846 definitions and main theorems are in place and have been thoroughly tested using
1847 Nitpick, we could start working on the proofs. Developing theories this way
1848 usually saves time, because faulty theorems and definitions are discovered much
1849 earlier in the process.
1851 \section{Option Reference}
1852 \label{option-reference}
1854 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1855 \def\qty#1{$\left<\textit{#1}\right>$}
1856 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1857 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1858 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1859 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1860 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1861 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1862 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1863 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1864 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1866 Nitpick's behavior can be influenced by various options, which can be specified
1867 in brackets after the \textbf{nitpick} command. Default values can be set
1868 using \textbf{nitpick\_\allowbreak params}. For example:
1871 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1874 The options are categorized as follows:\ mode of operation
1875 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1876 format (\S\ref{output-format}), automatic counterexample checks
1877 (\S\ref{authentication}), optimizations
1878 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1880 You can instruct Nitpick to run automatically on newly entered theorems by
1881 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1882 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
1883 and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1884 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1885 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1886 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1887 \textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
1888 Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
1891 The number of options can be overwhelming at first glance. Do not let that worry
1892 you: Nitpick's defaults have been chosen so that it almost always does the right
1893 thing, and the most important options have been covered in context in
1894 \S\ref{first-steps}.
1896 The descriptions below refer to the following syntactic quantities:
1899 \item[$\bullet$] \qtybf{string}: A string.
1900 \item[$\bullet$] \qtybf{string\_list\/}: A space-separated list of strings
1901 (e.g., ``\textit{ichi ni san}'').
1902 \item[$\bullet$] \qtybf{bool\/}: \textit{true} or \textit{false}.
1903 \item[$\bullet$] \qtybf{bool\_or\_smart\/}: \textit{true}, \textit{false}, or \textit{smart}.
1904 \item[$\bullet$] \qtybf{int\/}: An integer. Negative integers are prefixed with a hyphen.
1905 \item[$\bullet$] \qtybf{int\_or\_smart\/}: An integer or \textit{smart}.
1906 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1907 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1909 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1910 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1911 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1912 \item[$\bullet$] \qtybf{const\/}: The name of a HOL constant.
1913 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1914 \item[$\bullet$] \qtybf{term\_list\/}: A space-separated list of HOL terms (e.g.,
1915 ``$f~x$''~``$g~y$'').
1916 \item[$\bullet$] \qtybf{type}: A HOL type.
1919 Default values are indicated in square brackets. Boolean options have a negated
1920 counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
1921 Boolean options, ``= \textit{true}'' may be omitted.
1923 \subsection{Mode of Operation}
1924 \label{mode-of-operation}
1927 \optrue{blocking}{non\_blocking}
1928 Specifies whether the \textbf{nitpick} command should operate synchronously.
1929 The asynchronous (non-blocking) mode lets the user start proving the putative
1930 theorem while Nitpick looks for a counterexample, but it can also be more
1931 confusing. For technical reasons, automatic runs currently always block.
1933 \optrue{falsify}{satisfy}
1934 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1935 satisfying examples (models). This manual assumes throughout that
1936 \textit{falsify} is enabled.
1938 \opsmart{user\_axioms}{no\_user\_axioms}
1939 Specifies whether the user-defined axioms (specified using
1940 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1941 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1942 the constants that occur in the formula to falsify. The option is implicitly set
1943 to \textit{true} for automatic runs.
1945 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1946 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1947 these conditions are tagged as ``quasi genuine.'' The \textit{debug}
1948 (\S\ref{output-format}) option can be used to find out which axioms were
1952 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1953 (\S\ref{output-format}).}
1955 \optrue{assms}{no\_assms}
1956 Specifies whether the relevant assumptions in structured proofs should be
1957 considered. The option is implicitly enabled for automatic runs.
1960 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1962 \opfalse{overlord}{no\_overlord}
1963 Specifies whether Nitpick should put its temporary files in
1964 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1965 debugging Nitpick but also unsafe if several instances of the tool are run
1966 simultaneously. The files are identified by the extensions
1967 \texttt{.kki}, \texttt{.cnf}, \texttt{.out}, and
1968 \texttt{.err}; you may safely remove them after Nitpick has run.
1971 {\small See also \textit{debug} (\S\ref{output-format}).}
1974 \subsection{Scope of Search}
1975 \label{scope-of-search}
1978 \oparg{card}{type}{int\_seq}
1979 Specifies the sequence of cardinalities to use for a given type.
1980 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1981 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1984 {\small See also \textit{box} (\S\ref{scope-of-search}) and \textit{mono}
1985 (\S\ref{scope-of-search}).}
1987 \opdefault{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
1988 Specifies the default sequence of cardinalities to use. This can be overridden
1989 on a per-type basis using the \textit{card}~\qty{type} option described above.
1991 \oparg{max}{const}{int\_seq}
1992 Specifies the sequence of maximum multiplicities to use for a given
1993 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1994 number of distinct values that it can construct. Nonsensical values (e.g.,
1995 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1996 datatypes equipped with several constructors.
1998 \opnodefault{max}{int\_seq}
1999 Specifies the default sequence of maximum multiplicities to use for
2000 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
2001 basis using the \textit{max}~\qty{const} option described above.
2003 \opsmart{binary\_ints}{unary\_ints}
2004 Specifies whether natural numbers and integers should be encoded using a unary
2005 or binary notation. In unary mode, the cardinality fully specifies the subset
2006 used to approximate the type. For example:
2008 $$\hbox{\begin{tabular}{@{}rll@{}}%
2009 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
2010 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
2011 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
2016 $$\hbox{\begin{tabular}{@{}rll@{}}%
2017 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
2018 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
2021 In binary mode, the cardinality specifies the number of distinct values that can
2022 be constructed. Each of these value is represented by a bit pattern whose length
2023 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
2024 Nitpick attempts to choose the more appropriate encoding by inspecting the
2025 formula at hand, preferring the binary notation for problems involving
2026 multiplicative operators or large constants.
2028 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
2029 problems that refer to the types \textit{rat} or \textit{real} or the constants
2030 \textit{Suc}, \textit{gcd}, or \textit{lcm}.
2032 {\small See also \textit{bits} (\S\ref{scope-of-search}) and
2033 \textit{show\_datatypes} (\S\ref{output-format}).}
2035 \opdefault{bits}{int\_seq}{$\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{8},\mathbf{10},\mathbf{12}$}
2036 Specifies the number of bits to use to represent natural numbers and integers in
2037 binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
2039 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
2041 \opargboolorsmart{wf}{const}{non\_wf}
2042 Specifies whether the specified (co)in\-duc\-tively defined predicate is
2043 well-founded. The option can take the following values:
2046 \item[$\bullet$] \textbf{\textit{true}:} Tentatively treat the (co)in\-duc\-tive
2047 predicate as if it were well-founded. Since this is generally not sound when the
2048 predicate is not well-founded, the counterexamples are tagged as ``quasi
2051 \item[$\bullet$] \textbf{\textit{false}:} Treat the (co)in\-duc\-tive predicate
2052 as if it were not well-founded. The predicate is then unrolled as prescribed by
2053 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
2056 \item[$\bullet$] \textbf{\textit{smart}:} Try to prove that the inductive
2057 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
2058 \textit{size\_change} tactics. If this succeeds (or the predicate occurs with an
2059 appropriate polarity in the formula to falsify), use an efficient fixed point
2060 equation as specification of the predicate; otherwise, unroll the predicates
2061 according to the \textit{iter}~\qty{const} and \textit{iter} options.
2065 {\small See also \textit{iter} (\S\ref{scope-of-search}),
2066 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
2067 (\S\ref{timeouts}).}
2069 \opsmart{wf}{non\_wf}
2070 Specifies the default wellfoundedness setting to use. This can be overridden on
2071 a per-predicate basis using the \textit{wf}~\qty{const} option above.
2073 \oparg{iter}{const}{int\_seq}
2074 Specifies the sequence of iteration counts to use when unrolling a given
2075 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
2076 predicates that occur negatively and coinductive predicates that occur
2077 positively in the formula to falsify and that cannot be proved to be
2078 well-founded, but this behavior is influenced by the \textit{wf} option. The
2079 iteration counts are automatically bounded by the cardinality of the predicate's
2082 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
2083 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
2085 \opdefault{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
2086 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
2087 predicates. This can be overridden on a per-predicate basis using the
2088 \textit{iter} \qty{const} option above.
2090 \opdefault{bisim\_depth}{int\_seq}{$\mathbf{7}$}
2091 Specifies the sequence of iteration counts to use when unrolling the
2092 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
2093 of $-1$ means that no predicate is generated, in which case Nitpick performs an
2094 after-the-fact check to see if the known coinductive datatype values are
2095 bidissimilar. If two values are found to be bisimilar, the counterexample is
2096 tagged as ``quasi genuine.'' The iteration counts are automatically bounded by
2097 the sum of the cardinalities of the coinductive datatypes occurring in the
2100 \opargboolorsmart{box}{type}{dont\_box}
2101 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
2102 product type in an isomorphic datatype internally. Boxing is an effective mean
2103 to reduce the search space and speed up Nitpick, because the isomorphic datatype
2104 is approximated by a subset of the possible function or pair values.
2105 Like other drastic optimizations, it can also prevent the discovery of
2106 counterexamples. The option can take the following values:
2109 \item[$\bullet$] \textbf{\textit{true}:} Box the specified type whenever
2111 \item[$\bullet$] \textbf{\textit{false}:} Never box the type.
2112 \item[$\bullet$] \textbf{\textit{smart}:} Box the type only in contexts where it
2113 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
2114 higher-order functions are good candidates for boxing.
2118 {\small See also \textit{finitize} (\S\ref{scope-of-search}), \textit{verbose}
2119 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}).}
2121 \opsmart{box}{dont\_box}
2122 Specifies the default boxing setting to use. This can be overridden on a
2123 per-type basis using the \textit{box}~\qty{type} option described above.
2125 \opargboolorsmart{finitize}{type}{dont\_finitize}
2126 Specifies whether Nitpick should attempt to finitize a given type, which can be
2127 a function type or an infinite ``shallow datatype'' (an infinite datatype whose
2128 constructors don't appear in the problem).
2130 For function types, Nitpick performs a monotonicity analysis to detect functions
2131 that are constant at all but finitely many points (e.g., finite sets) and treats
2132 such occurrences specially, thereby increasing the precision. The option can
2133 then take the following values:
2136 \item[$\bullet$] \textbf{\textit{false}:} Don't attempt to finitize the type.
2137 \item[$\bullet$] \textbf{\textit{true}} or \textbf{\textit{smart}:} Finitize the
2138 type wherever possible.
2141 The semantics of the option is somewhat different for infinite shallow
2145 \item[$\bullet$] \textbf{\textit{true}:} Finitize the datatype. Since this is
2146 unsound, counterexamples generated under these conditions are tagged as ``quasi
2148 \item[$\bullet$] \textbf{\textit{false}:} Don't attempt to finitize the datatype.
2149 \item[$\bullet$] \textbf{\textit{smart}:} Perform a monotonicity analysis to
2150 detect whether the datatype can be safely finitized before finitizing it.
2153 Like other drastic optimizations, finitization can sometimes prevent the
2154 discovery of counterexamples.
2157 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{mono}
2158 (\S\ref{scope-of-search}), \textit{verbose} (\S\ref{output-format}), and
2159 \textit{debug} (\S\ref{output-format}).}
2161 \opsmart{finitize}{dont\_finitize}
2162 Specifies the default finitization setting to use. This can be overridden on a
2163 per-type basis using the \textit{finitize}~\qty{type} option described above.
2165 \opargboolorsmart{mono}{type}{non\_mono}
2166 Specifies whether the given type should be considered monotonic when enumerating
2167 scopes and finitizing types. If the option is set to \textit{smart}, Nitpick
2168 performs a monotonicity check on the type. Setting this option to \textit{true}
2169 can reduce the number of scopes tried, but it can also diminish the chance of
2170 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
2173 {\small See also \textit{card} (\S\ref{scope-of-search}),
2174 \textit{finitize} (\S\ref{scope-of-search}),
2175 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
2176 (\S\ref{output-format}).}
2178 \opsmart{mono}{non\_mono}
2179 Specifies the default monotonicity setting to use. This can be overridden on a
2180 per-type basis using the \textit{mono}~\qty{type} option described above.
2182 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
2183 Specifies whether type variables with the same sort constraints should be
2184 merged. Setting this option to \textit{true} can reduce the number of scopes
2185 tried and the size of the generated Kodkod formulas, but it also diminishes the
2186 theoretical chance of finding a counterexample.
2188 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
2190 \opargbool{std}{type}{non\_std}
2191 Specifies whether the given (recursive) datatype should be given standard
2192 models. Nonstandard models are unsound but can help debug structural induction
2193 proofs, as explained in \S\ref{inductive-properties}.
2195 \optrue{std}{non\_std}
2196 Specifies the default standardness to use. This can be overridden on a per-type
2197 basis using the \textit{std}~\qty{type} option described above.
2200 \subsection{Output Format}
2201 \label{output-format}
2204 \opfalse{verbose}{quiet}
2205 Specifies whether the \textbf{nitpick} command should explain what it does. This
2206 option is useful to determine which scopes are tried or which SAT solver is
2207 used. This option is implicitly disabled for automatic runs.
2209 \opfalse{debug}{no\_debug}
2210 Specifies whether Nitpick should display additional debugging information beyond
2211 what \textit{verbose} already displays. Enabling \textit{debug} also enables
2212 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
2213 option is implicitly disabled for automatic runs.
2216 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
2217 \textit{batch\_size} (\S\ref{optimizations}).}
2219 \opfalse{show\_datatypes}{hide\_datatypes}
2220 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
2221 be displayed as part of counterexamples. Such subsets are sometimes helpful when
2222 investigating whether a potential counterexample is genuine or spurious, but
2223 their potential for clutter is real.
2225 \opfalse{show\_consts}{hide\_consts}
2226 Specifies whether the values of constants occurring in the formula (including
2227 its axioms) should be displayed along with any counterexample. These values are
2228 sometimes helpful when investigating why a counterexample is
2229 genuine, but they can clutter the output.
2231 \opnodefault{show\_all}{bool}
2232 Abbreviation for \textit{show\_datatypes} and \textit{show\_consts}.
2234 \opdefault{max\_potential}{int}{$\mathbf{1}$}
2235 Specifies the maximum number of potential counterexamples to display. Setting
2236 this option to 0 speeds up the search for a genuine counterexample. This option
2237 is implicitly set to 0 for automatic runs. If you set this option to a value
2238 greater than 1, you will need an incremental SAT solver, such as
2239 \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that many of
2240 the counterexamples may be identical.
2243 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
2244 \textit{sat\_solver} (\S\ref{optimizations}).}
2246 \opdefault{max\_genuine}{int}{$\mathbf{1}$}
2247 Specifies the maximum number of genuine counterexamples to display. If you set
2248 this option to a value greater than 1, you will need an incremental SAT solver,
2249 such as \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that
2250 many of the counterexamples may be identical.
2253 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
2254 \textit{sat\_solver} (\S\ref{optimizations}).}
2256 \opnodefault{eval}{term\_list}
2257 Specifies the list of terms whose values should be displayed along with
2258 counterexamples. This option suffers from an ``observer effect'': Nitpick might
2259 find different counterexamples for different values of this option.
2261 \oparg{atoms}{type}{string\_list}
2262 Specifies the names to use to refer to the atoms of the given type. By default,
2263 Nitpick generates names of the form $a_1, \ldots, a_n$, where $a$ is the first
2264 letter of the type's name.
2266 \opnodefault{atoms}{string\_list}
2267 Specifies the default names to use to refer to atoms of any type. For example,
2268 to call the three atoms of type ${'}a$ \textit{ichi}, \textit{ni}, and
2269 \textit{san} instead of $a_1$, $a_2$, $a_3$, specify the option
2270 ``\textit{atoms}~${'}a$ = \textit{ichi~ni~san}''. The default names can be
2271 overridden on a per-type basis using the \textit{atoms}~\qty{type} option
2274 \oparg{format}{term}{int\_seq}
2275 Specifies how to uncurry the value displayed for a variable or constant.
2276 Uncurrying sometimes increases the readability of the output for high-arity
2277 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
2278 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
2279 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
2280 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
2281 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
2282 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
2283 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
2284 arguments that are not accounted for are left alone, as if the specification had
2285 been $1,\ldots,1,n_1,\ldots,n_k$.
2287 \opdefault{format}{int\_seq}{$\mathbf{1}$}
2288 Specifies the default format to use. Irrespective of the default format, the
2289 extra arguments to a Skolem constant corresponding to the outer bound variables
2290 are kept separated from the remaining arguments, the \textbf{for} arguments of
2291 an inductive definitions are kept separated from the remaining arguments, and
2292 the iteration counter of an unrolled inductive definition is shown alone. The
2293 default format can be overridden on a per-variable or per-constant basis using
2294 the \textit{format}~\qty{term} option described above.
2297 \subsection{Authentication}
2298 \label{authentication}
2301 \opfalse{check\_potential}{trust\_potential}
2302 Specifies whether potential counterexamples should be given to Isabelle's
2303 \textit{auto} tactic to assess their validity. If a potential counterexample is
2304 shown to be genuine, Nitpick displays a message to this effect and terminates.
2307 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
2309 \opfalse{check\_genuine}{trust\_genuine}
2310 Specifies whether genuine and quasi genuine counterexamples should be given to
2311 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
2312 counterexample is shown to be spurious, the user is kindly asked to send a bug
2313 report to the author at
2314 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
2317 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
2319 \opnodefault{expect}{string}
2320 Specifies the expected outcome, which must be one of the following:
2323 \item[$\bullet$] \textbf{\textit{genuine}:} Nitpick found a genuine counterexample.
2324 \item[$\bullet$] \textbf{\textit{quasi\_genuine}:} Nitpick found a ``quasi
2325 genuine'' counterexample (i.e., a counterexample that is genuine unless
2326 it contradicts a missing axiom or a dangerous option was used inappropriately).
2327 \item[$\bullet$] \textbf{\textit{potential}:} Nitpick found a potential counterexample.
2328 \item[$\bullet$] \textbf{\textit{none}:} Nitpick found no counterexample.
2329 \item[$\bullet$] \textbf{\textit{unknown}:} Nitpick encountered some problem (e.g.,
2330 Kodkod ran out of memory).
2333 Nitpick emits an error if the actual outcome differs from the expected outcome.
2334 This option is useful for regression testing.
2337 \subsection{Optimizations}
2338 \label{optimizations}
2340 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
2345 \opdefault{sat\_solver}{string}{smart}
2346 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2347 to be faster than their Java counterparts, but they can be more difficult to
2348 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2349 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2350 you will need an incremental SAT solver, such as \textit{MiniSat\_JNI}
2351 (recommended) or \textit{SAT4J}.
2353 The supported solvers are listed below:
2357 \item[$\bullet$] \textbf{\textit{MiniSat}:} MiniSat is an efficient solver
2358 written in \cpp{}. To use MiniSat, set the environment variable
2359 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2361 \footnote{Important note for Cygwin users: The path must be specified using
2362 native Windows syntax. Make sure to escape backslashes properly.%
2363 \label{cygwin-paths}}
2364 The \cpp{} sources and executables for MiniSat are available at
2365 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2366 and 2.0 beta (2007-07-21).
2368 \item[$\bullet$] \textbf{\textit{MiniSat\_JNI}:} The JNI (Java Native Interface)
2369 version of MiniSat is bundled with Kodkodi and is precompiled for the major
2370 platforms. It is also available from \texttt{native\-solver.\allowbreak tgz},
2371 which you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2372 version of MiniSat, the JNI version can be used incrementally.
2374 \item[$\bullet$] \textbf{\textit{PicoSAT}:} PicoSAT is an efficient solver
2375 written in C. You can install a standard version of
2376 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2377 that contains the \texttt{picosat} executable.%
2378 \footref{cygwin-paths}
2379 The C sources for PicoSAT are
2380 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2381 Nitpick has been tested with version 913.
2383 \item[$\bullet$] \textbf{\textit{zChaff}:} zChaff is an efficient solver written
2384 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2385 the directory that contains the \texttt{zchaff} executable.%
2386 \footref{cygwin-paths}
2387 The \cpp{} sources and executables for zChaff are available at
2388 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2389 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2391 \item[$\bullet$] \textbf{\textit{zChaff\_JNI}:} The JNI version of zChaff is
2392 bundled with Kodkodi and is precompiled for the major
2393 platforms. It is also available from \texttt{native\-solver.\allowbreak tgz},
2394 which you will find on Kodkod's web site \cite{kodkod-2009}.
2396 \item[$\bullet$] \textbf{\textit{RSat}:} RSat is an efficient solver written in
2397 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2398 directory that contains the \texttt{rsat} executable.%
2399 \footref{cygwin-paths}
2400 The \cpp{} sources for RSat are available at
2401 \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been tested with version
2404 \item[$\bullet$] \textbf{\textit{BerkMin}:} BerkMin561 is an efficient solver
2405 written in C. To use BerkMin, set the environment variable
2406 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2407 executable.\footref{cygwin-paths}
2408 The BerkMin executables are available at
2409 \url{http://eigold.tripod.com/BerkMin.html}.
2411 \item[$\bullet$] \textbf{\textit{BerkMin\_Alloy}:} Variant of BerkMin that is
2412 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2413 version of BerkMin, set the environment variable
2414 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2416 \footref{cygwin-paths}
2418 \item[$\bullet$] \textbf{\textit{Jerusat}:} Jerusat 1.3 is an efficient solver
2419 written in C. To use Jerusat, set the environment variable
2420 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2422 \footref{cygwin-paths}
2423 The C sources for Jerusat are available at
2424 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2426 \item[$\bullet$] \textbf{\textit{SAT4J}:} SAT4J is a reasonably efficient solver
2427 written in Java that can be used incrementally. It is bundled with Kodkodi and
2428 requires no further installation or configuration steps. Do not attempt to
2429 install the official SAT4J packages, because their API is incompatible with
2432 \item[$\bullet$] \textbf{\textit{SAT4J\_Light}:} Variant of SAT4J that is
2433 optimized for small problems. It can also be used incrementally.
2435 \item[$\bullet$] \textbf{\textit{smart}:} If \textit{sat\_solver} is set to
2436 \textit{smart}, Nitpick selects the first solver among MiniSat,
2437 PicoSAT, zChaff, RSat, BerkMin, BerkMin\_Alloy, Jerusat, MiniSat\_JNI, and zChaff\_JNI
2438 that is recognized by Isabelle. If none is found, it falls back on SAT4J, which
2439 should always be available. If \textit{verbose} (\S\ref{output-format}) is
2440 enabled, Nitpick displays which SAT solver was chosen.
2444 \opdefault{batch\_size}{int\_or\_smart}{smart}
2445 Specifies the maximum number of Kodkod problems that should be lumped together
2446 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2447 together ensures that Kodkodi is launched less often, but it makes the verbose
2448 output less readable and is sometimes detrimental to performance. If
2449 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2450 \textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
2452 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2453 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2454 be rewritten to use (automatically generated) discriminators and destructors.
2455 This optimization can drastically reduce the size of the Boolean formulas given
2459 {\small See also \textit{debug} (\S\ref{output-format}).}
2461 \optrue{specialize}{dont\_specialize}
2462 Specifies whether functions invoked with static arguments should be specialized.
2463 This optimization can drastically reduce the search space, especially for
2464 higher-order functions.
2467 {\small See also \textit{debug} (\S\ref{output-format}) and
2468 \textit{show\_consts} (\S\ref{output-format}).}
2470 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2471 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2472 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2473 predicates for which each the predicate occurs in at most one assumption of each
2474 introduction rule. Using the reflexive transitive closure is in principle
2475 equivalent to setting \textit{iter} to the cardinality of the predicate's
2476 domain, but it is usually more efficient.
2478 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2479 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2481 \optrue{fast\_descrs}{full\_descrs}
2482 Specifies whether Nitpick should optimize the definite and indefinite
2483 description operators (THE and SOME). The optimized versions usually help
2484 Nitpick generate more counterexamples or at least find them faster, but only the
2485 unoptimized versions are complete when all types occurring in the formula are
2488 {\small See also \textit{debug} (\S\ref{output-format}).}
2490 \optrue{peephole\_optim}{no\_peephole\_optim}
2491 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2492 peephole optimizer. These optimizations can make a significant difference.
2493 Unless you are tracking down a bug in Nitpick or distrust the peephole
2494 optimizer, you should leave this option enabled.
2496 \opdefault{max\_threads}{int}{0}
2497 Specifies the maximum number of threads to use in Kodkod. If this option is set
2498 to 0, Kodkod will compute an appropriate value based on the number of processor
2502 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2503 \textit{timeout} (\S\ref{timeouts}).}
2506 \subsection{Timeouts}
2510 \opdefault{timeout}{time}{$\mathbf{30}$ s}
2511 Specifies the maximum amount of time that the \textbf{nitpick} command should
2512 spend looking for a counterexample. Nitpick tries to honor this constraint as
2513 well as it can but offers no guarantees. For automatic runs,
2514 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2515 a time slot whose length is specified by the ``Auto Counterexample Time
2516 Limit'' option in Proof General.
2519 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2521 \opdefault{tac\_timeout}{time}{$\mathbf{500}$\,ms}
2522 Specifies the maximum amount of time that the \textit{auto} tactic should use
2523 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2524 and \textit{size\_change} should use when checking whether a (co)in\-duc\-tive
2525 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2526 can but offers no guarantees.
2529 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2530 \textit{check\_potential} (\S\ref{authentication}),
2531 and \textit{check\_genuine} (\S\ref{authentication}).}
2534 \section{Attribute Reference}
2535 \label{attribute-reference}
2537 Nitpick needs to consider the definitions of all constants occurring in a
2538 formula in order to falsify it. For constants introduced using the
2539 \textbf{definition} command, the definition is simply the associated
2540 \textit{\_def} axiom. In contrast, instead of using the internal representation
2541 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2542 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2543 equational specification entered by the user.
2545 Behind the scenes, Isabelle's built-in packages and theories rely on the
2546 following attributes to affect Nitpick's behavior:
2549 \flushitem{\textit{nitpick\_def}}
2552 This attribute specifies an alternative definition of a constant. The
2553 alternative definition should be logically equivalent to the constant's actual
2554 axiomatic definition and should be of the form
2556 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2558 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2561 \flushitem{\textit{nitpick\_simp}}
2564 This attribute specifies the equations that constitute the specification of a
2565 constant. For functions defined using the \textbf{primrec}, \textbf{function},
2566 and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
2567 \textit{simps} rules. The equations must be of the form
2569 \qquad $c~t_1~\ldots\ t_n \,=\, u.$
2571 \flushitem{\textit{nitpick\_psimp}}
2574 This attribute specifies the equations that constitute the partial specification
2575 of a constant. For functions defined using the \textbf{function} package, this
2576 corresponds to the \textit{psimps} rules. The conditional equations must be of
2579 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
2581 \flushitem{\textit{nitpick\_choice\_spec}}
2584 This attribute specifies the (free-form) specification of a constant defined
2585 using the \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command.
2589 When faced with a constant, Nitpick proceeds as follows:
2592 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2593 is not empty, Nitpick uses these rules as the specification of the constant.
2595 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2596 the constant is not empty, it uses these rules as the specification of the
2599 \item[3.] Otherwise, if the constant was defined using the
2600 \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command and the
2601 \textit{nitpick\_choice\_spec} set associated with the constant is not empty, it
2602 uses these theorems as the specification of the constant.
2604 \item[4.] Otherwise, it looks up the definition of the constant:
2607 \item[1.] If the \textit{nitpick\_def} set associated with the constant
2608 is not empty, it uses the latest rule added to the set as the definition of the
2609 constant; otherwise it uses the actual definition axiom.
2610 \item[2.] If the definition is of the form
2612 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
2614 then Nitpick assumes that the definition was made using an inductive package and
2615 based on the introduction rules marked with \textit{nitpick\_\allowbreak
2616 \allowbreak intros} tries to determine whether the definition is
2621 As an illustration, consider the inductive definition
2624 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2625 ``\textit{odd}~1'' $\,\mid$ \\
2626 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2629 By default, Nitpick uses the \textit{lfp}-based definition in conjunction with
2630 the introduction rules. To override this, we can specify an alternative
2631 definition as follows:
2634 \textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]:\kern.4em ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2637 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2638 = 1$. Alternatively, we can specify an equational specification of the constant:
2641 \textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2644 Such tweaks should be done with great care, because Nitpick will assume that the
2645 constant is completely defined by its equational specification. For example, if
2646 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
2647 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2648 (\S\ref{output-format}) option is extremely useful to understand what is going
2649 on when experimenting with \textit{nitpick\_} attributes.
2651 \section{Standard ML Interface}
2652 \label{standard-ml-interface}
2654 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2655 and debugging. Among the most interesting functions exported by Nitpick are
2656 those that let you invoke the tool programmatically and those that let you
2657 register and unregister custom coinductive datatypes as well as term
2660 \subsection{Invocation of Nitpick}
2661 \label{invocation-of-nitpick}
2663 The \textit{Nitpick} structure offers the following functions for invoking your
2664 favorite counterexample generator:
2667 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2668 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2669 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2670 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2671 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2674 The return value is a new proof state paired with an outcome string
2675 (``genuine'', ``quasi\_genuine'', ``potential'', ``none'', or ``unknown''). The
2676 \textit{params} type is a large record that lets you set Nitpick's options. The
2677 current default options can be retrieved by calling the following function
2678 defined in the \textit{Nitpick\_Isar} structure:
2681 $\textbf{val}\,~\textit{default\_params} :\,
2682 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2685 The second argument lets you override option values before they are parsed and
2686 put into a \textit{params} record. Here is an example:
2689 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout\/}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2690 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2691 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2692 & \textit{subgoal}\end{aligned}$
2697 \subsection{Registration of Coinductive Datatypes}
2698 \label{registration-of-coinductive-datatypes}
2700 If you have defined a custom coinductive datatype, you can tell Nitpick about
2701 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2702 uses for lazy lists. The interface for registering and unregistering coinductive
2703 datatypes consists of the following pair of functions defined in the
2704 \textit{Nitpick} structure:
2707 $\textbf{val}\,~\textit{register\_codatatype} :\,
2708 \textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2709 $\textbf{val}\,~\textit{unregister\_codatatype} :\,
2710 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2713 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2714 not been, you could have told Nitpick about it by adding the following line
2715 to your theory file:
2718 $\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
2719 & \textit{Nitpick.register\_codatatype} \\[-2pt]
2720 & \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
2721 & \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
2724 The \textit{register\_codatatype} function takes a coinductive type, its case
2725 function, and the list of its constructors. The case function must take its
2726 arguments in the order that the constructors are listed. If no case function
2727 with the correct signature is available, simply pass the empty string.
2729 On the other hand, if your goal is to cripple Nitpick, add the following line to
2730 your theory file and try to check a few conjectures about lazy lists:
2733 $\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
2734 \kern1pt'a~\textit{list\/}\textrm{''}\}\ \,{*}\}$
2737 Inductive datatypes can be registered as coinductive datatypes, given
2738 appropriate coinductive constructors. However, doing so precludes
2739 the use of the inductive constructors---Nitpick will generate an error if they
2742 \subsection{Registration of Term Postprocessors}
2743 \label{registration-of-term-postprocessors}
2745 It is possible to change the output of any term that Nitpick considers a
2746 datatype by registering a term postprocessor. The interface for registering and
2747 unregistering postprocessors consists of the following pair of functions defined
2748 in the \textit{Nitpick} structure:
2751 $\textbf{type}\,~\textit{term\_postprocessor}\,~{=} {}$ \\
2752 $\hbox{}\quad\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \rightarrow \textit{term}$ \\
2753 $\textbf{val}\,~\textit{register\_term\_postprocessors} : {}$ \\
2754 $\hbox{}\quad\textit{typ} \rightarrow \textit{term\_postprocessor} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2755 $\textbf{val}\,~\textit{unregister\_term\_postprocessors} :\,
2756 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2759 \S\ref{typedefs-quotient-types-records-rationals-and-reals} and
2760 \texttt{src/HOL/Library/Multiset.thy} illustrate this feature in context.
2762 \section{Known Bugs and Limitations}
2763 \label{known-bugs-and-limitations}
2765 Here are the known bugs and limitations in Nitpick at the time of writing:
2768 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2769 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2770 Nitpick to generate spurious counterexamples for theorems that refer to values
2771 for which the function is not defined. For example:
2774 \textbf{primrec} \textit{prec} \textbf{where} \\
2775 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2776 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2777 \textbf{nitpick} \\[2\smallskipamount]
2778 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2780 \\[2\smallskipamount]
2781 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2782 \textbf{by}~(\textit{auto simp}:~\textit{prec\_def})
2785 Such theorems are considered bad style because they rely on the internal
2786 representation of functions synthesized by Isabelle, which is an implementation
2789 \item[$\bullet$] Axioms that restrict the possible values of the
2790 \textit{undefined} constant are in general ignored.
2792 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2793 which can become invalid if you change the definition of an inductive predicate
2794 that is registered in the cache. To clear the cache,
2795 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2796 501$\,\textit{ms}$).
2798 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2799 \textbf{guess} command in a structured proof.
2801 \item[$\bullet$] The \textit{nitpick\_} attributes and the
2802 \textit{Nitpick.register\_} functions can cause havoc if used improperly.
2804 \item[$\bullet$] Although this has never been observed, arbitrary theorem
2805 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
2807 \item[$\bullet$] All constants, types, free variables, and schematic variables
2808 whose names start with \textit{Nitpick}{.} are reserved for internal use.
2812 \bibliography{../manual}{}
2813 \bibliographystyle{abbrv}