1 \documentclass[a4paper,12pt]{article}
2 \usepackage[T1]{fontenc}
5 \usepackage[english,french]{babel}
12 %\usepackage[scaled=.85]{beramono}
13 \usepackage{../iman,../pdfsetup}
16 %\evensidemargin=4.6mm
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}}
26 \def\lparr{\mathopen{(\mkern-4mu\mid}}
27 \def\rparr{\mathclose{\mid\mkern-4mu)}}
30 \def\undef{(\lambda x.\; \unk)}
31 %\def\unr{\textit{others}}
33 \def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
34 \def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
36 \hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
37 counter-example counter-examples data-type data-types co-data-type
38 co-data-types in-duc-tive co-in-duc-tive}
44 \selectlanguage{english}
46 \title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
47 Picking Nits \\[\smallskipamount]
48 \Large A User's Guide to Nitpick for Isabelle/HOL}
50 Jasmin Christian Blanchette \\
51 {\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\
58 \setlength{\parskip}{.7em plus .2em minus .1em}
59 \setlength{\parindent}{0pt}
60 \setlength{\abovedisplayskip}{\parskip}
61 \setlength{\abovedisplayshortskip}{.9\parskip}
62 \setlength{\belowdisplayskip}{\parskip}
63 \setlength{\belowdisplayshortskip}{.9\parskip}
65 % General-purpose enum environment with correct spacing
66 \newenvironment{enum}%
68 \setlength{\topsep}{.1\parskip}%
69 \setlength{\partopsep}{.1\parskip}%
70 \setlength{\itemsep}{\parskip}%
71 \advance\itemsep by-\parsep}}
74 \def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin
75 \advance\rightskip by\leftmargin}
76 \def\post{\vskip0pt plus1ex\endgroup}
78 \def\prew{\pre\advance\rightskip by-\leftmargin}
81 \section{Introduction}
84 Nitpick \cite{blanchette-nipkow-2009} 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 Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
115 machine called \texttt{java}. The examples presented in this manual can be found
116 in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.
118 Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
119 Nitpick also provides an automatic mode that can be enabled using the
120 ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof General. In this
121 mode, Nitpick is run on every newly entered theorem, much like Auto Quickcheck.
122 The collective time limit for Auto Nitpick and Auto Quickcheck can be set using
123 the ``Auto Counterexample Time Limit'' option.
126 \setbox\boxA=\hbox{\texttt{nospam}}
128 The known bugs and limitations at the time of writing are listed in
129 \S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
130 or this manual should be directed to
131 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
132 in.\allowbreak tum.\allowbreak de}.
134 \vskip2.5\smallskipamount
136 \textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
137 suggesting several textual improvements.
138 % and Perry James for reporting a typo.
140 \section{First Steps}
143 This section introduces Nitpick by presenting small examples. If possible, you
144 should try out the examples on your workstation. Your theory file should start
148 \textbf{theory}~\textit{Scratch} \\
149 \textbf{imports}~\textit{Main~Quotient\_Product~RealDef} \\
153 The results presented here were obtained using the JNI (Java Native Interface)
154 version of MiniSat and with multithreading disabled to reduce nondeterminism.
155 This was done by adding the line
158 \textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSat\_JNI}, \,\textit{max\_threads}~= 1]
161 after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
162 Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
163 be installed, as explained in \S\ref{optimizations}. If you have already
164 configured SAT solvers in Isabelle (e.g., for Refute), these will also be
165 available to Nitpick.
167 \subsection{Propositional Logic}
168 \label{propositional-logic}
170 Let's start with a trivial example from propositional logic:
173 \textbf{lemma}~``$P \longleftrightarrow Q$'' \\
177 You should get the following output:
181 Nitpick found a counterexample: \\[2\smallskipamount]
182 \hbox{}\qquad Free variables: \nopagebreak \\
183 \hbox{}\qquad\qquad $P = \textit{True}$ \\
184 \hbox{}\qquad\qquad $Q = \textit{False}$
187 Nitpick can also be invoked on individual subgoals, as in the example below:
190 \textbf{apply}~\textit{auto} \\[2\smallskipamount]
191 {\slshape goal (2 subgoals): \\
192 \phantom{0}1. $P\,\Longrightarrow\, Q$ \\
193 \phantom{0}2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
194 \textbf{nitpick}~1 \\[2\smallskipamount]
195 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
196 \hbox{}\qquad Free variables: \nopagebreak \\
197 \hbox{}\qquad\qquad $P = \textit{True}$ \\
198 \hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
199 \textbf{nitpick}~2 \\[2\smallskipamount]
200 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
201 \hbox{}\qquad Free variables: \nopagebreak \\
202 \hbox{}\qquad\qquad $P = \textit{False}$ \\
203 \hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
207 \subsection{Type Variables}
208 \label{type-variables}
210 If you are left unimpressed by the previous example, don't worry. The next
211 one is more mind- and computer-boggling:
214 \textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
216 \pagebreak[2] %% TYPESETTING
218 The putative lemma involves the definite description operator, {THE}, presented
219 in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
220 operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
221 lemma is merely asserting the indefinite description operator axiom with {THE}
222 substituted for {SOME}.
224 The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
225 containing type variables, Nitpick enumerates the possible domains for each type
226 variable, up to a given cardinality (8 by default), looking for a finite
230 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
232 Trying 8 scopes: \nopagebreak \\
233 \hbox{}\qquad \textit{card}~$'a$~= 1; \\
234 \hbox{}\qquad \textit{card}~$'a$~= 2; \\
235 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
236 \hbox{}\qquad \textit{card}~$'a$~= 8. \\[2\smallskipamount]
237 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
238 \hbox{}\qquad Free variables: \nopagebreak \\
239 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
240 \hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
244 Nitpick found a counterexample in which $'a$ has cardinality 3. (For
245 cardinalities 1 and 2, the formula holds.) In the counterexample, the three
246 values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
248 The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
249 \textit{verbose} is enabled. You can specify \textit{verbose} each time you
250 invoke \textbf{nitpick}, or you can set it globally using the command
253 \textbf{nitpick\_params} [\textit{verbose}]
256 This command also displays the current default values for all of the options
257 supported by Nitpick. The options are listed in \S\ref{option-reference}.
259 \subsection{Constants}
262 By just looking at Nitpick's output, it might not be clear why the
263 counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
264 this time telling it to show the values of the constants that occur in the
268 \textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
269 \textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
271 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
272 \hbox{}\qquad Free variables: \nopagebreak \\
273 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
274 \hbox{}\qquad\qquad $x = a_3$ \\
275 \hbox{}\qquad Constant: \nopagebreak \\
276 \hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
279 We can see more clearly now. Since the predicate $P$ isn't true for a unique
280 value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
281 $a_1$. Since $P~a_1$ is false, the entire formula is falsified.
283 As an optimization, Nitpick's preprocessor introduced the special constant
284 ``\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
285 $\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
286 satisfying $P~y$. We disable this optimization by passing the
287 \textit{full\_descrs} option:
290 \textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
292 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
293 \hbox{}\qquad Free variables: \nopagebreak \\
294 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
295 \hbox{}\qquad\qquad $x = a_3$ \\
296 \hbox{}\qquad Constant: \nopagebreak \\
297 \hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
300 As the result of another optimization, Nitpick directly assigned a value to the
301 subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
302 disable this second optimization by using the command
305 \textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
306 \textit{show\_consts}]
309 we finally get \textit{The}:
312 \slshape Constant: \nopagebreak \\
313 \hbox{}\qquad $\mathit{The} = \undef{}
314 (\!\begin{aligned}[t]%
315 & \{a_1, a_2, a_3\} := a_3,\> \{a_1, a_2\} := a_3,\> \{a_1, a_3\} := a_3, \\[-2pt] %% TYPESETTING
316 & \{a_1\} := a_1,\> \{a_2, a_3\} := a_1,\> \{a_2\} := a_2, \\[-2pt]
317 & \{a_3\} := a_3,\> \{\} := a_3)\end{aligned}$
320 Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
321 just like before.\footnote{The Isabelle/HOL notation $f(x :=
322 y)$ denotes the function that maps $x$ to $y$ and that otherwise behaves like
325 Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
326 unique $x$ such that'') at the front of our putative lemma's assumption:
329 \textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
332 The fix appears to work:
335 \textbf{nitpick} \\[2\smallskipamount]
336 \slshape Nitpick found no counterexample.
339 We can further increase our confidence in the formula by exhausting all
340 cardinalities up to 50:
343 \textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
344 can be entered as \texttt{-} (hyphen) or
345 \texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
346 \slshape Nitpick found no counterexample.
349 Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
352 \textbf{sledgehammer} \\[2\smallskipamount]
353 {\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
354 $\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
355 Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
356 \textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
357 {\slshape No subgoals!}% \\[2\smallskipamount]
361 This must be our lucky day.
363 \subsection{Skolemization}
364 \label{skolemization}
366 Are all invertible functions onto? Let's find out:
369 \textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
370 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
371 \textbf{nitpick} \\[2\smallskipamount]
373 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
374 \hbox{}\qquad Free variable: \nopagebreak \\
375 \hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
376 \hbox{}\qquad Skolem constants: \nopagebreak \\
377 \hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
378 \hbox{}\qquad\qquad $y = a_2$
381 Although $f$ is the only free variable occurring in the formula, Nitpick also
382 displays values for the bound variables $g$ and $y$. These values are available
383 to Nitpick because it performs skolemization as a preprocessing step.
385 In the previous example, skolemization only affected the outermost quantifiers.
386 This is not always the case, as illustrated below:
389 \textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
390 \textbf{nitpick} \\[2\smallskipamount]
392 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
393 \hbox{}\qquad Skolem constant: \nopagebreak \\
394 \hbox{}\qquad\qquad $\lambda x.\; f =
395 \undef{}(\!\begin{aligned}[t]
396 & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
397 & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
400 The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
401 $x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
402 function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
403 maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
405 The source of the Skolem constants is sometimes more obscure:
408 \textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
409 \textbf{nitpick} \\[2\smallskipamount]
411 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
412 \hbox{}\qquad Free variable: \nopagebreak \\
413 \hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
414 \hbox{}\qquad Skolem constants: \nopagebreak \\
415 \hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
416 \hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
419 What happened here is that Nitpick expanded the \textit{sym} constant to its
423 $\mathit{sym}~r \,\equiv\,
424 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
427 As their names suggest, the Skolem constants $\mathit{sym}.x$ and
428 $\mathit{sym}.y$ are simply the bound variables $x$ and $y$
429 from \textit{sym}'s definition.
431 \subsection{Natural Numbers and Integers}
432 \label{natural-numbers-and-integers}
434 Because of the axiom of infinity, the type \textit{nat} does not admit any
435 finite models. To deal with this, Nitpick's approach is to consider finite
436 subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined
437 value (displayed as `$\unk$'). The type \textit{int} is handled similarly.
438 Internally, undefined values lead to a three-valued logic.
440 Here is an example involving \textit{int\/}:
443 \textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
444 \textbf{nitpick} \\[2\smallskipamount]
445 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
446 \hbox{}\qquad Free variables: \nopagebreak \\
447 \hbox{}\qquad\qquad $i = 0$ \\
448 \hbox{}\qquad\qquad $j = 1$ \\
449 \hbox{}\qquad\qquad $m = 1$ \\
450 \hbox{}\qquad\qquad $n = 0$
453 Internally, Nitpick uses either a unary or a binary representation of numbers.
454 The unary representation is more efficient but only suitable for numbers very
455 close to zero. By default, Nitpick attempts to choose the more appropriate
456 encoding by inspecting the formula at hand. This behavior can be overridden by
457 passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For
458 binary notation, the number of bits to use can be specified using
459 the \textit{bits} option. For example:
462 \textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$]
465 With infinite types, we don't always have the luxury of a genuine counterexample
466 and must often content ourselves with a potential one. The tedious task of
467 finding out whether the potential counterexample is in fact genuine can be
468 outsourced to \textit{auto} by passing \textit{check\_potential}. For example:
471 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
472 \textbf{nitpick} [\textit{card~nat}~= 50, \textit{check\_potential}] \\[2\smallskipamount]
473 \slshape Warning: The conjecture either trivially holds for the given scopes or lies outside Nitpick's supported
474 fragment. Only potential counterexamples may be found. \\[2\smallskipamount]
475 Nitpick found a potential counterexample: \\[2\smallskipamount]
476 \hbox{}\qquad Free variable: \nopagebreak \\
477 \hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
478 Confirmation by ``\textit{auto}'': The above counterexample is genuine.
481 You might wonder why the counterexample is first reported as potential. The root
482 of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
483 \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
484 that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
485 \textit{False}; but otherwise, it does not know anything about values of $n \ge
486 \textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
487 \textit{True}. Since the assumption can never be satisfied, the putative lemma
488 can never be falsified.
490 Incidentally, if you distrust the so-called genuine counterexamples, you can
491 enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
492 aware that \textit{auto} will usually fail to prove that the counterexample is
495 Some conjectures involving elementary number theory make Nitpick look like a
496 giant with feet of clay:
499 \textbf{lemma} ``$P~\textit{Suc}$'' \\
500 \textbf{nitpick} \\[2\smallskipamount]
502 Nitpick found no counterexample.
505 On any finite set $N$, \textit{Suc} is a partial function; for example, if $N =
506 \{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\,
507 \ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when passed as
508 argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$. The next
512 \textbf{lemma} ``$P~(\textit{op}~{+}\Colon
513 \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
514 \textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
515 {\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
516 \hbox{}\qquad Free variable: \nopagebreak \\
517 \hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
518 \textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
519 {\slshape Nitpick found no counterexample.}
522 The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
523 $\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
526 Because numbers are infinite and are approximated using a three-valued logic,
527 there is usually no need to systematically enumerate domain sizes. If Nitpick
528 cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
529 unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
530 example above is an exception to this principle.) Nitpick nonetheless enumerates
531 all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
532 cardinalities are fast to handle and give rise to simpler counterexamples. This
533 is explained in more detail in \S\ref{scope-monotonicity}.
535 \subsection{Inductive Datatypes}
536 \label{inductive-datatypes}
538 Like natural numbers and integers, inductive datatypes with recursive
539 constructors admit no finite models and must be approximated by a subterm-closed
540 subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
541 Nitpick looks for all counterexamples that can be built using at most 10
544 Let's see with an example involving \textit{hd} (which returns the first element
545 of a list) and $@$ (which concatenates two lists):
548 \textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
549 \textbf{nitpick} \\[2\smallskipamount]
550 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
551 \hbox{}\qquad Free variables: \nopagebreak \\
552 \hbox{}\qquad\qquad $\textit{xs} = []$ \\
553 \hbox{}\qquad\qquad $\textit{y} = a_1$
556 To see why the counterexample is genuine, we enable \textit{show\_consts}
557 and \textit{show\_\allowbreak datatypes}:
560 {\slshape Datatype:} \\
561 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\
562 {\slshape Constants:} \\
563 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_1, a_1])$ \\
564 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$
567 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
570 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
571 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_1,
572 a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not
573 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
574 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
575 appending $[a_1, a_1]$ to itself gives $\unk$.
577 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
578 considers the following subsets:
580 \kern-.5\smallskipamount %% TYPESETTING
584 $\{[],\, [a_1],\, [a_2]\}$; \\
585 $\{[],\, [a_1],\, [a_3]\}$; \\
586 $\{[],\, [a_2],\, [a_3]\}$; \\
587 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\
588 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\
589 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\
590 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\
591 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\
592 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\
593 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\
594 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\
595 $\{[],\, [a_3],\, [a_3, a_3]\}$.
599 \kern-2\smallskipamount %% TYPESETTING
601 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
602 are listed and only those. As an example of a non-subterm-closed subset,
603 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe
604 that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin
605 \mathcal{S}$ as a subterm.
607 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
610 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
611 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
613 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
614 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
615 \hbox{}\qquad Free variables: \nopagebreak \\
616 \hbox{}\qquad\qquad $\textit{xs} = [a_1]$ \\
617 \hbox{}\qquad\qquad $\textit{ys} = [a_2]$ \\
618 \hbox{}\qquad Datatypes: \\
619 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
620 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$
623 Because datatypes are approximated using a three-valued logic, there is usually
624 no need to systematically enumerate cardinalities: If Nitpick cannot find a
625 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
626 unlikely that one could be found for smaller cardinalities.
628 \subsection{Typedefs, Quotient Types, Records, Rationals, and Reals}
629 \label{typedefs-quotient-types-records-rationals-and-reals}
631 Nitpick generally treats types declared using \textbf{typedef} as datatypes
632 whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
636 \textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
637 \textbf{by}~\textit{blast} \\[2\smallskipamount]
638 \textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
639 \textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
640 \textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
641 \textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
642 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
643 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
644 \hbox{}\qquad Free variables: \nopagebreak \\
645 \hbox{}\qquad\qquad $P = \{\Abs{0},\, \Abs{1}\}$ \\
646 \hbox{}\qquad\qquad $x = \Abs{2}$ \\
647 \hbox{}\qquad Datatypes: \\
648 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
649 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$
652 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
654 Quotient types are handled in much the same way. The following fragment defines
655 the integer type \textit{my\_int} by encoding the integer $x$ by a pair of
656 natural numbers $(m, n)$ such that $x + n = m$:
659 \textbf{fun} \textit{my\_int\_rel} \textbf{where} \\
660 ``$\textit{my\_int\_rel}~(x,\, y)~(u,\, v) = (x + v = u + y)$'' \\[2\smallskipamount]
662 \textbf{quotient\_type}~\textit{my\_int} = ``$\textit{nat} \times \textit{nat\/}$''$\;{/}\;$\textit{my\_int\_rel} \\
663 \textbf{by}~(\textit{auto simp add\/}:\ \textit{equivp\_def expand\_fun\_eq}) \\[2\smallskipamount]
665 \textbf{definition}~\textit{add\_raw}~\textbf{where} \\
666 ``$\textit{add\_raw} \,\equiv\, \lambda(x,\, y)~(u,\, v).\; (x + (u\Colon\textit{nat}), y + (v\Colon\textit{nat}))$'' \\[2\smallskipamount]
668 \textbf{quotient\_definition} ``$\textit{add\/}\Colon\textit{my\_int} \Rightarrow \textit{my\_int} \Rightarrow \textit{my\_int\/}$'' \textbf{is} \textit{add\_raw} \\[2\smallskipamount]
670 \textbf{lemma} ``$\textit{add}~x~y = \textit{add}~x~x$'' \\
671 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
672 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
673 \hbox{}\qquad Free variables: \nopagebreak \\
674 \hbox{}\qquad\qquad $x = \Abs{(0,\, 0)}$ \\
675 \hbox{}\qquad\qquad $y = \Abs{(1,\, 0)}$ \\
676 \hbox{}\qquad Datatypes: \\
677 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, \unr\}$ \\
678 \hbox{}\qquad\qquad $\textit{nat} \times \textit{nat}~[\textsl{boxed\/}] = \{(0,\, 0),\> (1,\, 0),\> \unr\}$ \\
679 \hbox{}\qquad\qquad $\textit{my\_int} = \{\Abs{(0,\, 0)},\> \Abs{(1,\, 0)},\> \unr\}$
682 In the counterexample, $\Abs{(0,\, 0)}$ and $\Abs{(1,\, 0)}$ represent the
683 integers $0$ and $1$, respectively. Other representants would have been
684 possible---e.g., $\Abs{(5,\, 5)}$ and $\Abs{(12,\, 11)}$. If we are going to
685 use \textit{my\_int} extensively, it pays off to install a term postprocessor
686 that converts the pair notation to the standard mathematical notation:
689 $\textbf{ML}~\,\{{*} \\
691 %& ({*}~\,\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \\[-2pt]
692 %& \phantom{(*}~\,{\rightarrow}\;\textit{term}~\,{*}) \\[-2pt]
693 & \textbf{fun}\,~\textit{my\_int\_postproc}~\_~\_~\_~T~(\textit{Const}~\_~\$~(\textit{Const}~\_~\$~\textit{t1}~\$~\textit{t2\/})) = {} \\[-2pt]
694 & \phantom{fun}\,~\textit{HOLogic.mk\_number}~T~(\textit{snd}~(\textit{HOLogic.dest\_number~t1}) \\[-2pt]
695 & \phantom{fun\,~\textit{HOLogic.mk\_number}~T~(}{-}~\textit{snd}~(\textit{HOLogic.dest\_number~t2\/})) \\[-2pt]
696 & \phantom{fun}\!{\mid}\,~\textit{my\_int\_postproc}~\_~\_~\_~\_~t = t \\[-2pt]
697 {*}\}\end{aligned}$ \\[2\smallskipamount]
698 $\textbf{setup}~\,\{{*} \\
700 & \textit{Nitpick.register\_term\_postprocessor}~\!\begin{aligned}[t]
701 & @\{\textrm{typ}~\textit{my\_int}\}~\textit{my\_int\_postproc}\end{aligned} \\[-2pt]
705 Records are also handled as datatypes with a single constructor:
708 \textbf{record} \textit{point} = \\
709 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
710 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
711 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
712 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
713 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
714 \hbox{}\qquad Free variables: \nopagebreak \\
715 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
716 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
717 \hbox{}\qquad Datatypes: \\
718 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
719 \hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
720 & \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
721 & \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
724 Finally, Nitpick provides rudimentary support for rationals and reals using a
728 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
729 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
730 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
731 \hbox{}\qquad Free variables: \nopagebreak \\
732 \hbox{}\qquad\qquad $x = 1/2$ \\
733 \hbox{}\qquad\qquad $y = -1/2$ \\
734 \hbox{}\qquad Datatypes: \\
735 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
736 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
737 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
740 \subsection{Inductive and Coinductive Predicates}
741 \label{inductive-and-coinductive-predicates}
743 Inductively defined predicates (and sets) are particularly problematic for
744 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
745 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
746 the problem is that they are defined using a least fixed point construction.
748 Nitpick's philosophy is that not all inductive predicates are equal. Consider
749 the \textit{even} predicate below:
752 \textbf{inductive}~\textit{even}~\textbf{where} \\
753 ``\textit{even}~0'' $\,\mid$ \\
754 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
757 This predicate enjoys the desirable property of being well-founded, which means
758 that the introduction rules don't give rise to infinite chains of the form
761 $\cdots\,\Longrightarrow\, \textit{even}~k''
762 \,\Longrightarrow\, \textit{even}~k'
763 \,\Longrightarrow\, \textit{even}~k.$
766 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
770 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
771 \,\Longrightarrow\, \textit{even}~(k - 2)
772 \,\Longrightarrow\, \textit{even}~k.$
775 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
776 fixed point computation.%
777 \footnote{If an inductive predicate is
778 well-founded, then it has exactly one fixed point, which is simultaneously the
779 least and the greatest fixed point. In these circumstances, the computation of
780 the least fixed point amounts to the computation of an arbitrary fixed point,
781 which can be performed using a straightforward recursive equation.}
782 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
783 just as Isabelle's \textbf{function} package usually discharges termination
784 proof obligations automatically.
786 Let's try an example:
789 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
790 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
791 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
792 Nitpick can compute it efficiently. \\[2\smallskipamount]
794 \hbox{}\qquad \textit{card nat}~= 50. \\[2\smallskipamount]
795 Nitpick found a potential counterexample for \textit{card nat}~= 50: \\[2\smallskipamount]
796 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
797 Nitpick could not find a better counterexample. \\[2\smallskipamount]
801 No genuine counterexample is possible because Nitpick cannot rule out the
802 existence of a natural number $n \ge 50$ such that both $\textit{even}~n$ and
803 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
804 existential quantifier:
807 \textbf{lemma} ``$\exists n \mathbin{\le} 49.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
808 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}] \\[2\smallskipamount]
809 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
810 \hbox{}\qquad Empty assignment
813 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
814 we use the following definition instead?
817 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
818 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
819 ``$\textit{even}'~2$'' $\,\mid$ \\
820 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
823 This definition is not well-founded: From $\textit{even}'~0$ and
824 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
825 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
827 Let's check a property involving $\textit{even}'$. To make up for the
828 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
829 \textit{nat}'s cardinality to a mere 10:
832 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
833 \lnot\;\textit{even}'~n$'' \\
834 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
836 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
837 Nitpick might need to unroll it. \\[2\smallskipamount]
839 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
840 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
841 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
842 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
843 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
844 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
845 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
846 \hbox{}\qquad Constant: \nopagebreak \\
847 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
848 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
849 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
850 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
854 Nitpick's output is very instructive. First, it tells us that the predicate is
855 unrolled, meaning that it is computed iteratively from the empty set. Then it
856 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
859 The output also shows how each iteration contributes to $\textit{even}'$. The
860 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
861 predicate depends on an iteration counter. Iteration 0 provides the basis
862 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
863 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
864 iterations would not contribute any new elements.
866 Some values are marked with superscripted question
867 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
868 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
869 \textit{True} or $\unk$, never \textit{False}.
871 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
872 iterations. However, these numbers are bounded by the cardinality of the
873 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
874 ever needed to compute the value of a \textit{nat} predicate. You can specify
875 the number of iterations using the \textit{iter} option, as explained in
876 \S\ref{scope-of-search}.
878 In the next formula, $\textit{even}'$ occurs both positively and negatively:
881 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
882 \textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
883 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
884 \hbox{}\qquad Free variable: \nopagebreak \\
885 \hbox{}\qquad\qquad $n = 1$ \\
886 \hbox{}\qquad Constants: \nopagebreak \\
887 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
888 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
889 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
892 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
893 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
894 fixed point (not necessarily the least one). It is used to falsify
895 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
896 $\textit{even}'~(n - 2)$.
898 Coinductive predicates are handled dually. For example:
901 \textbf{coinductive} \textit{nats} \textbf{where} \\
902 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
903 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
904 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
905 \slshape Nitpick found a counterexample:
906 \\[2\smallskipamount]
907 \hbox{}\qquad Constants: \nopagebreak \\
908 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
909 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
910 & \unr\})\end{aligned}$ \\
911 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
914 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
915 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
916 inductive predicates for which each the predicate occurs in at most one
917 assumption of each introduction rule. For example:
920 \textbf{inductive} \textit{odd} \textbf{where} \\
921 ``$\textit{odd}~1$'' $\,\mid$ \\
922 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
923 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
924 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
925 \slshape Nitpick found a counterexample:
926 \\[2\smallskipamount]
927 \hbox{}\qquad Free variable: \nopagebreak \\
928 \hbox{}\qquad\qquad $n = 1$ \\
929 \hbox{}\qquad Constants: \nopagebreak \\
930 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
931 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
932 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
934 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
935 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
937 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
938 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
939 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
943 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
944 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
945 elements from known ones. The set $\textit{odd}$ consists of all the values
946 reachable through the reflexive transitive closure of
947 $\textit{odd}_{\textrm{step}}$ starting with any element from
948 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
949 transitive closure to encode linear predicates is normally either more thorough
950 or more efficient than unrolling (depending on the value of \textit{iter}), but
951 for those cases where it isn't you can disable it by passing the
952 \textit{dont\_star\_linear\_preds} option.
954 \subsection{Coinductive Datatypes}
955 \label{coinductive-datatypes}
957 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
958 datatypes, the \textit{Coinductive\_List} theory from Andreas Lochbihler's
959 \textit{Coinductive} AFP entry \cite{lochbihler-2010} provides a coinductive
960 ``lazy list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick
961 supports these lazy lists seamlessly and provides a hook, described in
962 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
965 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
966 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
967 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
968 1, 2, 3, \ldots]$ can be defined as lazy lists using the
969 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
970 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
971 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
973 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
974 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
978 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
979 \textbf{nitpick} \\[2\smallskipamount]
980 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
981 \hbox{}\qquad Free variables: \nopagebreak \\
982 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
983 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
986 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
987 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
988 infinite list $[a_1, a_1, a_1, \ldots]$.
990 The next example is more interesting:
993 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
994 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
995 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
996 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
997 some scopes. \\[2\smallskipamount]
999 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 1,
1000 and \textit{bisim\_depth}~= 0. \\
1001 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1002 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 8,
1003 and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
1004 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
1005 \textit{card}~``\kern1pt$'a~\textit{list\/}$''~= 2, and \textit{bisim\_\allowbreak
1007 \\[2\smallskipamount]
1008 \hbox{}\qquad Free variables: \nopagebreak \\
1009 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
1010 \hbox{}\qquad\qquad $\textit{b} = a_2$ \\
1011 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1012 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
1016 The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
1017 $\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
1018 $[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment
1019 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
1020 the segment leading to the binder is the stem.
1022 A salient property of coinductive datatypes is that two objects are considered
1023 equal if and only if they lead to the same observations. For example, the lazy
1024 lists $\textrm{THE}~\omega.\; \omega =
1025 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
1026 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
1027 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
1028 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
1029 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
1030 concept of equality for coinductive datatypes is called bisimulation and is
1031 defined coinductively.
1033 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
1034 the Kodkod problem to ensure that distinct objects lead to different
1035 observations. This precaution is somewhat expensive and often unnecessary, so it
1036 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
1037 bisimilarity check is then performed \textsl{after} the counterexample has been
1038 found to ensure correctness. If this after-the-fact check fails, the
1039 counterexample is tagged as ``quasi genuine'' and Nitpick recommends to try
1040 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
1041 check for the previous example saves approximately 150~milli\-seconds; the speed
1042 gains can be more significant for larger scopes.
1044 The next formula illustrates the need for bisimilarity (either as a Kodkod
1045 predicate or as an after-the-fact check) to prevent spurious counterexamples:
1048 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
1049 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1050 \textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
1051 \slshape Nitpick found a quasi genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
1052 \hbox{}\qquad Free variables: \nopagebreak \\
1053 \hbox{}\qquad\qquad $a = a_1$ \\
1054 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
1055 \textit{LCons}~a_1~\omega$ \\
1056 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1057 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
1058 \hbox{}\qquad\qquad $'a~\textit{llist} =
1059 \{\!\begin{aligned}[t]
1060 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
1061 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
1062 \\[2\smallskipamount]
1063 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1064 that the counterexample is genuine. \\[2\smallskipamount]
1065 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1066 \slshape Nitpick found no counterexample.
1069 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1070 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1072 A compromise between leaving out the bisimilarity predicate from the Kodkod
1073 problem and performing the after-the-fact check is to specify a lower
1074 nonnegative \textit{bisim\_depth} value than the default one provided by
1075 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1076 be distinguished from each other by their prefixes of length $K$. Be aware that
1077 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1078 finding any counterexamples.
1083 Nitpick normally maps function and product types directly to the corresponding
1084 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1085 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1086 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1087 off to treat these types in the same way as plain datatypes, by approximating
1088 them by a subset of a given cardinality. This technique is called ``boxing'' and
1089 is particularly useful for functions passed as arguments to other functions, for
1090 high-arity functions, and for large tuples. Under the hood, boxing involves
1091 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1092 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1094 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1095 using de Bruijn's notation:
1098 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1101 The $\textit{lift}~t~k$ function increments all variables with indices greater
1102 than or equal to $k$ by one:
1105 \textbf{primrec} \textit{lift} \textbf{where} \\
1106 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1107 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1108 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1111 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1112 term $t$ has a loose variable with index $k$ or more:
1115 \textbf{primrec}~\textit{loose} \textbf{where} \\
1116 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1117 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1118 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1121 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1125 \textbf{primrec}~\textit{subst} \textbf{where} \\
1126 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1127 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1128 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1129 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1132 A substitution is a function that maps variable indices to terms. Observe that
1133 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1134 away, because the recursive call for the \textit{Lam} case involves an altered
1135 version. Also notice the \textit{lift} call, which increments the variable
1136 indices when moving under a \textit{Lam}.
1138 A reasonable property to expect of substitution is that it should leave closed
1139 terms unchanged. Alas, even this simple property does not hold:
1142 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1143 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1145 Trying 8 scopes: \nopagebreak \\
1146 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1147 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1148 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1149 \hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
1150 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1151 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1152 \hbox{}\qquad Free variables: \nopagebreak \\
1153 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1154 & 0 := \textit{Var}~0,\>
1155 1 := \textit{Var}~0,\>
1156 2 := \textit{Var}~0, \\[-2pt]
1157 & 3 := \textit{Var}~0,\>
1158 4 := \textit{Var}~0,\>
1159 5 := \textit{Var}~0)\end{aligned}$ \\
1160 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1161 Total time: $4679$ ms.
1164 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1165 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1166 $\lambda$-term notation, $t$~is
1167 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1168 The bug is in \textit{subst\/}: The $\textit{lift}~(\sigma~m)~1$ call should be
1169 replaced with $\textit{lift}~(\sigma~m)~0$.
1171 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1172 cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1173 For the formula of interest, knowing 6 values of that type was enough to find
1174 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1175 considered, a hopeless undertaking:
1178 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1179 {\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
1183 Boxing can be enabled or disabled globally or on a per-type basis using the
1184 \textit{box} option. Nitpick usually performs reasonable choices about which
1185 types should be boxed, but option tweaking sometimes helps. A related optimization,
1186 ``finalization,'' attempts to wrap functions that constant at all but finitely
1187 many points (e.g., finite sets); see the documentation for the \textit{finalize}
1188 option in \S\ref{scope-of-search} for details.
1192 \subsection{Scope Monotonicity}
1193 \label{scope-monotonicity}
1195 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1196 and \textit{max}) controls which scopes are actually tested. In general, to
1197 exhaust all models below a certain cardinality bound, the number of scopes that
1198 Nitpick must consider increases exponentially with the number of type variables
1199 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1200 cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
1201 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1203 Fortunately, many formulas exhibit a property called \textsl{scope
1204 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1205 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1207 Consider the formula
1210 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1213 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1214 $'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
1215 exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
1216 that any counterexample found with a small scope would still be a counterexample
1217 in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
1218 by the larger scope. Nitpick comes to the same conclusion after a careful
1219 inspection of the formula and the relevant definitions:
1222 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1224 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1225 Nitpick might be able to skip some scopes.
1226 \\[2\smallskipamount]
1228 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1229 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1230 \textit{list\/}''~= 1, \\
1231 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 1, and
1232 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 1. \\
1233 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1234 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1235 \textit{list\/}''~= 2, \\
1236 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 2, and
1237 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 2. \\
1238 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1239 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
1240 \textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
1241 \textit{list\/}''~= 8, \\
1242 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 8, and
1243 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 8.
1244 \\[2\smallskipamount]
1245 Nitpick found a counterexample for
1246 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1247 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1248 \textit{list\/}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 5, and
1249 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 5:
1250 \\[2\smallskipamount]
1251 \hbox{}\qquad Free variables: \nopagebreak \\
1252 \hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
1253 \hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
1254 Total time: 1636 ms.
1257 In theory, it should be sufficient to test a single scope:
1260 \textbf{nitpick}~[\textit{card}~= 8]
1263 However, this is often less efficient in practice and may lead to overly complex
1266 If the monotonicity check fails but we believe that the formula is monotonic (or
1267 we don't mind missing some counterexamples), we can pass the
1268 \textit{mono} option. To convince yourself that this option is risky,
1269 simply consider this example from \S\ref{skolemization}:
1272 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1273 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1274 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1275 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1276 \textbf{nitpick} \\[2\smallskipamount]
1278 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1279 \hbox{}\qquad $\vdots$
1282 (It turns out the formula holds if and only if $\textit{card}~'a \le
1283 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1284 monotonicity checks can be disabled by passing \textit{non\_mono}
1285 (\S\ref{optimizations}).
1287 As insinuated in \S\ref{natural-numbers-and-integers} and
1288 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1289 are normally monotonic and treated as such. The same is true for record types,
1290 \textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
1291 cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
1292 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1293 consider only 8~scopes instead of $32\,768$.
1295 \subsection{Inductive Properties}
1296 \label{inductive-properties}
1298 Inductive properties are a particular pain to prove, because the failure to
1299 establish an induction step can mean several things:
1302 \item The property is invalid.
1303 \item The property is valid but is too weak to support the induction step.
1304 \item The property is valid and strong enough; it's just that we haven't found
1308 Depending on which scenario applies, we would take the appropriate course of
1312 \item Repair the statement of the property so that it becomes valid.
1313 \item Generalize the property and/or prove auxiliary properties.
1314 \item Work harder on a proof.
1317 How can we distinguish between the three scenarios? Nitpick's normal mode of
1318 operation can often detect scenario 1, and Isabelle's automatic tactics help with
1319 scenario 3. Using appropriate techniques, it is also often possible to use
1320 Nitpick to identify scenario 2. Consider the following transition system,
1321 in which natural numbers represent states:
1324 \textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
1325 ``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
1326 ``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
1327 ``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
1330 We will try to prove that only even numbers are reachable:
1333 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
1336 Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
1337 so let's attempt a proof by induction:
1340 \textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
1341 \textbf{apply}~\textit{auto}
1344 This leaves us in the following proof state:
1347 {\slshape goal (2 subgoals): \\
1348 \phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
1349 \phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
1353 If we run Nitpick on the first subgoal, it still won't find any
1354 counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
1355 is helpless. However, notice the $n \in \textit{reach}$ assumption, which
1356 strengthens the induction hypothesis but is not immediately usable in the proof.
1357 If we remove it and invoke Nitpick, this time we get a counterexample:
1360 \textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
1361 \textbf{nitpick} \\[2\smallskipamount]
1362 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1363 \hbox{}\qquad Skolem constant: \nopagebreak \\
1364 \hbox{}\qquad\qquad $n = 0$
1367 Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
1368 to strength the lemma:
1371 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
1374 Unfortunately, the proof by induction still gets stuck, except that Nitpick now
1375 finds the counterexample $n = 2$. We generalize the lemma further to
1378 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
1381 and this time \textit{arith} can finish off the subgoals.
1383 A similar technique can be employed for structural induction. The
1384 following mini formalization of full binary trees will serve as illustration:
1387 \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]
1388 \textbf{primrec}~\textit{labels}~\textbf{where} \\
1389 ``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
1390 ``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
1391 \textbf{primrec}~\textit{swap}~\textbf{where} \\
1392 ``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
1393 \phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
1394 ``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
1397 The \textit{labels} function returns the set of labels occurring on leaves of a
1398 tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
1399 labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
1400 obtained by swapping $a$ and $b$:
1403 \textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
1406 Nitpick can't find any counterexample, so we proceed with induction
1407 (this time favoring a more structured style):
1410 \textbf{proof}~(\textit{induct}~$t$) \\
1411 \hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
1413 \hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
1416 Nitpick can't find any counterexample at this point either, but it makes the
1417 following suggestion:
1421 Hint: To check that the induction hypothesis is general enough, try this command:
1422 \textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
1425 If we follow the hint, we get a ``nonstandard'' counterexample for the step:
1428 \slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
1429 \hbox{}\qquad Free variables: \nopagebreak \\
1430 \hbox{}\qquad\qquad $a = a_1$ \\
1431 \hbox{}\qquad\qquad $b = a_2$ \\
1432 \hbox{}\qquad\qquad $t = \xi_1$ \\
1433 \hbox{}\qquad\qquad $u = \xi_2$ \\
1434 \hbox{}\qquad Datatype: \nopagebreak \\
1435 \hbox{}\qquad\qquad $\alpha~\textit{btree} = \{\xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \textit{Branch}~\xi_1~\xi_2\}$ \\
1436 \hbox{}\qquad {\slshape Constants:} \nopagebreak \\
1437 \hbox{}\qquad\qquad $\textit{labels} = \undef
1438 (\!\begin{aligned}[t]%
1439 & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
1440 & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
1441 \hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \undef
1442 (\!\begin{aligned}[t]%
1443 & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
1444 & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount]
1445 The existence of a nonstandard model suggests that the induction hypothesis is not general enough or may even
1446 be wrong. See the Nitpick manual's ``Inductive Properties'' section for details (``\textit{isabelle doc nitpick}'').
1449 Reading the Nitpick manual is a most excellent idea.
1450 But what's going on? The \textit{non\_std} option told the tool to look for
1451 nonstandard models of binary trees, which means that new ``nonstandard'' trees
1452 $\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees
1453 generated by the \textit{Leaf} and \textit{Branch} constructors.%
1454 \footnote{Notice the similarity between allowing nonstandard trees here and
1455 allowing unreachable states in the preceding example (by removing the ``$n \in
1456 \textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
1457 set of objects over which the induction is performed while doing the step
1458 in order to test the induction hypothesis's strength.}
1459 Unlike standard trees, these new trees contain cycles. We will see later that
1460 every property of acyclic trees that can be proved without using induction also
1461 holds for cyclic trees. Hence,
1464 \textsl{If the induction
1465 hypothesis is strong enough, the induction step will hold even for nonstandard
1466 objects, and Nitpick won't find any nonstandard counterexample.}
1469 But here the tool find some nonstandard trees $t = \xi_1$
1470 and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in
1471 \textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$.
1472 Because neither tree contains both $a$ and $b$, the induction hypothesis tells
1473 us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
1474 and as a result we know nothing about the labels of the tree
1475 $\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
1476 $\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
1477 labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
1478 \textit{labels}$ $(\textit{swap}~u~a~b)$.
1480 The solution is to ensure that we always know what the labels of the subtrees
1481 are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
1482 $t$ in the statement of the lemma:
1485 \textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
1486 \phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
1487 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
1488 \phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
1489 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
1492 This time, Nitpick won't find any nonstandard counterexample, and we can perform
1493 the induction step using \textit{auto}.
1495 \section{Case Studies}
1496 \label{case-studies}
1498 As a didactic device, the previous section focused mostly on toy formulas whose
1499 validity can easily be assessed just by looking at the formula. We will now
1500 review two somewhat more realistic case studies that are within Nitpick's
1501 reach:\ a context-free grammar modeled by mutually inductive sets and a
1502 functional implementation of AA trees. The results presented in this
1503 section were produced with the following settings:
1506 \textbf{nitpick\_params} [\textit{max\_potential}~= 0]
1509 \subsection{A Context-Free Grammar}
1510 \label{a-context-free-grammar}
1512 Our first case study is taken from section 7.4 in the Isabelle tutorial
1513 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1514 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1517 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1518 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1519 $A$ & $::=$ & $aS \mid bAA$ \\
1520 $B$ & $::=$ & $bS \mid aBB$
1524 The intuition behind the grammar is that $A$ generates all string with one more
1525 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1527 The alphabet consists exclusively of $a$'s and $b$'s:
1530 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1533 Strings over the alphabet are represented by \textit{alphabet list}s.
1534 Nonterminals in the grammar become sets of strings. The production rules
1535 presented above can be expressed as a mutually inductive definition:
1538 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1539 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1540 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1541 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1542 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1543 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1544 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1547 The conversion of the grammar into the inductive definition was done manually by
1548 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1551 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1552 d'\^etre}. A good approach is to state desirable properties of the specification
1553 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1554 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1555 counterexamples will point to bugs in the specification. For our grammar
1556 example, we will proceed in two steps, separating the soundness and the
1557 completeness of the set $S$. First, soundness:
1560 \textbf{theorem}~\textit{S\_sound\/}: \\
1561 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1562 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1563 \textbf{nitpick} \\[2\smallskipamount]
1564 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1565 \hbox{}\qquad Free variable: \nopagebreak \\
1566 \hbox{}\qquad\qquad $w = [b]$
1569 It would seem that $[b] \in S$. How could this be? An inspection of the
1570 introduction rules reveals that the only rule with a right-hand side of the form
1571 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1575 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1578 On closer inspection, we can see that this rule is wrong. To match the
1579 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1583 \textbf{nitpick} \\[2\smallskipamount]
1584 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1585 \hbox{}\qquad Free variable: \nopagebreak \\
1586 \hbox{}\qquad\qquad $w = [a, a, b]$
1589 Some detective work is necessary to find out what went wrong here. To get $[a,
1590 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1594 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1597 Now, this formula must be wrong: The same assumption occurs twice, and the
1598 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1599 the assumptions should have been a $w$.
1601 With the correction made, we don't get any counterexample from Nitpick. Let's
1602 move on and check completeness:
1605 \textbf{theorem}~\textit{S\_complete}: \\
1606 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1607 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1608 \longrightarrow w \in S$'' \\
1609 \textbf{nitpick} \\[2\smallskipamount]
1610 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1611 \hbox{}\qquad Free variable: \nopagebreak \\
1612 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1615 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1616 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1617 the introduction rules we have are probably correct. Perhaps we simply lack an
1618 introduction rule. Comparing the grammar with the inductive definition, our
1619 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1620 without which the grammar cannot generate two or more $b$'s in a row. So we add
1624 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1627 With this last change, we don't get any counterexamples from Nitpick for either
1628 soundness or completeness. We can even generalize our result to cover $A$ and
1632 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1633 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1634 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1635 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1636 \textbf{nitpick} \\[2\smallskipamount]
1637 \slshape Nitpick ran out of time after checking 7 of 8 scopes.
1640 \subsection{AA Trees}
1643 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1644 similar performance to red-black trees, but with a simpler implementation
1645 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1646 a total order $<$. We start by defining the datatype and some basic extractor
1650 \textbf{datatype} $'a$~\textit{aa\_tree} = \\
1651 \hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}'' \\[2\smallskipamount]
1652 \textbf{primrec} \textit{data} \textbf{where} \\
1653 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1654 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1655 \textbf{primrec} \textit{dataset} \textbf{where} \\
1656 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1657 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1658 \textbf{primrec} \textit{level} \textbf{where} \\
1659 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1660 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1661 \textbf{primrec} \textit{left} \textbf{where} \\
1662 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1663 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1664 \textbf{primrec} \textit{right} \textbf{where} \\
1665 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1666 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1669 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1670 as follows \cite{wikipedia-2009-aa-trees}:
1672 \kern.2\parskip %% TYPESETTING
1675 Each node has a level field, and the following invariants must remain true for
1676 the tree to be valid:
1680 \kern-.4\parskip %% TYPESETTING
1685 \item[1.] The level of a leaf node is one.
1686 \item[2.] The level of a left child is strictly less than that of its parent.
1687 \item[3.] The level of a right child is less than or equal to that of its parent.
1688 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1689 \item[5.] Every node of level greater than one must have two children.
1694 \kern.4\parskip %% TYPESETTING
1696 The \textit{wf} predicate formalizes this description:
1699 \textbf{primrec} \textit{wf} \textbf{where} \\
1700 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1701 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1702 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1703 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1704 \phantom{``$($}$\textrm{else}$ \\
1705 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1706 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1707 \mathrel{\land} \textit{level}~u \le k$ \\
1708 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1711 Rebalancing the tree upon insertion and removal of elements is performed by two
1712 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1715 \textbf{primrec} \textit{skew} \textbf{where} \\
1716 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1717 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1718 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1719 \textit{level}~t~\textrm{then}$ \\
1720 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1721 (\textit{right}~t)~u)$ \\
1722 \phantom{``(}$\textrm{else}$ \\
1723 \phantom{``(\quad}$N~x~k~t~u)$''
1727 \textbf{primrec} \textit{split} \textbf{where} \\
1728 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1729 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1730 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1731 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1732 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1733 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1734 \phantom{``(}$\textrm{else}$ \\
1735 \phantom{``(\quad}$N~x~k~t~u)$''
1738 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1739 of elements stored in the tree:
1742 \textbf{theorem}~\textit{dataset\_skew\_split\/}:\\
1743 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1744 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1745 \textbf{nitpick} \\[2\smallskipamount]
1746 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1749 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1750 should not alter the tree:
1753 \textbf{theorem}~\textit{wf\_skew\_split\/}:\\
1754 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1755 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1756 \textbf{nitpick} \\[2\smallskipamount]
1757 {\slshape Nitpick found no counterexample.}
1760 Insertion is implemented recursively. It preserves the sort order:
1763 \textbf{primrec}~\textit{insort} \textbf{where} \\
1764 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1765 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1766 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1767 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1770 Notice that we deliberately commented out the application of \textit{skew} and
1771 \textit{split}. Let's see if this causes any problems:
1774 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1775 \textbf{nitpick} \\[2\smallskipamount]
1776 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1777 \hbox{}\qquad Free variables: \nopagebreak \\
1778 \hbox{}\qquad\qquad $t = N~a_1~1~\Lambda~\Lambda$ \\
1779 \hbox{}\qquad\qquad $x = a_2$
1782 It's hard to see why this is a counterexample. To improve readability, we will
1783 restrict the theorem to \textit{nat}, so that we don't need to look up the value
1784 of the $\textit{op}~{<}$ constant to find out which element is smaller than the
1785 other. In addition, we will tell Nitpick to display the value of
1786 $\textit{insort}~t~x$ using the \textit{eval} option. This gives
1789 \textbf{theorem} \textit{wf\_insort\_nat\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1790 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1791 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1792 \hbox{}\qquad Free variables: \nopagebreak \\
1793 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1794 \hbox{}\qquad\qquad $x = 0$ \\
1795 \hbox{}\qquad Evaluated term: \\
1796 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1799 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1800 where both have a level of 1. This violates the second AA tree invariant, which
1801 states that a left child's level must be less than its parent's. This shouldn't
1802 come as a surprise, considering that we commented out the tree rebalancing code.
1803 Reintroducing the code seems to solve the problem:
1806 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1807 \textbf{nitpick} \\[2\smallskipamount]
1808 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1811 Insertion should transform the set of elements represented by the tree in the
1815 \textbf{theorem} \textit{dataset\_insort\/}:\kern.4em
1816 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1817 \textbf{nitpick} \\[2\smallskipamount]
1818 {\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
1821 We could continue like this and sketch a complete theory of AA trees. Once the
1822 definitions and main theorems are in place and have been thoroughly tested using
1823 Nitpick, we could start working on the proofs. Developing theories this way
1824 usually saves time, because faulty theorems and definitions are discovered much
1825 earlier in the process.
1827 \section{Option Reference}
1828 \label{option-reference}
1830 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1831 \def\qty#1{$\left<\textit{#1}\right>$}
1832 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1833 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1834 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1835 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1836 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1837 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1838 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1839 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1840 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1842 Nitpick's behavior can be influenced by various options, which can be specified
1843 in brackets after the \textbf{nitpick} command. Default values can be set
1844 using \textbf{nitpick\_\allowbreak params}. For example:
1847 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1850 The options are categorized as follows:\ mode of operation
1851 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1852 format (\S\ref{output-format}), automatic counterexample checks
1853 (\S\ref{authentication}), optimizations
1854 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1856 You can instruct Nitpick to run automatically on newly entered theorems by
1857 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1858 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
1859 and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1860 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1861 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1862 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1863 \textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
1864 Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
1867 The number of options can be overwhelming at first glance. Do not let that worry
1868 you: Nitpick's defaults have been chosen so that it almost always does the right
1869 thing, and the most important options have been covered in context in
1870 \S\ref{first-steps}.
1872 The descriptions below refer to the following syntactic quantities:
1875 \item[$\bullet$] \qtybf{string}: A string.
1876 \item[$\bullet$] \qtybf{bool\/}: \textit{true} or \textit{false}.
1877 \item[$\bullet$] \qtybf{bool\_or\_smart\/}: \textit{true}, \textit{false}, or \textit{smart}.
1878 \item[$\bullet$] \qtybf{int\/}: An integer. Negative integers are prefixed with a hyphen.
1879 \item[$\bullet$] \qtybf{int\_or\_smart\/}: An integer or \textit{smart}.
1880 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1881 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1883 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1884 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1885 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1886 \item[$\bullet$] \qtybf{const\/}: The name of a HOL constant.
1887 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1888 \item[$\bullet$] \qtybf{term\_list\/}: A space-separated list of HOL terms (e.g.,
1889 ``$f~x$''~``$g~y$'').
1890 \item[$\bullet$] \qtybf{type}: A HOL type.
1893 Default values are indicated in square brackets. Boolean options have a negated
1894 counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
1895 Boolean options, ``= \textit{true}'' may be omitted.
1897 \subsection{Mode of Operation}
1898 \label{mode-of-operation}
1901 \optrue{blocking}{non\_blocking}
1902 Specifies whether the \textbf{nitpick} command should operate synchronously.
1903 The asynchronous (non-blocking) mode lets the user start proving the putative
1904 theorem while Nitpick looks for a counterexample, but it can also be more
1905 confusing. For technical reasons, automatic runs currently always block.
1907 \optrue{falsify}{satisfy}
1908 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1909 satisfying examples (models). This manual assumes throughout that
1910 \textit{falsify} is enabled.
1912 \opsmart{user\_axioms}{no\_user\_axioms}
1913 Specifies whether the user-defined axioms (specified using
1914 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1915 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1916 the constants that occur in the formula to falsify. The option is implicitly set
1917 to \textit{true} for automatic runs.
1919 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1920 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1921 these conditions are tagged as ``quasi genuine.'' The \textit{debug}
1922 (\S\ref{output-format}) option can be used to find out which axioms were
1926 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1927 (\S\ref{output-format}).}
1929 \optrue{assms}{no\_assms}
1930 Specifies whether the relevant assumptions in structured proofs should be
1931 considered. The option is implicitly enabled for automatic runs.
1934 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1936 \opfalse{overlord}{no\_overlord}
1937 Specifies whether Nitpick should put its temporary files in
1938 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1939 debugging Nitpick but also unsafe if several instances of the tool are run
1940 simultaneously. The files are identified by the extensions
1941 \texttt{.kki}, \texttt{.cnf}, \texttt{.out}, and
1942 \texttt{.err}; you may safely remove them after Nitpick has run.
1945 {\small See also \textit{debug} (\S\ref{output-format}).}
1948 \subsection{Scope of Search}
1949 \label{scope-of-search}
1952 \oparg{card}{type}{int\_seq}
1953 Specifies the sequence of cardinalities to use for a given type.
1954 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1955 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1958 {\small See also \textit{box} (\S\ref{scope-of-search}) and \textit{mono}
1959 (\S\ref{scope-of-search}).}
1961 \opdefault{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
1962 Specifies the default sequence of cardinalities to use. This can be overridden
1963 on a per-type basis using the \textit{card}~\qty{type} option described above.
1965 \oparg{max}{const}{int\_seq}
1966 Specifies the sequence of maximum multiplicities to use for a given
1967 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1968 number of distinct values that it can construct. Nonsensical values (e.g.,
1969 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1970 datatypes equipped with several constructors.
1972 \opnodefault{max}{int\_seq}
1973 Specifies the default sequence of maximum multiplicities to use for
1974 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
1975 basis using the \textit{max}~\qty{const} option described above.
1977 \opsmart{binary\_ints}{unary\_ints}
1978 Specifies whether natural numbers and integers should be encoded using a unary
1979 or binary notation. In unary mode, the cardinality fully specifies the subset
1980 used to approximate the type. For example:
1982 $$\hbox{\begin{tabular}{@{}rll@{}}%
1983 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
1984 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
1985 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
1990 $$\hbox{\begin{tabular}{@{}rll@{}}%
1991 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
1992 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
1995 In binary mode, the cardinality specifies the number of distinct values that can
1996 be constructed. Each of these value is represented by a bit pattern whose length
1997 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
1998 Nitpick attempts to choose the more appropriate encoding by inspecting the
1999 formula at hand, preferring the binary notation for problems involving
2000 multiplicative operators or large constants.
2002 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
2003 problems that refer to the types \textit{rat} or \textit{real} or the constants
2004 \textit{Suc}, \textit{gcd}, or \textit{lcm}.
2006 {\small See also \textit{bits} (\S\ref{scope-of-search}) and
2007 \textit{show\_datatypes} (\S\ref{output-format}).}
2009 \opdefault{bits}{int\_seq}{$\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{8},\mathbf{10},\mathbf{12}$}
2010 Specifies the number of bits to use to represent natural numbers and integers in
2011 binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
2013 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
2015 \opargboolorsmart{wf}{const}{non\_wf}
2016 Specifies whether the specified (co)in\-duc\-tively defined predicate is
2017 well-founded. The option can take the following values:
2020 \item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
2021 predicate as if it were well-founded. Since this is generally not sound when the
2022 predicate is not well-founded, the counterexamples are tagged as ``quasi
2025 \item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
2026 as if it were not well-founded. The predicate is then unrolled as prescribed by
2027 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
2030 \item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
2031 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
2032 \textit{size\_change} tactics. If this succeeds (or the predicate occurs with an
2033 appropriate polarity in the formula to falsify), use an efficient fixed point
2034 equation as specification of the predicate; otherwise, unroll the predicates
2035 according to the \textit{iter}~\qty{const} and \textit{iter} options.
2039 {\small See also \textit{iter} (\S\ref{scope-of-search}),
2040 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
2041 (\S\ref{timeouts}).}
2043 \opsmart{wf}{non\_wf}
2044 Specifies the default wellfoundedness setting to use. This can be overridden on
2045 a per-predicate basis using the \textit{wf}~\qty{const} option above.
2047 \oparg{iter}{const}{int\_seq}
2048 Specifies the sequence of iteration counts to use when unrolling a given
2049 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
2050 predicates that occur negatively and coinductive predicates that occur
2051 positively in the formula to falsify and that cannot be proved to be
2052 well-founded, but this behavior is influenced by the \textit{wf} option. The
2053 iteration counts are automatically bounded by the cardinality of the predicate's
2056 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
2057 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
2059 \opdefault{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
2060 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
2061 predicates. This can be overridden on a per-predicate basis using the
2062 \textit{iter} \qty{const} option above.
2064 \opdefault{bisim\_depth}{int\_seq}{$\mathbf{7}$}
2065 Specifies the sequence of iteration counts to use when unrolling the
2066 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
2067 of $-1$ means that no predicate is generated, in which case Nitpick performs an
2068 after-the-fact check to see if the known coinductive datatype values are
2069 bidissimilar. If two values are found to be bisimilar, the counterexample is
2070 tagged as ``quasi genuine.'' The iteration counts are automatically bounded by
2071 the sum of the cardinalities of the coinductive datatypes occurring in the
2074 \opargboolorsmart{box}{type}{dont\_box}
2075 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
2076 product type in an isomorphic datatype internally. Boxing is an effective mean
2077 to reduce the search space and speed up Nitpick, because the isomorphic datatype
2078 is approximated by a subset of the possible function or pair values.
2079 Like other drastic optimizations, it can also prevent the discovery of
2080 counterexamples. The option can take the following values:
2083 \item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
2085 \item[$\bullet$] \textbf{\textit{false}}: Never box the type.
2086 \item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
2087 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
2088 higher-order functions are good candidates for boxing.
2092 {\small See also \textit{finitize} (\S\ref{scope-of-search}), \textit{verbose}
2093 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}).}
2095 \opsmart{box}{dont\_box}
2096 Specifies the default boxing setting to use. This can be overridden on a
2097 per-type basis using the \textit{box}~\qty{type} option described above.
2099 \opargboolorsmart{finitize}{type}{dont\_finitize}
2100 Specifies whether Nitpick should attempt to finitize a given type, which can be
2101 a function type or an infinite ``shallow datatype'' (an infinite datatype whose
2102 constructors don't appear in the problem).
2104 For function types, Nitpick performs a monotonicity analysis to detect functions
2105 that are constant at all but finitely many points (e.g., finite sets) and treats
2106 such occurrences specially, thereby increasing the precision. The option can
2107 then take the following values:
2110 \item[$\bullet$] \textbf{\textit{false}}: Don't attempt to finitize the type.
2111 \item[$\bullet$] \textbf{\textit{true}} or \textbf{\textit{smart}}: Finitize the
2112 type wherever possible.
2115 The semantics of the option is somewhat different for infinite shallow
2119 \item[$\bullet$] \textbf{\textit{true}}: Finitize the datatype. Since this is
2120 unsound, counterexamples generated under these conditions are tagged as ``quasi
2122 \item[$\bullet$] \textbf{\textit{false}}: Don't attempt to finitize the datatype.
2123 \item[$\bullet$] \textbf{\textit{smart}}: Perform a monotonicity analysis to
2124 detect whether the datatype can be safely finitized before finitizing it.
2127 Like other drastic optimizations, finitization can sometimes prevent the
2128 discovery of counterexamples.
2131 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{mono}
2132 (\S\ref{scope-of-search}), \textit{verbose} (\S\ref{output-format}), and
2133 \textit{debug} (\S\ref{output-format}).}
2135 \opsmart{finitize}{dont\_finitize}
2136 Specifies the default finitization setting to use. This can be overridden on a
2137 per-type basis using the \textit{finitize}~\qty{type} option described above.
2139 \opargboolorsmart{mono}{type}{non\_mono}
2140 Specifies whether the given type should be considered monotonic when enumerating
2141 scopes and finitizing types. If the option is set to \textit{smart}, Nitpick
2142 performs a monotonicity check on the type. Setting this option to \textit{true}
2143 can reduce the number of scopes tried, but it can also diminish the chance of
2144 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
2147 {\small See also \textit{card} (\S\ref{scope-of-search}),
2148 \textit{finitize} (\S\ref{scope-of-search}),
2149 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
2150 (\S\ref{output-format}).}
2152 \opsmart{mono}{non\_mono}
2153 Specifies the default monotonicity setting to use. This can be overridden on a
2154 per-type basis using the \textit{mono}~\qty{type} option described above.
2156 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
2157 Specifies whether type variables with the same sort constraints should be
2158 merged. Setting this option to \textit{true} can reduce the number of scopes
2159 tried and the size of the generated Kodkod formulas, but it also diminishes the
2160 theoretical chance of finding a counterexample.
2162 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
2164 \opargbool{std}{type}{non\_std}
2165 Specifies whether the given (recursive) datatype should be given standard
2166 models. Nonstandard models are unsound but can help debug structural induction
2167 proofs, as explained in \S\ref{inductive-properties}.
2169 \optrue{std}{non\_std}
2170 Specifies the default standardness to use. This can be overridden on a per-type
2171 basis using the \textit{std}~\qty{type} option described above.
2174 \subsection{Output Format}
2175 \label{output-format}
2178 \opfalse{verbose}{quiet}
2179 Specifies whether the \textbf{nitpick} command should explain what it does. This
2180 option is useful to determine which scopes are tried or which SAT solver is
2181 used. This option is implicitly disabled for automatic runs.
2183 \opfalse{debug}{no\_debug}
2184 Specifies whether Nitpick should display additional debugging information beyond
2185 what \textit{verbose} already displays. Enabling \textit{debug} also enables
2186 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
2187 option is implicitly disabled for automatic runs.
2190 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
2191 \textit{batch\_size} (\S\ref{optimizations}).}
2193 \opfalse{show\_datatypes}{hide\_datatypes}
2194 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
2195 be displayed as part of counterexamples. Such subsets are sometimes helpful when
2196 investigating whether a potential counterexample is genuine or spurious, but
2197 their potential for clutter is real.
2199 \opfalse{show\_consts}{hide\_consts}
2200 Specifies whether the values of constants occurring in the formula (including
2201 its axioms) should be displayed along with any counterexample. These values are
2202 sometimes helpful when investigating why a counterexample is
2203 genuine, but they can clutter the output.
2205 \opfalse{show\_all}{dont\_show\_all}
2206 Enabling this option effectively enables \textit{show\_datatypes} and
2207 \textit{show\_consts}.
2209 \opdefault{max\_potential}{int}{$\mathbf{1}$}
2210 Specifies the maximum number of potential counterexamples to display. Setting
2211 this option to 0 speeds up the search for a genuine counterexample. This option
2212 is implicitly set to 0 for automatic runs. If you set this option to a value
2213 greater than 1, you will need an incremental SAT solver, such as
2214 \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that many of
2215 the counterexamples may be identical.
2218 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
2219 \textit{sat\_solver} (\S\ref{optimizations}).}
2221 \opdefault{max\_genuine}{int}{$\mathbf{1}$}
2222 Specifies the maximum number of genuine counterexamples to display. If you set
2223 this option to a value greater than 1, you will need an incremental SAT solver,
2224 such as \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that
2225 many of the counterexamples may be identical.
2228 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
2229 \textit{sat\_solver} (\S\ref{optimizations}).}
2231 \opnodefault{eval}{term\_list}
2232 Specifies the list of terms whose values should be displayed along with
2233 counterexamples. This option suffers from an ``observer effect'': Nitpick might
2234 find different counterexamples for different values of this option.
2236 \oparg{format}{term}{int\_seq}
2237 Specifies how to uncurry the value displayed for a variable or constant.
2238 Uncurrying sometimes increases the readability of the output for high-arity
2239 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
2240 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
2241 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
2242 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
2243 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
2244 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
2245 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
2246 arguments that are not accounted for are left alone, as if the specification had
2247 been $1,\ldots,1,n_1,\ldots,n_k$.
2249 \opdefault{format}{int\_seq}{$\mathbf{1}$}
2250 Specifies the default format to use. Irrespective of the default format, the
2251 extra arguments to a Skolem constant corresponding to the outer bound variables
2252 are kept separated from the remaining arguments, the \textbf{for} arguments of
2253 an inductive definitions are kept separated from the remaining arguments, and
2254 the iteration counter of an unrolled inductive definition is shown alone. The
2255 default format can be overridden on a per-variable or per-constant basis using
2256 the \textit{format}~\qty{term} option described above.
2259 \subsection{Authentication}
2260 \label{authentication}
2263 \opfalse{check\_potential}{trust\_potential}
2264 Specifies whether potential counterexamples should be given to Isabelle's
2265 \textit{auto} tactic to assess their validity. If a potential counterexample is
2266 shown to be genuine, Nitpick displays a message to this effect and terminates.
2269 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
2271 \opfalse{check\_genuine}{trust\_genuine}
2272 Specifies whether genuine and quasi genuine counterexamples should be given to
2273 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
2274 counterexample is shown to be spurious, the user is kindly asked to send a bug
2275 report to the author at
2276 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
2279 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
2281 \opnodefault{expect}{string}
2282 Specifies the expected outcome, which must be one of the following:
2285 \item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
2286 \item[$\bullet$] \textbf{\textit{quasi\_genuine}}: Nitpick found a ``quasi
2287 genuine'' counterexample (i.e., a counterexample that is genuine unless
2288 it contradicts a missing axiom or a dangerous option was used inappropriately).
2289 \item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
2290 \item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
2291 \item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
2292 Kodkod ran out of memory).
2295 Nitpick emits an error if the actual outcome differs from the expected outcome.
2296 This option is useful for regression testing.
2299 \subsection{Optimizations}
2300 \label{optimizations}
2302 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
2307 \opdefault{sat\_solver}{string}{smart}
2308 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2309 to be faster than their Java counterparts, but they can be more difficult to
2310 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2311 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2312 you will need an incremental SAT solver, such as \textit{MiniSat\_JNI}
2313 (recommended) or \textit{SAT4J}.
2315 The supported solvers are listed below:
2321 \item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
2322 written in \cpp{}. To use MiniSat, set the environment variable
2323 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2325 \footnote{Important note for Cygwin users: The path must be specified using
2326 native Windows syntax. Make sure to escape backslashes properly.%
2327 \label{cygwin-paths}}
2328 The \cpp{} sources and executables for MiniSat are available at
2329 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2330 and 2.0 beta (2007-07-21).
2332 \item[$\bullet$] \textbf{\textit{MiniSat\_JNI}}: The JNI (Java Native Interface)
2333 version of MiniSat is bundled with Kodkodi and is precompiled for the major
2334 platforms. It is also available from \texttt{native\-solver.\allowbreak tgz},
2335 which you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2336 version of MiniSat, the JNI version can be used incrementally.
2338 \item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
2339 written in C. You can install a standard version of
2340 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2341 that contains the \texttt{picosat} executable.%
2342 \footref{cygwin-paths}
2343 The C sources for PicoSAT are
2344 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2345 Nitpick has been tested with version 913.
2347 \item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
2348 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2349 the directory that contains the \texttt{zchaff} executable.%
2350 \footref{cygwin-paths}
2351 The \cpp{} sources and executables for zChaff are available at
2352 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2353 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2355 \item[$\bullet$] \textbf{\textit{zChaff\_JNI}}: The JNI version of zChaff is
2356 bundled with Kodkodi and is precompiled for the major
2357 platforms. It is also available from \texttt{native\-solver.\allowbreak tgz},
2358 which you will find on Kodkod's web site \cite{kodkod-2009}.
2360 \item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
2361 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2362 directory that contains the \texttt{rsat} executable.%
2363 \footref{cygwin-paths}
2364 The \cpp{} sources for RSat are available at
2365 \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been tested with version
2368 \item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
2369 written in C. To use BerkMin, set the environment variable
2370 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2371 executable.\footref{cygwin-paths}
2372 The BerkMin executables are available at
2373 \url{http://eigold.tripod.com/BerkMin.html}.
2375 \item[$\bullet$] \textbf{\textit{BerkMin\_Alloy}}: Variant of BerkMin that is
2376 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2377 version of BerkMin, set the environment variable
2378 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2380 \footref{cygwin-paths}
2382 \item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
2383 written in C. To use Jerusat, set the environment variable
2384 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2386 \footref{cygwin-paths}
2387 The C sources for Jerusat are available at
2388 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2390 \item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
2391 written in Java that can be used incrementally. It is bundled with Kodkodi and
2392 requires no further installation or configuration steps. Do not attempt to
2393 install the official SAT4J packages, because their API is incompatible with
2396 \item[$\bullet$] \textbf{\textit{SAT4J\_Light}}: Variant of SAT4J that is
2397 optimized for small problems. It can also be used incrementally.
2399 \item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
2400 experimental solver written in \cpp. To use HaifaSat, set the environment
2401 variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
2402 \texttt{HaifaSat} executable.%
2403 \footref{cygwin-paths}
2404 The \cpp{} sources for HaifaSat are available at
2405 \url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
2407 \item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
2408 \textit{smart}, Nitpick selects the first solver among MiniSat,
2409 PicoSAT, zChaff, RSat, BerkMin, BerkMin\_Alloy, Jerusat, MiniSat\_JNI, and zChaff\_JNI
2410 that is recognized by Isabelle. If none is found, it falls back on SAT4J, which
2411 should always be available. If \textit{verbose} (\S\ref{output-format}) is
2412 enabled, Nitpick displays which SAT solver was chosen.
2416 \opdefault{batch\_size}{int\_or\_smart}{smart}
2417 Specifies the maximum number of Kodkod problems that should be lumped together
2418 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2419 together ensures that Kodkodi is launched less often, but it makes the verbose
2420 output less readable and is sometimes detrimental to performance. If
2421 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2422 \textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
2424 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2425 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2426 be rewritten to use (automatically generated) discriminators and destructors.
2427 This optimization can drastically reduce the size of the Boolean formulas given
2431 {\small See also \textit{debug} (\S\ref{output-format}).}
2433 \optrue{specialize}{dont\_specialize}
2434 Specifies whether functions invoked with static arguments should be specialized.
2435 This optimization can drastically reduce the search space, especially for
2436 higher-order functions.
2439 {\small See also \textit{debug} (\S\ref{output-format}) and
2440 \textit{show\_consts} (\S\ref{output-format}).}
2442 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2443 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2444 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2445 predicates for which each the predicate occurs in at most one assumption of each
2446 introduction rule. Using the reflexive transitive closure is in principle
2447 equivalent to setting \textit{iter} to the cardinality of the predicate's
2448 domain, but it is usually more efficient.
2450 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2451 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2453 \optrue{fast\_descrs}{full\_descrs}
2454 Specifies whether Nitpick should optimize the definite and indefinite
2455 description operators (THE and SOME). The optimized versions usually help
2456 Nitpick generate more counterexamples or at least find them faster, but only the
2457 unoptimized versions are complete when all types occurring in the formula are
2460 {\small See also \textit{debug} (\S\ref{output-format}).}
2462 \optrue{peephole\_optim}{no\_peephole\_optim}
2463 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2464 peephole optimizer. These optimizations can make a significant difference.
2465 Unless you are tracking down a bug in Nitpick or distrust the peephole
2466 optimizer, you should leave this option enabled.
2468 \opdefault{max\_threads}{int}{0}
2469 Specifies the maximum number of threads to use in Kodkod. If this option is set
2470 to 0, Kodkod will compute an appropriate value based on the number of processor
2474 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2475 \textit{timeout} (\S\ref{timeouts}).}
2478 \subsection{Timeouts}
2482 \opdefault{timeout}{time}{$\mathbf{30}$ s}
2483 Specifies the maximum amount of time that the \textbf{nitpick} command should
2484 spend looking for a counterexample. Nitpick tries to honor this constraint as
2485 well as it can but offers no guarantees. For automatic runs,
2486 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2487 a time slot whose length is specified by the ``Auto Counterexample Time
2488 Limit'' option in Proof General.
2491 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2493 \opdefault{tac\_timeout}{time}{$\mathbf{500}$\,ms}
2494 Specifies the maximum amount of time that the \textit{auto} tactic should use
2495 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2496 and \textit{size\_change} should use when checking whether a (co)in\-duc\-tive
2497 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2498 can but offers no guarantees.
2501 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2502 \textit{check\_potential} (\S\ref{authentication}),
2503 and \textit{check\_genuine} (\S\ref{authentication}).}
2506 \section{Attribute Reference}
2507 \label{attribute-reference}
2509 Nitpick needs to consider the definitions of all constants occurring in a
2510 formula in order to falsify it. For constants introduced using the
2511 \textbf{definition} command, the definition is simply the associated
2512 \textit{\_def} axiom. In contrast, instead of using the internal representation
2513 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2514 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2515 equational specification entered by the user.
2517 Behind the scenes, Isabelle's built-in packages and theories rely on the
2518 following attributes to affect Nitpick's behavior:
2521 \flushitem{\textit{nitpick\_def}}
2524 This attribute specifies an alternative definition of a constant. The
2525 alternative definition should be logically equivalent to the constant's actual
2526 axiomatic definition and should be of the form
2528 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2530 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2533 \flushitem{\textit{nitpick\_simp}}
2536 This attribute specifies the equations that constitute the specification of a
2537 constant. For functions defined using the \textbf{primrec}, \textbf{function},
2538 and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
2539 \textit{simps} rules. The equations must be of the form
2541 \qquad $c~t_1~\ldots\ t_n \,=\, u.$
2543 \flushitem{\textit{nitpick\_psimp}}
2546 This attribute specifies the equations that constitute the partial specification
2547 of a constant. For functions defined using the \textbf{function} package, this
2548 corresponds to the \textit{psimps} rules. The conditional equations must be of
2551 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
2553 \flushitem{\textit{nitpick\_intro}}
2556 This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
2557 For predicates defined using the \textbf{inductive} or \textbf{coinductive}
2558 command, this corresponds to the \textit{intros} rules. The introduction rules
2561 \qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
2562 \ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk$ \\
2563 \hbox{}\qquad ${\Longrightarrow}\;\, c\ u_1\ \ldots\ u_n$,
2565 where the $P_i$'s are side conditions that do not involve $c$ and $M$ is an
2566 optional monotonic operator. The order of the assumptions is irrelevant.
2568 \flushitem{\textit{nitpick\_choice\_spec}}
2571 This attribute specifies the (free-form) specification of a constant defined
2572 using the \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command.
2576 When faced with a constant, Nitpick proceeds as follows:
2579 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2580 is not empty, Nitpick uses these rules as the specification of the constant.
2582 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2583 the constant is not empty, it uses these rules as the specification of the
2586 \item[3.] Otherwise, if the constant was defined using the
2587 \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command and the
2588 \textit{nitpick\_choice\_spec} set associated with the constant is not empty, it
2589 uses these theorems as the specification of the constant.
2591 \item[4.] Otherwise, it looks up the definition of the constant:
2594 \item[1.] If the \textit{nitpick\_def} set associated with the constant
2595 is not empty, it uses the latest rule added to the set as the definition of the
2596 constant; otherwise it uses the actual definition axiom.
2597 \item[2.] If the definition is of the form
2599 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
2601 then Nitpick assumes that the definition was made using an inductive package and
2602 based on the introduction rules marked with \textit{nitpick\_\allowbreak
2603 \allowbreak intros} tries to determine whether the definition is
2608 As an illustration, consider the inductive definition
2611 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2612 ``\textit{odd}~1'' $\,\mid$ \\
2613 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2616 Isabelle automatically attaches the \textit{nitpick\_intro} attribute to
2617 the above rules. Nitpick then uses the \textit{lfp}-based definition in
2618 conjunction with these rules. To override this, we can specify an alternative
2619 definition as follows:
2622 \textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]:\kern.4em ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2625 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2626 = 1$. Alternatively, we can specify an equational specification of the constant:
2629 \textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2632 Such tweaks should be done with great care, because Nitpick will assume that the
2633 constant is completely defined by its equational specification. For example, if
2634 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
2635 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2636 (\S\ref{output-format}) option is extremely useful to understand what is going
2637 on when experimenting with \textit{nitpick\_} attributes.
2639 \section{Standard ML Interface}
2640 \label{standard-ml-interface}
2642 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2643 and debugging. Among the most interesting functions exported by Nitpick are
2644 those that let you invoke the tool programmatically and those that let you
2645 register and unregister custom coinductive datatypes as well as term
2648 \subsection{Invocation of Nitpick}
2649 \label{invocation-of-nitpick}
2651 The \textit{Nitpick} structure offers the following functions for invoking your
2652 favorite counterexample generator:
2655 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2656 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2657 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2658 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2659 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2662 The return value is a new proof state paired with an outcome string
2663 (``genuine'', ``quasi\_genuine'', ``potential'', ``none'', or ``unknown''). The
2664 \textit{params} type is a large record that lets you set Nitpick's options. The
2665 current default options can be retrieved by calling the following function
2666 defined in the \textit{Nitpick\_Isar} structure:
2669 $\textbf{val}\,~\textit{default\_params} :\,
2670 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2673 The second argument lets you override option values before they are parsed and
2674 put into a \textit{params} record. Here is an example:
2677 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout\/}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2678 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2679 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2680 & \textit{subgoal}\end{aligned}$
2685 \subsection{Registration of Coinductive Datatypes}
2686 \label{registration-of-coinductive-datatypes}
2688 If you have defined a custom coinductive datatype, you can tell Nitpick about
2689 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2690 uses for lazy lists. The interface for registering and unregistering coinductive
2691 datatypes consists of the following pair of functions defined in the
2692 \textit{Nitpick} structure:
2695 $\textbf{val}\,~\textit{register\_codatatype} :\,
2696 \textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2697 $\textbf{val}\,~\textit{unregister\_codatatype} :\,
2698 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2701 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2702 not been, you could have told Nitpick about it by adding the following line
2703 to your theory file:
2706 $\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
2707 & \textit{Nitpick.register\_codatatype} \\[-2pt]
2708 & \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
2709 & \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
2712 The \textit{register\_codatatype} function takes a coinductive type, its case
2713 function, and the list of its constructors. The case function must take its
2714 arguments in the order that the constructors are listed. If no case function
2715 with the correct signature is available, simply pass the empty string.
2717 On the other hand, if your goal is to cripple Nitpick, add the following line to
2718 your theory file and try to check a few conjectures about lazy lists:
2721 $\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
2722 \kern1pt'a~\textit{list\/}\textrm{''}\}\ \,{*}\}$
2725 Inductive datatypes can be registered as coinductive datatypes, given
2726 appropriate coinductive constructors. However, doing so precludes
2727 the use of the inductive constructors---Nitpick will generate an error if they
2730 \subsection{Registration of Term Postprocessors}
2731 \label{registration-of-term-postprocessors}
2733 It is possible to change the output of any term that Nitpick considers a
2734 datatype by registering a term postprocessor. The interface for registering and
2735 unregistering postprocessors consists of the following pair of functions defined
2736 in the \textit{Nitpick} structure:
2739 $\textbf{type}\,~\textit{term\_postprocessor}\,~{=} {}$ \\
2740 $\hbox{}\quad\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \rightarrow \textit{term}$ \\
2741 $\textbf{val}\,~\textit{register\_term\_postprocessors} : {}$ \\
2742 $\hbox{}\quad\textit{typ} \rightarrow \textit{term\_postprocessor} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2743 $\textbf{val}\,~\textit{unregister\_term\_postprocessors} :\,
2744 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2747 \S\ref{typedefs-quotient-types-records-rationals-and-reals} and
2748 \texttt{src/HOL/Library/Multiset.thy} illustrate this feature in context.
2750 \section{Known Bugs and Limitations}
2751 \label{known-bugs-and-limitations}
2753 Here are the known bugs and limitations in Nitpick at the time of writing:
2756 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2757 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2758 Nitpick to generate spurious counterexamples for theorems that refer to values
2759 for which the function is not defined. For example:
2762 \textbf{primrec} \textit{prec} \textbf{where} \\
2763 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2764 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2765 \textbf{nitpick} \\[2\smallskipamount]
2766 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2768 \\[2\smallskipamount]
2769 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2770 \textbf{by}~(\textit{auto simp}:~\textit{prec\_def})
2773 Such theorems are considered bad style because they rely on the internal
2774 representation of functions synthesized by Isabelle, which is an implementation
2777 \item[$\bullet$] Axioms that restrict the possible values of the
2778 \textit{undefined} constant are in general ignored.
2780 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2781 which can become invalid if you change the definition of an inductive predicate
2782 that is registered in the cache. To clear the cache,
2783 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2784 501$\,\textit{ms}$).
2786 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2787 \textbf{guess} command in a structured proof.
2789 \item[$\bullet$] The \textit{nitpick\_} attributes and the
2790 \textit{Nitpick.register\_} functions can cause havoc if used improperly.
2792 \item[$\bullet$] Although this has never been observed, arbitrary theorem
2793 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
2795 \item[$\bullet$] All constants, types, free variables, and schematic variables
2796 whose names start with \textit{Nitpick}{.} are reserved for internal use.
2800 \bibliography{../manual}{}
2801 \bibliographystyle{abbrv}