added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
1 \documentclass[a4paper,12pt]{article}
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22 \def\Colon{\mathord{:\mkern-1.5mu:}}
23 %\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}}
24 %\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}}
25 \def\lparr{\mathopen{(\mkern-4mu\mid}}
26 \def\rparr{\mathclose{\mid\mkern-4mu)}}
29 \def\undef{(\lambda x.\; \unk)}
30 %\def\unr{\textit{others}}
32 \def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
33 \def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
35 \hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
36 counter-example counter-examples data-type data-types co-data-type
37 co-data-types in-duc-tive co-in-duc-tive}
43 \selectlanguage{english}
45 \title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
46 Picking Nits \\[\smallskipamount]
47 \Large A User's Guide to Nitpick for Isabelle/HOL}
49 Jasmin Christian Blanchette \\
50 {\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\
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80 \section{Introduction}
83 Nitpick \cite{blanchette-nipkow-2009} is a counterexample generator for
84 Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
85 combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
86 quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
87 first-order relational model finder developed by the Software Design Group at
88 MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
89 borrows many ideas and code fragments, but it benefits from Kodkod's
90 optimizations and a new encoding scheme. The name Nitpick is shamelessly
91 appropriated from a now retired Alloy precursor.
93 Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
94 theorem and wait a few seconds. Nonetheless, there are situations where knowing
95 how it works under the hood and how it reacts to various options helps
96 increase the test coverage. This manual also explains how to install the tool on
97 your workstation. Should the motivation fail you, think of the many hours of
98 hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.
100 Another common use of Nitpick is to find out whether the axioms of a locale are
101 satisfiable, while the locale is being developed. To check this, it suffices to
105 \textbf{lemma}~``$\textit{False}$'' \\
106 \textbf{nitpick}~[\textit{show\_all}]
109 after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
110 must find a model for the axioms. If it finds no model, we have an indication
111 that the axioms might be unsatisfiable.
113 Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
114 machine called \texttt{java}. The examples presented in this manual can be found
115 in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.
117 Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
118 Nitpick also provides an automatic mode that can be enabled using the
119 ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof General. In this
120 mode, Nitpick is run on every newly entered theorem, much like Auto Quickcheck.
121 The collective time limit for Auto Nitpick and Auto Quickcheck can be set using
122 the ``Auto Counterexample Time Limit'' option.
125 \setbox\boxA=\hbox{\texttt{nospam}}
127 The known bugs and limitations at the time of writing are listed in
128 \S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
129 or this manual should be directed to
130 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
131 in.\allowbreak tum.\allowbreak de}.
133 \vskip2.5\smallskipamount
135 \textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
136 suggesting several textual improvements.
137 % and Perry James for reporting a typo.
139 \section{First Steps}
142 This section introduces Nitpick by presenting small examples. If possible, you
143 should try out the examples on your workstation. Your theory file should start
147 \textbf{theory}~\textit{Scratch} \\
148 \textbf{imports}~\textit{Main} \\
152 The results presented here were obtained using the JNI version of MiniSat and
153 with multithreading disabled to reduce nondeterminism. This was done by adding
157 \textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSatJNI}, \,\textit{max\_threads}~= 1]
160 after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
161 Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
162 be installed, as explained in \S\ref{optimizations}. If you have already
163 configured SAT solvers in Isabelle (e.g., for Refute), these will also be
164 available to Nitpick.
166 \subsection{Propositional Logic}
167 \label{propositional-logic}
169 Let's start with a trivial example from propositional logic:
172 \textbf{lemma}~``$P \longleftrightarrow Q$'' \\
176 You should get the following output:
180 Nitpick found a counterexample: \\[2\smallskipamount]
181 \hbox{}\qquad Free variables: \nopagebreak \\
182 \hbox{}\qquad\qquad $P = \textit{True}$ \\
183 \hbox{}\qquad\qquad $Q = \textit{False}$
186 Nitpick can also be invoked on individual subgoals, as in the example below:
189 \textbf{apply}~\textit{auto} \\[2\smallskipamount]
190 {\slshape goal (2 subgoals): \\
191 \phantom{0}1. $P\,\Longrightarrow\, Q$ \\
192 \phantom{0}2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
193 \textbf{nitpick}~1 \\[2\smallskipamount]
194 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
195 \hbox{}\qquad Free variables: \nopagebreak \\
196 \hbox{}\qquad\qquad $P = \textit{True}$ \\
197 \hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
198 \textbf{nitpick}~2 \\[2\smallskipamount]
199 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
200 \hbox{}\qquad Free variables: \nopagebreak \\
201 \hbox{}\qquad\qquad $P = \textit{False}$ \\
202 \hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
206 \subsection{Type Variables}
207 \label{type-variables}
209 If you are left unimpressed by the previous example, don't worry. The next
210 one is more mind- and computer-boggling:
213 \textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
215 \pagebreak[2] %% TYPESETTING
217 The putative lemma involves the definite description operator, {THE}, presented
218 in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
219 operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
220 lemma is merely asserting the indefinite description operator axiom with {THE}
221 substituted for {SOME}.
223 The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
224 containing type variables, Nitpick enumerates the possible domains for each type
225 variable, up to a given cardinality (8 by default), looking for a finite
229 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
231 Trying 8 scopes: \nopagebreak \\
232 \hbox{}\qquad \textit{card}~$'a$~= 1; \\
233 \hbox{}\qquad \textit{card}~$'a$~= 2; \\
234 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
235 \hbox{}\qquad \textit{card}~$'a$~= 8. \\[2\smallskipamount]
236 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
237 \hbox{}\qquad Free variables: \nopagebreak \\
238 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
239 \hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
243 Nitpick found a counterexample in which $'a$ has cardinality 3. (For
244 cardinalities 1 and 2, the formula holds.) In the counterexample, the three
245 values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
247 The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
248 \textit{verbose} is enabled. You can specify \textit{verbose} each time you
249 invoke \textbf{nitpick}, or you can set it globally using the command
252 \textbf{nitpick\_params} [\textit{verbose}]
255 This command also displays the current default values for all of the options
256 supported by Nitpick. The options are listed in \S\ref{option-reference}.
258 \subsection{Constants}
261 By just looking at Nitpick's output, it might not be clear why the
262 counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
263 this time telling it to show the values of the constants that occur in the
267 \textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
268 \textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
270 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
271 \hbox{}\qquad Free variables: \nopagebreak \\
272 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
273 \hbox{}\qquad\qquad $x = a_3$ \\
274 \hbox{}\qquad Constant: \nopagebreak \\
275 \hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
278 We can see more clearly now. Since the predicate $P$ isn't true for a unique
279 value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
280 $a_1$. Since $P~a_1$ is false, the entire formula is falsified.
282 As an optimization, Nitpick's preprocessor introduced the special constant
283 ``\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
284 $\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
285 satisfying $P~y$. We disable this optimization by passing the
286 \textit{full\_descrs} option:
289 \textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
291 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
292 \hbox{}\qquad Free variables: \nopagebreak \\
293 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
294 \hbox{}\qquad\qquad $x = a_3$ \\
295 \hbox{}\qquad Constant: \nopagebreak \\
296 \hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
299 As the result of another optimization, Nitpick directly assigned a value to the
300 subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
301 disable this second optimization by using the command
304 \textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
305 \textit{show\_consts}]
308 we finally get \textit{The}:
311 \slshape Constant: \nopagebreak \\
312 \hbox{}\qquad $\mathit{The} = \undef{}
313 (\!\begin{aligned}[t]%
314 & \{\} := a_3,\> \{a_3\} := a_3,\> \{a_2\} := a_2, \\[-2pt] %% TYPESETTING
315 & \{a_2, a_3\} := a_1,\> \{a_1\} := a_1,\> \{a_1, a_3\} := a_3, \\[-2pt]
316 & \{a_1, a_2\} := a_3,\> \{a_1, a_2, a_3\} := a_3)\end{aligned}$
319 Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
320 just like before.\footnote{The Isabelle/HOL notation $f(x :=
321 y)$ denotes the function that maps $x$ to $y$ and that otherwise behaves like
324 Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
325 unique $x$ such that'') at the front of our putative lemma's assumption:
328 \textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
331 The fix appears to work:
334 \textbf{nitpick} \\[2\smallskipamount]
335 \slshape Nitpick found no counterexample.
338 We can further increase our confidence in the formula by exhausting all
339 cardinalities up to 50:
342 \textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
343 can be entered as \texttt{-} (hyphen) or
344 \texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
345 \slshape Nitpick found no counterexample.
348 Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
351 \textbf{sledgehammer} \\[2\smallskipamount]
352 {\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
353 $\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
354 Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
355 \textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
356 {\slshape No subgoals!}% \\[2\smallskipamount]
360 This must be our lucky day.
362 \subsection{Skolemization}
363 \label{skolemization}
365 Are all invertible functions onto? Let's find out:
368 \textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
369 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
370 \textbf{nitpick} \\[2\smallskipamount]
372 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
373 \hbox{}\qquad Free variable: \nopagebreak \\
374 \hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
375 \hbox{}\qquad Skolem constants: \nopagebreak \\
376 \hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
377 \hbox{}\qquad\qquad $y = a_2$
380 Although $f$ is the only free variable occurring in the formula, Nitpick also
381 displays values for the bound variables $g$ and $y$. These values are available
382 to Nitpick because it performs skolemization as a preprocessing step.
384 In the previous example, skolemization only affected the outermost quantifiers.
385 This is not always the case, as illustrated below:
388 \textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
389 \textbf{nitpick} \\[2\smallskipamount]
391 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
392 \hbox{}\qquad Skolem constant: \nopagebreak \\
393 \hbox{}\qquad\qquad $\lambda x.\; f =
394 \undef{}(\!\begin{aligned}[t]
395 & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
396 & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
399 The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
400 $x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
401 function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
402 maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
404 The source of the Skolem constants is sometimes more obscure:
407 \textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
408 \textbf{nitpick} \\[2\smallskipamount]
410 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
411 \hbox{}\qquad Free variable: \nopagebreak \\
412 \hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
413 \hbox{}\qquad Skolem constants: \nopagebreak \\
414 \hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
415 \hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
418 What happened here is that Nitpick expanded the \textit{sym} constant to its
422 $\mathit{sym}~r \,\equiv\,
423 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
426 As their names suggest, the Skolem constants $\mathit{sym}.x$ and
427 $\mathit{sym}.y$ are simply the bound variables $x$ and $y$
428 from \textit{sym}'s definition.
430 Although skolemization is a useful optimization, you can disable it by invoking
431 Nitpick with \textit{dont\_skolemize}. See \S\ref{optimizations} for details.
433 \subsection{Natural Numbers and Integers}
434 \label{natural-numbers-and-integers}
436 Because of the axiom of infinity, the type \textit{nat} does not admit any
437 finite models. To deal with this, Nitpick's approach is to consider finite
438 subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined
439 value (displayed as `$\unk$'). The type \textit{int} is handled similarly.
440 Internally, undefined values lead to a three-valued logic.
442 Here is an example involving \textit{int}:
445 \textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
446 \textbf{nitpick} \\[2\smallskipamount]
447 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
448 \hbox{}\qquad Free variables: \nopagebreak \\
449 \hbox{}\qquad\qquad $i = 0$ \\
450 \hbox{}\qquad\qquad $j = 1$ \\
451 \hbox{}\qquad\qquad $m = 1$ \\
452 \hbox{}\qquad\qquad $n = 0$
455 Internally, Nitpick uses either a unary or a binary representation of numbers.
456 The unary representation is more efficient but only suitable for numbers very
457 close to zero. By default, Nitpick attempts to choose the more appropriate
458 encoding by inspecting the formula at hand. This behavior can be overridden by
459 passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For
460 binary notation, the number of bits to use can be specified using
461 the \textit{bits} option. For example:
464 \textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$]
467 With infinite types, we don't always have the luxury of a genuine counterexample
468 and must often content ourselves with a potential one. The tedious task of
469 finding out whether the potential counterexample is in fact genuine can be
470 outsourced to \textit{auto} by passing \textit{check\_potential}. For example:
473 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
474 \textbf{nitpick} [\textit{card~nat}~= 100, \textit{check\_potential}] \\[2\smallskipamount]
475 \slshape 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} [\textit{card} = 1--6] \\[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 $\textit{y} = a_3$
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}~= $\{[],\, [a_3, a_3],\, [a_3],\, \unr\}$ \\
562 {\slshape Constants:} \\
563 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_3, a_3])$ \\
564 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_3, a_3] := a_3,\> [a_3] := a_3)$
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 $[a_3,
572 a_3]$ to $[a_3]$ would normally give $[a_3, a_3, a_3]$, 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 $[a_3, a_3]$ 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 $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_3]\}$, and observe
604 that $[a_1, a_3]$ (i.e., $a_1 \mathbin{\#} [a_3]$) has $[a_3] \notin
605 \mathcal{S}$ 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 $\textit{xs} = [a_2]$ \\
617 \hbox{}\qquad\qquad $\textit{ys} = [a_3]$ \\
618 \hbox{}\qquad Datatypes: \\
619 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
620 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_3],\, [a_2],\, \unr\}$
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, Records, Rationals, and Reals}
629 \label{typedefs-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 $P = \{\Abs{1},\, \Abs{0}\}$ \\
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 $\textit{three} = \{\Abs{2},\, \Abs{1},\, \Abs{0},\, \unr\}$
653 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
656 Records, which are implemented as \textbf{typedef}s behind the scenes, are
657 handled in much the same way:
660 \textbf{record} \textit{point} = \\
661 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
662 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
663 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
664 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
665 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
666 \hbox{}\qquad Free variables: \nopagebreak \\
667 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
668 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
669 \hbox{}\qquad Datatypes: \\
670 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
671 \hbox{}\qquad\qquad $\textit{point} = \{\lparr\textit{Xcoord} = 1,\>
672 \textit{Ycoord} = 1\rparr,\> \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr,\, \unr\}$\kern-1pt %% QUIET
675 Finally, Nitpick provides rudimentary support for rationals and reals using a
679 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
680 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
681 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
682 \hbox{}\qquad Free variables: \nopagebreak \\
683 \hbox{}\qquad\qquad $x = 1/2$ \\
684 \hbox{}\qquad\qquad $y = -1/2$ \\
685 \hbox{}\qquad Datatypes: \\
686 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
687 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
688 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
691 \subsection{Inductive and Coinductive Predicates}
692 \label{inductive-and-coinductive-predicates}
694 Inductively defined predicates (and sets) are particularly problematic for
695 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
696 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
697 the problem is that they are defined using a least fixed point construction.
699 Nitpick's philosophy is that not all inductive predicates are equal. Consider
700 the \textit{even} predicate below:
703 \textbf{inductive}~\textit{even}~\textbf{where} \\
704 ``\textit{even}~0'' $\,\mid$ \\
705 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
708 This predicate enjoys the desirable property of being well-founded, which means
709 that the introduction rules don't give rise to infinite chains of the form
712 $\cdots\,\Longrightarrow\, \textit{even}~k''
713 \,\Longrightarrow\, \textit{even}~k'
714 \,\Longrightarrow\, \textit{even}~k.$
717 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
721 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
722 \,\Longrightarrow\, \textit{even}~(k - 2)
723 \,\Longrightarrow\, \textit{even}~k.$
726 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
727 fixed point computation.%
728 \footnote{If an inductive predicate is
729 well-founded, then it has exactly one fixed point, which is simultaneously the
730 least and the greatest fixed point. In these circumstances, the computation of
731 the least fixed point amounts to the computation of an arbitrary fixed point,
732 which can be performed using a straightforward recursive equation.}
733 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
734 just as Isabelle's \textbf{function} package usually discharges termination
735 proof obligations automatically.
737 Let's try an example:
740 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
741 \textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
742 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
743 Nitpick can compute it efficiently. \\[2\smallskipamount]
745 \hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
746 Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
747 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
748 Nitpick could not find a better counterexample. \\[2\smallskipamount]
752 No genuine counterexample is possible because Nitpick cannot rule out the
753 existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
754 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
755 existential quantifier:
758 \textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
759 \textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}] \\[2\smallskipamount]
760 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
761 \hbox{}\qquad Empty assignment
764 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
765 we use the following definition instead?
768 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
769 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
770 ``$\textit{even}'~2$'' $\,\mid$ \\
771 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
774 This definition is not well-founded: From $\textit{even}'~0$ and
775 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
776 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
778 Let's check a property involving $\textit{even}'$. To make up for the
779 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
780 \textit{nat}'s cardinality to a mere 10:
783 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
784 \lnot\;\textit{even}'~n$'' \\
785 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
787 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
788 Nitpick might need to unroll it. \\[2\smallskipamount]
790 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
791 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
792 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
793 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
794 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
795 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
796 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
797 \hbox{}\qquad Constant: \nopagebreak \\
798 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
799 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
800 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
801 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
805 Nitpick's output is very instructive. First, it tells us that the predicate is
806 unrolled, meaning that it is computed iteratively from the empty set. Then it
807 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
810 The output also shows how each iteration contributes to $\textit{even}'$. The
811 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
812 predicate depends on an iteration counter. Iteration 0 provides the basis
813 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
814 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
815 iterations would not contribute any new elements.
817 Some values are marked with superscripted question
818 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
819 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
820 \textit{True} or $\unk$, never \textit{False}.
822 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
823 iterations. However, these numbers are bounded by the cardinality of the
824 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
825 ever needed to compute the value of a \textit{nat} predicate. You can specify
826 the number of iterations using the \textit{iter} option, as explained in
827 \S\ref{scope-of-search}.
829 In the next formula, $\textit{even}'$ occurs both positively and negatively:
832 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
833 \textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
834 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
835 \hbox{}\qquad Free variable: \nopagebreak \\
836 \hbox{}\qquad\qquad $n = 1$ \\
837 \hbox{}\qquad Constants: \nopagebreak \\
838 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
839 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
840 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
843 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
844 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
845 fixed point (not necessarily the least one). It is used to falsify
846 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
847 $\textit{even}'~(n - 2)$.
849 Coinductive predicates are handled dually. For example:
852 \textbf{coinductive} \textit{nats} \textbf{where} \\
853 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
854 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
855 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
856 \slshape Nitpick found a counterexample:
857 \\[2\smallskipamount]
858 \hbox{}\qquad Constants: \nopagebreak \\
859 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
860 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
861 & \unr\})\end{aligned}$ \\
862 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
865 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
866 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
867 inductive predicates for which each the predicate occurs in at most one
868 assumption of each introduction rule. For example:
871 \textbf{inductive} \textit{odd} \textbf{where} \\
872 ``$\textit{odd}~1$'' $\,\mid$ \\
873 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
874 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
875 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
876 \slshape Nitpick found a counterexample:
877 \\[2\smallskipamount]
878 \hbox{}\qquad Free variable: \nopagebreak \\
879 \hbox{}\qquad\qquad $n = 1$ \\
880 \hbox{}\qquad Constants: \nopagebreak \\
881 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
882 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
883 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
885 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
886 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
888 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
889 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
890 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
894 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
895 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
896 elements from known ones. The set $\textit{odd}$ consists of all the values
897 reachable through the reflexive transitive closure of
898 $\textit{odd}_{\textrm{step}}$ starting with any element from
899 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
900 transitive closure to encode linear predicates is normally either more thorough
901 or more efficient than unrolling (depending on the value of \textit{iter}), but
902 for those cases where it isn't you can disable it by passing the
903 \textit{dont\_star\_linear\_preds} option.
905 \subsection{Coinductive Datatypes}
906 \label{coinductive-datatypes}
908 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
909 datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
910 list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
911 these lazy lists seamlessly and provides a hook, described in
912 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
915 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
916 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
917 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
918 1, 2, 3, \ldots]$ can be defined as lazy lists using the
919 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
920 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
921 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
923 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
924 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
928 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
929 \textbf{nitpick} \\[2\smallskipamount]
930 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
931 \hbox{}\qquad Free variables: \nopagebreak \\
932 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
933 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
936 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
937 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
938 infinite list $[a_1, a_1, a_1, \ldots]$.
940 The next example is more interesting:
943 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
944 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
945 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
946 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
947 some scopes. \\[2\smallskipamount]
949 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 1,
950 and \textit{bisim\_depth}~= 0. \\
951 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
952 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 8,
953 and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
954 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
955 \textit{card}~``\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
957 \\[2\smallskipamount]
958 \hbox{}\qquad Free variables: \nopagebreak \\
959 \hbox{}\qquad\qquad $\textit{a} = a_2$ \\
960 \hbox{}\qquad\qquad $\textit{b} = a_1$ \\
961 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
962 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_1~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega)$ \\[2\smallskipamount]
966 The lazy list $\textit{xs}$ is simply $[a_2, a_2, a_2, \ldots]$, whereas
967 $\textit{ys}$ is $[a_1, a_2, a_2, a_2, \ldots]$, i.e., a lasso-shaped list with
968 $[a_1]$ as its stem and $[a_2]$ as its cycle. In general, the list segment
969 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
970 the segment leading to the binder is the stem.
972 A salient property of coinductive datatypes is that two objects are considered
973 equal if and only if they lead to the same observations. For example, the lazy
974 lists $\textrm{THE}~\omega.\; \omega =
975 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
976 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
977 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
978 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
979 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
980 concept of equality for coinductive datatypes is called bisimulation and is
981 defined coinductively.
983 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
984 the Kodkod problem to ensure that distinct objects lead to different
985 observations. This precaution is somewhat expensive and often unnecessary, so it
986 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
987 bisimilarity check is then performed \textsl{after} the counterexample has been
988 found to ensure correctness. If this after-the-fact check fails, the
989 counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
990 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
991 check for the previous example saves approximately 150~milli\-seconds; the speed
992 gains can be more significant for larger scopes.
994 The next formula illustrates the need for bisimilarity (either as a Kodkod
995 predicate or as an after-the-fact check) to prevent spurious counterexamples:
998 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
999 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1000 \textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
1001 \slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
1002 \hbox{}\qquad Free variables: \nopagebreak \\
1003 \hbox{}\qquad\qquad $a = a_2$ \\
1004 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
1005 \textit{LCons}~a_2~\omega$ \\
1006 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
1007 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
1008 \hbox{}\qquad\qquad $'a~\textit{llist} =
1009 \{\!\begin{aligned}[t]
1010 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega, \\[-2pt]
1011 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega,\> \unr\}\end{aligned}$
1012 \\[2\smallskipamount]
1013 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1014 that the counterexample is genuine. \\[2\smallskipamount]
1015 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1016 \slshape Nitpick found no counterexample.
1019 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1020 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1022 A compromise between leaving out the bisimilarity predicate from the Kodkod
1023 problem and performing the after-the-fact check is to specify a lower
1024 nonnegative \textit{bisim\_depth} value than the default one provided by
1025 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1026 be distinguished from each other by their prefixes of length $K$. Be aware that
1027 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1028 finding any counterexamples.
1033 Nitpick normally maps function and product types directly to the corresponding
1034 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1035 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1036 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1037 off to treat these types in the same way as plain datatypes, by approximating
1038 them by a subset of a given cardinality. This technique is called ``boxing'' and
1039 is particularly useful for functions passed as arguments to other functions, for
1040 high-arity functions, and for large tuples. Under the hood, boxing involves
1041 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1042 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1044 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1045 using de Bruijn's notation:
1048 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1051 The $\textit{lift}~t~k$ function increments all variables with indices greater
1052 than or equal to $k$ by one:
1055 \textbf{primrec} \textit{lift} \textbf{where} \\
1056 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1057 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1058 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1061 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1062 term $t$ has a loose variable with index $k$ or more:
1065 \textbf{primrec}~\textit{loose} \textbf{where} \\
1066 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1067 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1068 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1071 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1075 \textbf{primrec}~\textit{subst} \textbf{where} \\
1076 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1077 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1078 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1079 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1082 A substitution is a function that maps variable indices to terms. Observe that
1083 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1084 away, because the recursive call for the \textit{Lam} case involves an altered
1085 version. Also notice the \textit{lift} call, which increments the variable
1086 indices when moving under a \textit{Lam}.
1088 A reasonable property to expect of substitution is that it should leave closed
1089 terms unchanged. Alas, even this simple property does not hold:
1092 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1093 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1095 Trying 8 scopes: \nopagebreak \\
1096 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1097 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1098 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1099 \hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
1100 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1101 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1102 \hbox{}\qquad Free variables: \nopagebreak \\
1103 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1104 & 0 := \textit{Var}~0,\>
1105 1 := \textit{Var}~0,\>
1106 2 := \textit{Var}~0, \\[-2pt]
1107 & 3 := \textit{Var}~0,\>
1108 4 := \textit{Var}~0,\>
1109 5 := \textit{Var}~0)\end{aligned}$ \\
1110 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1111 Total time: $4679$ ms.
1114 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1115 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1116 $\lambda$-term notation, $t$~is
1117 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1118 The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
1119 replaced with $\textit{lift}~(\sigma~m)~0$.
1121 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1122 cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1123 For the formula of interest, knowing 6 values of that type was enough to find
1124 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1125 considered, a hopeless undertaking:
1128 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1129 {\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
1133 Boxing can be enabled or disabled globally or on a per-type basis using the
1134 \textit{box} option. Moreover, setting the cardinality of a function or
1135 product type implicitly enables boxing for that type. Nitpick usually performs
1136 reasonable choices about which types should be boxed, but option tweaking
1141 \subsection{Scope Monotonicity}
1142 \label{scope-monotonicity}
1144 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1145 and \textit{max}) controls which scopes are actually tested. In general, to
1146 exhaust all models below a certain cardinality bound, the number of scopes that
1147 Nitpick must consider increases exponentially with the number of type variables
1148 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1149 cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
1150 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1152 Fortunately, many formulas exhibit a property called \textsl{scope
1153 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1154 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1156 Consider the formula
1159 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1162 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1163 $'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
1164 exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
1165 that any counterexample found with a small scope would still be a counterexample
1166 in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
1167 by the larger scope. Nitpick comes to the same conclusion after a careful
1168 inspection of the formula and the relevant definitions:
1171 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1173 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1174 Nitpick might be able to skip some scopes.
1175 \\[2\smallskipamount]
1177 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1178 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1179 \textit{list}''~= 1, \\
1180 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
1181 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
1182 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1183 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1184 \textit{list}''~= 2, \\
1185 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
1186 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
1187 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1188 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
1189 \textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
1190 \textit{list}''~= 8, \\
1191 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
1192 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
1193 \\[2\smallskipamount]
1194 Nitpick found a counterexample for
1195 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1196 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1197 \textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
1198 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
1199 \\[2\smallskipamount]
1200 \hbox{}\qquad Free variables: \nopagebreak \\
1201 \hbox{}\qquad\qquad $\textit{xs} = [a_4, a_5]$ \\
1202 \hbox{}\qquad\qquad $\textit{ys} = [b_3, b_3]$ \\[2\smallskipamount]
1203 Total time: 1636 ms.
1206 In theory, it should be sufficient to test a single scope:
1209 \textbf{nitpick}~[\textit{card}~= 8]
1212 However, this is often less efficient in practice and may lead to overly complex
1215 If the monotonicity check fails but we believe that the formula is monotonic (or
1216 we don't mind missing some counterexamples), we can pass the
1217 \textit{mono} option. To convince yourself that this option is risky,
1218 simply consider this example from \S\ref{skolemization}:
1221 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1222 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1223 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1224 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1225 \textbf{nitpick} \\[2\smallskipamount]
1227 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1228 \hbox{}\qquad $\vdots$
1231 (It turns out the formula holds if and only if $\textit{card}~'a \le
1232 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1233 monotonicity checks can be disabled by passing \textit{non\_mono}
1234 (\S\ref{optimizations}).
1236 As insinuated in \S\ref{natural-numbers-and-integers} and
1237 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1238 are normally monotonic and treated as such. The same is true for record types,
1239 \textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
1240 cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
1241 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1242 consider only 8~scopes instead of $32\,768$.
1244 \subsection{Inductive Properties}
1245 \label{inductive-properties}
1247 Inductive properties are a particular pain to prove, because the failure to
1248 establish an induction step can mean several things:
1251 \item The property is invalid.
1252 \item The property is valid but is too weak to support the induction step.
1253 \item The property is valid and strong enough; it's just that we haven't found
1257 Depending on which scenario applies, we would take the appropriate course of
1261 \item Repair the statement of the property so that it becomes valid.
1262 \item Generalize the property and/or prove auxiliary properties.
1263 \item Work harder on a proof.
1266 How can we distinguish between the three scenarios? Nitpick's normal mode of
1267 operation can often detect scenario 1, and Isabelle's automatic tactics help with
1268 scenario 3. Using appropriate techniques, it is also often possible to use
1269 Nitpick to identify scenario 2. Consider the following transition system,
1270 in which natural numbers represent states:
1273 \textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
1274 ``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
1275 ``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
1276 ``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
1279 We will try to prove that only even numbers are reachable:
1282 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
1285 Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
1286 so let's attempt a proof by induction:
1289 \textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
1290 \textbf{apply}~\textit{auto}
1293 This leaves us in the following proof state:
1296 {\slshape goal (2 subgoals): \\
1297 \phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
1298 \phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
1302 If we run Nitpick on the first subgoal, it still won't find any
1303 counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
1304 is helpless. However, notice the $n \in \textit{reach}$ assumption, which
1305 strengthens the induction hypothesis but is not immediately usable in the proof.
1306 If we remove it and invoke Nitpick, this time we get a counterexample:
1309 \textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
1310 \textbf{nitpick} \\[2\smallskipamount]
1311 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1312 \hbox{}\qquad Skolem constant: \nopagebreak \\
1313 \hbox{}\qquad\qquad $n = 0$
1316 Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
1317 to strength the lemma:
1320 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
1323 Unfortunately, the proof by induction still gets stuck, except that Nitpick now
1324 finds the counterexample $n = 2$. We generalize the lemma further to
1327 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
1330 and this time \textit{arith} can finish off the subgoals.
1332 A similar technique can be employed for structural induction. The
1333 following mini-formalization of full binary trees will serve as illustration:
1336 \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]
1337 \textbf{primrec}~\textit{labels}~\textbf{where} \\
1338 ``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
1339 ``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
1340 \textbf{primrec}~\textit{swap}~\textbf{where} \\
1341 ``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
1342 \phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
1343 ``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
1346 The \textit{labels} function returns the set of labels occurring on leaves of a
1347 tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
1348 labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
1349 obtained by swapping $a$ and $b$:
1352 \textbf{lemma} $``\lbrakk a \in \textit{labels}~t;\, b \in \textit{labels}~t;\, a \not= b\rbrakk {}$ \\
1353 \phantom{\textbf{lemma} ``}$\,{\Longrightarrow}{\;\,} \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
1356 Nitpick can't find any counterexample, so we proceed with induction
1357 (this time favoring a more structured style):
1360 \textbf{proof}~(\textit{induct}~$t$) \\
1361 \hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
1363 \hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
1366 Nitpick can't find any counterexample at this point either, but it makes the
1367 following suggestion:
1371 Hint: To check that the induction hypothesis is general enough, try the following command:
1372 \textbf{nitpick}~[\textit{non\_std} ``${\kern1pt'a}~\textit{bin\_tree}$'', \textit{show\_consts}].
1375 If we follow the hint, we get a ``nonstandard'' counterexample for the step:
1378 \slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1379 \hbox{}\qquad Free variables: \nopagebreak \\
1380 \hbox{}\qquad\qquad $a = a_4$ \\
1381 \hbox{}\qquad\qquad $b = a_3$ \\
1382 \hbox{}\qquad\qquad $t = \xi_3$ \\
1383 \hbox{}\qquad\qquad $u = \xi_4$ \\
1384 \hbox{}\qquad {\slshape Constants:} \nopagebreak \\
1385 \hbox{}\qquad\qquad $\textit{labels} = \undef
1386 (\!\begin{aligned}[t]%
1387 & \xi_3 := \{a_4\},\> \xi_4 := \{a_1, a_3\}, \\[-2pt] %% TYPESETTING
1388 & \textit{Branch}~\xi_3~\xi_3 := \{a_4\}, \\[-2pt]
1389 & \textit{Branch}~\xi_3~\xi_4 := \{a_1, a_3, a_4\})\end{aligned}$ \\
1390 \hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \undef
1391 (\!\begin{aligned}[t]%
1392 & \xi_3 := \xi_3,\> \xi_4 := \xi_3, \\[-2pt]
1393 & \textit{Branch}~\xi_3~\xi_3 := \textit{Branch}~\xi_3~\xi_3, \\[-2pt]
1394 & \textit{Branch}~\xi_4~\xi_3 := \textit{Branch}~\xi_3~\xi_3)\end{aligned}$ \\[2\smallskipamount]
1395 The existence of a nonstandard model suggests that the induction hypothesis is not general enough or perhaps
1396 even wrong. See the ``Inductive Properties'' section of the Nitpick manual for details (``\textit{isabelle doc nitpick}'').
1399 Reading the Nitpick manual is a most excellent idea.
1400 But what's going on? The \textit{non\_std} ``${\kern1pt'a}~\textit{bin\_tree}$''
1401 option told the tool to look for nonstandard models of binary trees, which
1402 means that new ``nonstandard'' trees $\xi_1, \xi_2, \ldots$, are now allowed in
1403 addition to the standard trees generated by the \textit{Leaf} and
1404 \textit{Branch} constructors.%
1405 \footnote{Notice the similarity between allowing nonstandard trees here and
1406 allowing unreachable states in the preceding example (by removing the ``$n \in
1407 \textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
1408 set of objects over which the induction is performed while doing the step
1409 so as to test the induction hypothesis's strength.}
1410 The new trees are so nonstandard that we know nothing about them, except what
1411 the induction hypothesis states and what can be proved about all trees without
1412 relying on induction or case distinction. The key observation is,
1415 \textsl{If the induction
1416 hypothesis is strong enough, the induction step will hold even for nonstandard
1417 objects, and Nitpick won't find any nonstandard counterexample.}
1420 But here, Nitpick did find some nonstandard trees $t = \xi_3$
1421 and $u = \xi_4$ such that $a \in \textit{labels}~t$, $b \notin
1422 \textit{labels}~t$, $a \notin \textit{labels}~u$, and $b \in \textit{labels}~u$.
1423 Because neither tree contains both $a$ and $b$, the induction hypothesis tells
1424 us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
1425 and as a result we know nothing about the labels of the tree
1426 $\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
1427 $\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
1428 labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
1429 \textit{labels}$ $(\textit{swap}~u~a~b)$.
1431 The solution is to ensure that we always know what the labels of the subtrees
1432 are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
1433 $t$ in the statement of the lemma:
1436 \textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
1437 \phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
1438 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
1439 \phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
1440 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
1443 This time, Nitpick won't find any nonstandard counterexample, and we can perform
1444 the induction step using \textbf{auto}.
1446 \section{Case Studies}
1447 \label{case-studies}
1449 As a didactic device, the previous section focused mostly on toy formulas whose
1450 validity can easily be assessed just by looking at the formula. We will now
1451 review two somewhat more realistic case studies that are within Nitpick's
1452 reach:\ a context-free grammar modeled by mutually inductive sets and a
1453 functional implementation of AA trees. The results presented in this
1454 section were produced with the following settings:
1457 \textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
1460 \subsection{A Context-Free Grammar}
1461 \label{a-context-free-grammar}
1463 Our first case study is taken from section 7.4 in the Isabelle tutorial
1464 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1465 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1468 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1469 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1470 $A$ & $::=$ & $aS \mid bAA$ \\
1471 $B$ & $::=$ & $bS \mid aBB$
1475 The intuition behind the grammar is that $A$ generates all string with one more
1476 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1478 The alphabet consists exclusively of $a$'s and $b$'s:
1481 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1484 Strings over the alphabet are represented by \textit{alphabet list}s.
1485 Nonterminals in the grammar become sets of strings. The production rules
1486 presented above can be expressed as a mutually inductive definition:
1489 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1490 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1491 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1492 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1493 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1494 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1495 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1498 The conversion of the grammar into the inductive definition was done manually by
1499 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1502 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1503 d'\^etre}. A good approach is to state desirable properties of the specification
1504 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1505 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1506 counterexamples will point to bugs in the specification. For our grammar
1507 example, we will proceed in two steps, separating the soundness and the
1508 completeness of the set $S$. First, soundness:
1511 \textbf{theorem}~\textit{S\_sound}: \\
1512 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1513 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1514 \textbf{nitpick} \\[2\smallskipamount]
1515 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1516 \hbox{}\qquad Free variable: \nopagebreak \\
1517 \hbox{}\qquad\qquad $w = [b]$
1520 It would seem that $[b] \in S$. How could this be? An inspection of the
1521 introduction rules reveals that the only rule with a right-hand side of the form
1522 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1526 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1529 On closer inspection, we can see that this rule is wrong. To match the
1530 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1534 \textbf{nitpick} \\[2\smallskipamount]
1535 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1536 \hbox{}\qquad Free variable: \nopagebreak \\
1537 \hbox{}\qquad\qquad $w = [a, a, b]$
1540 Some detective work is necessary to find out what went wrong here. To get $[a,
1541 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1545 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1548 Now, this formula must be wrong: The same assumption occurs twice, and the
1549 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1550 the assumptions should have been a $w$.
1552 With the correction made, we don't get any counterexample from Nitpick. Let's
1553 move on and check completeness:
1556 \textbf{theorem}~\textit{S\_complete}: \\
1557 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1558 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1559 \longrightarrow w \in S$'' \\
1560 \textbf{nitpick} \\[2\smallskipamount]
1561 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1562 \hbox{}\qquad Free variable: \nopagebreak \\
1563 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1566 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1567 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1568 the introduction rules we have are probably correct. Perhaps we simply lack an
1569 introduction rule. Comparing the grammar with the inductive definition, our
1570 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1571 without which the grammar cannot generate two or more $b$'s in a row. So we add
1575 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1578 With this last change, we don't get any counterexamples from Nitpick for either
1579 soundness or completeness. We can even generalize our result to cover $A$ and
1583 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1584 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1585 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1586 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1587 \textbf{nitpick} \\[2\smallskipamount]
1588 \slshape Nitpick found no counterexample.
1591 \subsection{AA Trees}
1594 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1595 similar performance to red-black trees, but with a simpler implementation
1596 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1597 a total order $<$. We start by defining the datatype and some basic extractor
1601 \textbf{datatype} $'a$~\textit{aa\_tree} = \\
1602 \hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}'' \\[2\smallskipamount]
1603 \textbf{primrec} \textit{data} \textbf{where} \\
1604 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1605 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1606 \textbf{primrec} \textit{dataset} \textbf{where} \\
1607 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1608 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1609 \textbf{primrec} \textit{level} \textbf{where} \\
1610 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1611 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1612 \textbf{primrec} \textit{left} \textbf{where} \\
1613 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1614 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1615 \textbf{primrec} \textit{right} \textbf{where} \\
1616 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1617 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1620 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1621 as follows \cite{wikipedia-2009-aa-trees}:
1623 \kern.2\parskip %% TYPESETTING
1626 Each node has a level field, and the following invariants must remain true for
1627 the tree to be valid:
1631 \kern-.4\parskip %% TYPESETTING
1636 \item[1.] The level of a leaf node is one.
1637 \item[2.] The level of a left child is strictly less than that of its parent.
1638 \item[3.] The level of a right child is less than or equal to that of its parent.
1639 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1640 \item[5.] Every node of level greater than one must have two children.
1645 \kern.4\parskip %% TYPESETTING
1647 The \textit{wf} predicate formalizes this description:
1650 \textbf{primrec} \textit{wf} \textbf{where} \\
1651 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1652 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1653 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1654 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1655 \phantom{``$($}$\textrm{else}$ \\
1656 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1657 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1658 \mathrel{\land} \textit{level}~u \le k$ \\
1659 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1662 Rebalancing the tree upon insertion and removal of elements is performed by two
1663 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1666 \textbf{primrec} \textit{skew} \textbf{where} \\
1667 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1668 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1669 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1670 \textit{level}~t~\textrm{then}$ \\
1671 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1672 (\textit{right}~t)~u)$ \\
1673 \phantom{``(}$\textrm{else}$ \\
1674 \phantom{``(\quad}$N~x~k~t~u)$''
1678 \textbf{primrec} \textit{split} \textbf{where} \\
1679 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1680 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1681 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1682 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1683 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1684 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1685 \phantom{``(}$\textrm{else}$ \\
1686 \phantom{``(\quad}$N~x~k~t~u)$''
1689 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1690 of elements stored in the tree:
1693 \textbf{theorem}~\textit{dataset\_skew\_split}:\\
1694 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1695 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1696 \textbf{nitpick} \\[2\smallskipamount]
1697 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1700 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1701 should not alter the tree:
1704 \textbf{theorem}~\textit{wf\_skew\_split}:\\
1705 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1706 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1707 \textbf{nitpick} \\[2\smallskipamount]
1708 {\slshape Nitpick found no counterexample.}
1711 Insertion is implemented recursively. It preserves the sort order:
1714 \textbf{primrec}~\textit{insort} \textbf{where} \\
1715 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1716 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1717 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1718 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1721 Notice that we deliberately commented out the application of \textit{skew} and
1722 \textit{split}. Let's see if this causes any problems:
1725 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1726 \textbf{nitpick} \\[2\smallskipamount]
1727 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1728 \hbox{}\qquad Free variables: \nopagebreak \\
1729 \hbox{}\qquad\qquad $t = N~a_3~1~\Lambda~\Lambda$ \\
1730 \hbox{}\qquad\qquad $x = a_4$
1733 It's hard to see why this is a counterexample. To improve readability, we will
1734 restrict the theorem to \textit{nat}, so that we don't need to look up the value
1735 of the $\textit{op}~{<}$ constant to find out which element is smaller than the
1736 other. In addition, we will tell Nitpick to display the value of
1737 $\textit{insort}~t~x$ using the \textit{eval} option. This gives
1740 \textbf{theorem} \textit{wf\_insort\_nat}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1741 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1742 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1743 \hbox{}\qquad Free variables: \nopagebreak \\
1744 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1745 \hbox{}\qquad\qquad $x = 0$ \\
1746 \hbox{}\qquad Evaluated term: \\
1747 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1750 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1751 where both have a level of 1. This violates the second AA tree invariant, which
1752 states that a left child's level must be less than its parent's. This shouldn't
1753 come as a surprise, considering that we commented out the tree rebalancing code.
1754 Reintroducing the code seems to solve the problem:
1757 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1758 \textbf{nitpick} \\[2\smallskipamount]
1759 {\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
1762 Insertion should transform the set of elements represented by the tree in the
1766 \textbf{theorem} \textit{dataset\_insort}:\kern.4em
1767 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1768 \textbf{nitpick} \\[2\smallskipamount]
1769 {\slshape Nitpick ran out of time after checking 5 of 8 scopes.}
1772 We could continue like this and sketch a complete theory of AA trees without
1773 performing a single proof. Once the definitions and main theorems are in place
1774 and have been thoroughly tested using Nitpick, we could start working on the
1775 proofs. Developing theories this way usually saves time, because faulty theorems
1776 and definitions are discovered much earlier in the process.
1778 \section{Option Reference}
1779 \label{option-reference}
1781 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1782 \def\qty#1{$\left<\textit{#1}\right>$}
1783 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1784 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1785 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1786 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1787 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1788 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1789 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1790 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1791 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1793 Nitpick's behavior can be influenced by various options, which can be specified
1794 in brackets after the \textbf{nitpick} command. Default values can be set
1795 using \textbf{nitpick\_\allowbreak params}. For example:
1798 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1801 The options are categorized as follows:\ mode of operation
1802 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1803 format (\S\ref{output-format}), automatic counterexample checks
1804 (\S\ref{authentication}), optimizations
1805 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1807 You can instruct Nitpick to run automatically on newly entered theorems by
1808 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1809 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
1810 and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1811 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1812 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1813 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1814 \textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
1815 Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
1818 The number of options can be overwhelming at first glance. Do not let that worry
1819 you: Nitpick's defaults have been chosen so that it almost always does the right
1820 thing, and the most important options have been covered in context in
1821 \S\ref{first-steps}.
1823 The descriptions below refer to the following syntactic quantities:
1826 \item[$\bullet$] \qtybf{string}: A string.
1827 \item[$\bullet$] \qtybf{bool}: \textit{true} or \textit{false}.
1828 \item[$\bullet$] \qtybf{bool\_or\_smart}: \textit{true}, \textit{false}, or \textit{smart}.
1829 \item[$\bullet$] \qtybf{int}: An integer. Negative integers are prefixed with a hyphen.
1830 \item[$\bullet$] \qtybf{int\_or\_smart}: An integer or \textit{smart}.
1831 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1832 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1834 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1835 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1836 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1837 \item[$\bullet$] \qtybf{const}: The name of a HOL constant.
1838 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1839 \item[$\bullet$] \qtybf{term\_list}: A space-separated list of HOL terms (e.g.,
1840 ``$f~x$''~``$g~y$'').
1841 \item[$\bullet$] \qtybf{type}: A HOL type.
1844 Default values are indicated in square brackets. Boolean options have a negated
1845 counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
1846 Boolean options, ``= \textit{true}'' may be omitted.
1848 \subsection{Mode of Operation}
1849 \label{mode-of-operation}
1852 \optrue{blocking}{non\_blocking}
1853 Specifies whether the \textbf{nitpick} command should operate synchronously.
1854 The asynchronous (non-blocking) mode lets the user start proving the putative
1855 theorem while Nitpick looks for a counterexample, but it can also be more
1856 confusing. For technical reasons, automatic runs currently always block.
1858 \optrue{falsify}{satisfy}
1859 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1860 satisfying examples (models). This manual assumes throughout that
1861 \textit{falsify} is enabled.
1863 \opsmart{user\_axioms}{no\_user\_axioms}
1864 Specifies whether the user-defined axioms (specified using
1865 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1866 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1867 the constants that occur in the formula to falsify. The option is implicitly set
1868 to \textit{true} for automatic runs.
1870 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1871 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1872 these conditions are tagged as ``likely genuine.'' The \textit{debug}
1873 (\S\ref{output-format}) option can be used to find out which axioms were
1877 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1878 (\S\ref{output-format}).}
1880 \optrue{assms}{no\_assms}
1881 Specifies whether the relevant assumptions in structured proof should be
1882 considered. The option is implicitly enabled for automatic runs.
1885 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1887 \opfalse{overlord}{no\_overlord}
1888 Specifies whether Nitpick should put its temporary files in
1889 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1890 debugging Nitpick but also unsafe if several instances of the tool are run
1894 {\small See also \textit{debug} (\S\ref{output-format}).}
1897 \subsection{Scope of Search}
1898 \label{scope-of-search}
1901 \oparg{card}{type}{int\_seq}
1902 Specifies the sequence of cardinalities to use for a given type.
1903 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1904 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1905 Although function and product types are normally mapped directly to the
1906 corresponding Kodkod concepts, setting
1907 the cardinality of such types is also allowed and implicitly enables ``boxing''
1908 for them, as explained in the description of the \textit{box}~\qty{type}
1909 and \textit{box} (\S\ref{scope-of-search}) options.
1912 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1914 \opdefault{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
1915 Specifies the default sequence of cardinalities to use. This can be overridden
1916 on a per-type basis using the \textit{card}~\qty{type} option described above.
1918 \oparg{max}{const}{int\_seq}
1919 Specifies the sequence of maximum multiplicities to use for a given
1920 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1921 number of distinct values that it can construct. Nonsensical values (e.g.,
1922 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1923 datatypes equipped with several constructors.
1925 \opnodefault{max}{int\_seq}
1926 Specifies the default sequence of maximum multiplicities to use for
1927 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
1928 basis using the \textit{max}~\qty{const} option described above.
1930 \opsmart{binary\_ints}{unary\_ints}
1931 Specifies whether natural numbers and integers should be encoded using a unary
1932 or binary notation. In unary mode, the cardinality fully specifies the subset
1933 used to approximate the type. For example:
1935 $$\hbox{\begin{tabular}{@{}rll@{}}%
1936 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
1937 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
1938 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
1943 $$\hbox{\begin{tabular}{@{}rll@{}}%
1944 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
1945 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
1948 In binary mode, the cardinality specifies the number of distinct values that can
1949 be constructed. Each of these value is represented by a bit pattern whose length
1950 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
1951 Nitpick attempts to choose the more appropriate encoding by inspecting the
1952 formula at hand, preferring the binary notation for problems involving
1953 multiplicative operators or large constants.
1955 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
1956 problems that refer to the types \textit{rat} or \textit{real} or the constants
1957 \textit{Suc}, \textit{gcd}, or \textit{lcm}.
1959 {\small See also \textit{bits} (\S\ref{scope-of-search}) and
1960 \textit{show\_datatypes} (\S\ref{output-format}).}
1962 \opdefault{bits}{int\_seq}{$\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{8},\mathbf{10},\mathbf{12}$}
1963 Specifies the number of bits to use to represent natural numbers and integers in
1964 binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
1966 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
1968 \opargboolorsmart{wf}{const}{non\_wf}
1969 Specifies whether the specified (co)in\-duc\-tively defined predicate is
1970 well-founded. The option can take the following values:
1973 \item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
1974 predicate as if it were well-founded. Since this is generally not sound when the
1975 predicate is not well-founded, the counterexamples are tagged as ``likely
1978 \item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
1979 as if it were not well-founded. The predicate is then unrolled as prescribed by
1980 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
1983 \item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
1984 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
1985 \textit{size\_change} tactics. If this succeeds (or the predicate occurs with an
1986 appropriate polarity in the formula to falsify), use an efficient fixed point
1987 equation as specification of the predicate; otherwise, unroll the predicates
1988 according to the \textit{iter}~\qty{const} and \textit{iter} options.
1992 {\small See also \textit{iter} (\S\ref{scope-of-search}),
1993 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
1994 (\S\ref{timeouts}).}
1996 \opsmart{wf}{non\_wf}
1997 Specifies the default wellfoundedness setting to use. This can be overridden on
1998 a per-predicate basis using the \textit{wf}~\qty{const} option above.
2000 \oparg{iter}{const}{int\_seq}
2001 Specifies the sequence of iteration counts to use when unrolling a given
2002 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
2003 predicates that occur negatively and coinductive predicates that occur
2004 positively in the formula to falsify and that cannot be proved to be
2005 well-founded, but this behavior is influenced by the \textit{wf} option. The
2006 iteration counts are automatically bounded by the cardinality of the predicate's
2009 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
2010 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
2012 \opdefault{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
2013 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
2014 predicates. This can be overridden on a per-predicate basis using the
2015 \textit{iter} \qty{const} option above.
2017 \opdefault{bisim\_depth}{int\_seq}{$\mathbf{7}$}
2018 Specifies the sequence of iteration counts to use when unrolling the
2019 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
2020 of $-1$ means that no predicate is generated, in which case Nitpick performs an
2021 after-the-fact check to see if the known coinductive datatype values are
2022 bidissimilar. If two values are found to be bisimilar, the counterexample is
2023 tagged as ``likely genuine.'' The iteration counts are automatically bounded by
2024 the sum of the cardinalities of the coinductive datatypes occurring in the
2027 \opargboolorsmart{box}{type}{dont\_box}
2028 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
2029 product type in an isomorphic datatype internally. Boxing is an effective mean
2030 to reduce the search space and speed up Nitpick, because the isomorphic datatype
2031 is approximated by a subset of the possible function or pair values;
2032 like other drastic optimizations, it can also prevent the discovery of
2033 counterexamples. The option can take the following values:
2036 \item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
2038 \item[$\bullet$] \textbf{\textit{false}}: Never box the type.
2039 \item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
2040 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
2041 higher-order functions are good candidates for boxing.
2044 Setting the \textit{card}~\qty{type} option for a function or product type
2045 implicitly enables boxing for that type.
2048 {\small See also \textit{verbose} (\S\ref{output-format})
2049 and \textit{debug} (\S\ref{output-format}).}
2051 \opsmart{box}{dont\_box}
2052 Specifies the default boxing setting to use. This can be overridden on a
2053 per-type basis using the \textit{box}~\qty{type} option described above.
2055 \opargboolorsmart{mono}{type}{non\_mono}
2056 Specifies whether the given type should be considered monotonic when
2057 enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a
2058 monotonicity check on the type. Setting this option to \textit{true} can reduce
2059 the number of scopes tried, but it also diminishes the theoretical chance of
2060 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
2063 {\small See also \textit{card} (\S\ref{scope-of-search}),
2064 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
2065 (\S\ref{output-format}).}
2067 \opsmart{mono}{non\_box}
2068 Specifies the default monotonicity setting to use. This can be overridden on a
2069 per-type basis using the \textit{mono}~\qty{type} option described above.
2071 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
2072 Specifies whether type variables with the same sort constraints should be
2073 merged. Setting this option to \textit{true} can reduce the number of scopes
2074 tried and the size of the generated Kodkod formulas, but it also diminishes the
2075 theoretical chance of finding a counterexample.
2077 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
2079 \opargbool{std}{type}{non\_std}
2080 Specifies whether the given type should be given standard models.
2081 Nonstandard models are unsound but can help debug inductive arguments,
2082 as explained in \S\ref{inductive-properties}.
2084 \optrue{std}{non\_std}
2085 Specifies the default standardness to use. This can be overridden on a per-type
2086 basis using the \textit{std}~\qty{type} option described above.
2089 \subsection{Output Format}
2090 \label{output-format}
2093 \opfalse{verbose}{quiet}
2094 Specifies whether the \textbf{nitpick} command should explain what it does. This
2095 option is useful to determine which scopes are tried or which SAT solver is
2096 used. This option is implicitly disabled for automatic runs.
2098 \opfalse{debug}{no\_debug}
2099 Specifies whether Nitpick should display additional debugging information beyond
2100 what \textit{verbose} already displays. Enabling \textit{debug} also enables
2101 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
2102 option is implicitly disabled for automatic runs.
2105 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
2106 \textit{batch\_size} (\S\ref{optimizations}).}
2108 \optrue{show\_skolems}{hide\_skolem}
2109 Specifies whether the values of Skolem constants should be displayed as part of
2110 counterexamples. Skolem constants correspond to bound variables in the original
2111 formula and usually help us to understand why the counterexample falsifies the
2115 {\small See also \textit{skolemize} (\S\ref{optimizations}).}
2117 \opfalse{show\_datatypes}{hide\_datatypes}
2118 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
2119 be displayed as part of counterexamples. Such subsets are sometimes helpful when
2120 investigating whether a potential counterexample is genuine or spurious, but
2121 their potential for clutter is real.
2123 \opfalse{show\_consts}{hide\_consts}
2124 Specifies whether the values of constants occurring in the formula (including
2125 its axioms) should be displayed along with any counterexample. These values are
2126 sometimes helpful when investigating why a counterexample is
2127 genuine, but they can clutter the output.
2129 \opfalse{show\_all}{dont\_show\_all}
2130 Enabling this option effectively enables \textit{show\_skolems},
2131 \textit{show\_datatypes}, and \textit{show\_consts}.
2133 \opdefault{max\_potential}{int}{$\mathbf{1}$}
2134 Specifies the maximum number of potential counterexamples to display. Setting
2135 this option to 0 speeds up the search for a genuine counterexample. This option
2136 is implicitly set to 0 for automatic runs. If you set this option to a value
2137 greater than 1, you will need an incremental SAT solver: For efficiency, it is
2138 recommended to install the JNI version of MiniSat and set \textit{sat\_solver} =
2139 \textit{MiniSatJNI}. Also be aware that many of the counterexamples may look
2140 identical, unless the \textit{show\_all} (\S\ref{output-format}) option is
2144 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
2145 \textit{sat\_solver} (\S\ref{optimizations}).}
2147 \opdefault{max\_genuine}{int}{$\mathbf{1}$}
2148 Specifies the maximum number of genuine counterexamples to display. If you set
2149 this option to a value greater than 1, you will need an incremental SAT solver:
2150 For efficiency, it is recommended to install the JNI version of MiniSat and set
2151 \textit{sat\_solver} = \textit{MiniSatJNI}. Also be aware that many of the
2152 counterexamples may look identical, unless the \textit{show\_all}
2153 (\S\ref{output-format}) option is enabled.
2156 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
2157 \textit{sat\_solver} (\S\ref{optimizations}).}
2159 \opnodefault{eval}{term\_list}
2160 Specifies the list of terms whose values should be displayed along with
2161 counterexamples. This option suffers from an ``observer effect'': Nitpick might
2162 find different counterexamples for different values of this option.
2164 \oparg{format}{term}{int\_seq}
2165 Specifies how to uncurry the value displayed for a variable or constant.
2166 Uncurrying sometimes increases the readability of the output for high-arity
2167 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
2168 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
2169 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
2170 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
2171 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
2172 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
2173 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
2174 arguments that are not accounted for are left alone, as if the specification had
2175 been $1,\ldots,1,n_1,\ldots,n_k$.
2178 {\small See also \textit{uncurry} (\S\ref{optimizations}).}
2180 \opdefault{format}{int\_seq}{$\mathbf{1}$}
2181 Specifies the default format to use. Irrespective of the default format, the
2182 extra arguments to a Skolem constant corresponding to the outer bound variables
2183 are kept separated from the remaining arguments, the \textbf{for} arguments of
2184 an inductive definitions are kept separated from the remaining arguments, and
2185 the iteration counter of an unrolled inductive definition is shown alone. The
2186 default format can be overridden on a per-variable or per-constant basis using
2187 the \textit{format}~\qty{term} option described above.
2190 %% MARK: Authentication
2191 \subsection{Authentication}
2192 \label{authentication}
2195 \opfalse{check\_potential}{trust\_potential}
2196 Specifies whether potential counterexamples should be given to Isabelle's
2197 \textit{auto} tactic to assess their validity. If a potential counterexample is
2198 shown to be genuine, Nitpick displays a message to this effect and terminates.
2201 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
2203 \opfalse{check\_genuine}{trust\_genuine}
2204 Specifies whether genuine and likely genuine counterexamples should be given to
2205 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
2206 counterexample is shown to be spurious, the user is kindly asked to send a bug
2207 report to the author at
2208 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
2211 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
2213 \opnodefault{expect}{string}
2214 Specifies the expected outcome, which must be one of the following:
2217 \item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
2218 \item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a ``likely
2219 genuine'' counterexample (i.e., a counterexample that is genuine unless
2220 it contradicts a missing axiom or a dangerous option was used inappropriately).
2221 \item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
2222 \item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
2223 \item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
2224 Kodkod ran out of memory).
2227 Nitpick emits an error if the actual outcome differs from the expected outcome.
2228 This option is useful for regression testing.
2231 \subsection{Optimizations}
2232 \label{optimizations}
2234 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
2239 \opdefault{sat\_solver}{string}{smart}
2240 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2241 to be faster than their Java counterparts, but they can be more difficult to
2242 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2243 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2244 you will need an incremental SAT solver, such as \textit{MiniSatJNI}
2245 (recommended) or \textit{SAT4J}.
2247 The supported solvers are listed below:
2251 \item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
2252 written in \cpp{}. To use MiniSat, set the environment variable
2253 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2254 executable. The \cpp{} sources and executables for MiniSat are available at
2255 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2256 and 2.0 beta (2007-07-21).
2258 \item[$\bullet$] \textbf{\textit{MiniSatJNI}}: The JNI (Java Native Interface)
2259 version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which
2260 you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2261 version of MiniSat, the JNI version can be used incrementally.
2264 %%% "It is bundled with Kodkodi and requires no further installation or
2265 %%% configuration steps. Alternatively,"
2266 \item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
2267 written in C. You can install a standard version of
2268 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2269 that contains the \texttt{picosat} executable. The C sources for PicoSAT are
2270 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2271 Nitpick has been tested with version 913.
2273 \item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
2274 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2275 the directory that contains the \texttt{zchaff} executable. The \cpp{} sources
2276 and executables for zChaff are available at
2277 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2278 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2280 \item[$\bullet$] \textbf{\textit{zChaffJNI}}: The JNI version of zChaff is
2281 bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on
2282 Kodkod's web site \cite{kodkod-2009}.
2284 \item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
2285 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2286 directory that contains the \texttt{rsat} executable. The \cpp{} sources for
2287 RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been
2288 tested with version 2.01.
2290 \item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
2291 written in C. To use BerkMin, set the environment variable
2292 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2293 executable. The BerkMin executables are available at
2294 \url{http://eigold.tripod.com/BerkMin.html}.
2296 \item[$\bullet$] \textbf{\textit{BerkMinAlloy}}: Variant of BerkMin that is
2297 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2298 version of BerkMin, set the environment variable
2299 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2302 \item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
2303 written in C. To use Jerusat, set the environment variable
2304 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2305 executable. The C sources for Jerusat are available at
2306 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2308 \item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
2309 written in Java that can be used incrementally. It is bundled with Kodkodi and
2310 requires no further installation or configuration steps. Do not attempt to
2311 install the official SAT4J packages, because their API is incompatible with
2314 \item[$\bullet$] \textbf{\textit{SAT4JLight}}: Variant of SAT4J that is
2315 optimized for small problems. It can also be used incrementally.
2317 \item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
2318 experimental solver written in \cpp. To use HaifaSat, set the environment
2319 variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
2320 \texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at
2321 \url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
2323 \item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
2324 \textit{smart}, Nitpick selects the first solver among MiniSat,
2325 PicoSAT, zChaff, RSat, BerkMin, BerkMinAlloy, Jerusat, MiniSatJNI, and zChaffJNI
2326 that is recognized by Isabelle. If none is found, it falls back on SAT4J, which
2327 should always be available. If \textit{verbose} (\S\ref{output-format}) is
2328 enabled, Nitpick displays which SAT solver was chosen.
2332 \opdefault{batch\_size}{int\_or\_smart}{smart}
2333 Specifies the maximum number of Kodkod problems that should be lumped together
2334 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2335 together ensures that Kodkodi is launched less often, but it makes the verbose
2336 output less readable and is sometimes detrimental to performance. If
2337 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2338 \textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
2340 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2341 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2342 be rewritten to use (automatically generated) discriminators and destructors.
2343 This optimization can drastically reduce the size of the Boolean formulas given
2347 {\small See also \textit{debug} (\S\ref{output-format}).}
2349 \optrue{specialize}{dont\_specialize}
2350 Specifies whether functions invoked with static arguments should be specialized.
2351 This optimization can drastically reduce the search space, especially for
2352 higher-order functions.
2355 {\small See also \textit{debug} (\S\ref{output-format}) and
2356 \textit{show\_consts} (\S\ref{output-format}).}
2358 \optrue{skolemize}{dont\_skolemize}
2359 Specifies whether the formula should be skolemized. For performance reasons,
2360 (positive) $\forall$-quanti\-fiers that occur in the scope of a higher-order
2361 (positive) $\exists$-quanti\-fier are left unchanged.
2364 {\small See also \textit{debug} (\S\ref{output-format}) and
2365 \textit{show\_skolems} (\S\ref{output-format}).}
2367 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2368 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2369 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2370 predicates for which each the predicate occurs in at most one assumption of each
2371 introduction rule. Using the reflexive transitive closure is in principle
2372 equivalent to setting \textit{iter} to the cardinality of the predicate's
2373 domain, but it is usually more efficient.
2375 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2376 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2378 \optrue{uncurry}{dont\_uncurry}
2379 Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no
2380 tangible effect on efficiency, but it creates opportunities for the boxing
2384 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug}
2385 (\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).}
2387 \optrue{fast\_descrs}{full\_descrs}
2388 Specifies whether Nitpick should optimize the definite and indefinite
2389 description operators (THE and SOME). The optimized versions usually help
2390 Nitpick generate more counterexamples or at least find them faster, but only the
2391 unoptimized versions are complete when all types occurring in the formula are
2394 {\small See also \textit{debug} (\S\ref{output-format}).}
2396 \optrue{peephole\_optim}{no\_peephole\_optim}
2397 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2398 peephole optimizer. These optimizations can make a significant difference.
2399 Unless you are tracking down a bug in Nitpick or distrust the peephole
2400 optimizer, you should leave this option enabled.
2402 \opdefault{sym\_break}{int}{20}
2403 Specifies an upper bound on the number of relations for which Kodkod generates
2404 symmetry breaking predicates. According to the Kodkod documentation
2405 \cite{kodkod-2009-options}, ``in general, the higher this value, the more
2406 symmetries will be broken, and the faster the formula will be solved. But,
2407 setting the value too high may have the opposite effect and slow down the
2410 \opdefault{sharing\_depth}{int}{3}
2411 Specifies the depth to which Kodkod should check circuits for equivalence during
2412 the translation to SAT. The default of 3 is the same as in Alloy. The minimum
2413 allowed depth is 1. Increasing the sharing may result in a smaller SAT problem,
2414 but can also slow down Kodkod.
2416 \opfalse{flatten\_props}{dont\_flatten\_props}
2417 Specifies whether Kodkod should try to eliminate intermediate Boolean variables.
2418 Although this might sound like a good idea, in practice it can drastically slow
2421 \opdefault{max\_threads}{int}{0}
2422 Specifies the maximum number of threads to use in Kodkod. If this option is set
2423 to 0, Kodkod will compute an appropriate value based on the number of processor
2427 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2428 \textit{timeout} (\S\ref{timeouts}).}
2431 \subsection{Timeouts}
2435 \opdefault{timeout}{time}{$\mathbf{30}$ s}
2436 Specifies the maximum amount of time that the \textbf{nitpick} command should
2437 spend looking for a counterexample. Nitpick tries to honor this constraint as
2438 well as it can but offers no guarantees. For automatic runs,
2439 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2440 a time slot whose length is specified by the ``Auto Counterexample Time
2441 Limit'' option in Proof General.
2444 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2446 \opdefault{tac\_timeout}{time}{$\mathbf{500}$\,ms}
2447 Specifies the maximum amount of time that the \textit{auto} tactic should use
2448 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2449 and \textit{size\_change} should use when checking whether a (co)in\-duc\-tive
2450 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2451 can but offers no guarantees.
2454 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2455 \textit{check\_potential} (\S\ref{authentication}),
2456 and \textit{check\_genuine} (\S\ref{authentication}).}
2459 \section{Attribute Reference}
2460 \label{attribute-reference}
2462 Nitpick needs to consider the definitions of all constants occurring in a
2463 formula in order to falsify it. For constants introduced using the
2464 \textbf{definition} command, the definition is simply the associated
2465 \textit{\_def} axiom. In contrast, instead of using the internal representation
2466 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2467 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2468 equational specification entered by the user.
2470 Behind the scenes, Isabelle's built-in packages and theories rely on the
2471 following attributes to affect Nitpick's behavior:
2474 \flushitem{\textit{nitpick\_def}}
2477 This attribute specifies an alternative definition of a constant. The
2478 alternative definition should be logically equivalent to the constant's actual
2479 axiomatic definition and should be of the form
2481 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2483 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2486 \flushitem{\textit{nitpick\_simp}}
2489 This attribute specifies the equations that constitute the specification of a
2490 constant. For functions defined using the \textbf{primrec}, \textbf{function},
2491 and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
2492 \textit{simps} rules. The equations must be of the form
2494 \qquad $c~t_1~\ldots\ t_n \,=\, u.$
2496 \flushitem{\textit{nitpick\_psimp}}
2499 This attribute specifies the equations that constitute the partial specification
2500 of a constant. For functions defined using the \textbf{function} package, this
2501 corresponds to the \textit{psimps} rules. The conditional equations must be of
2504 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
2506 \flushitem{\textit{nitpick\_intro}}
2509 This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
2510 For predicates defined using the \textbf{inductive} or \textbf{coinductive}
2511 command, this corresponds to the \textit{intros} rules. The introduction rules
2514 \qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
2515 \ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\
2518 where the $P_i$'s are side conditions that do not involve $c$ and $M$ is an
2519 optional monotonic operator. The order of the assumptions is irrelevant.
2523 When faced with a constant, Nitpick proceeds as follows:
2526 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2527 is not empty, Nitpick uses these rules as the specification of the constant.
2529 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2530 the constant is not empty, it uses these rules as the specification of the
2533 \item[3.] Otherwise, it looks up the definition of the constant:
2536 \item[1.] If the \textit{nitpick\_def} set associated with the constant
2537 is not empty, it uses the latest rule added to the set as the definition of the
2538 constant; otherwise it uses the actual definition axiom.
2539 \item[2.] If the definition is of the form
2541 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
2543 then Nitpick assumes that the definition was made using an inductive package and
2544 based on the introduction rules marked with \textit{nitpick\_\allowbreak
2545 ind\_\allowbreak intros} tries to determine whether the definition is
2550 As an illustration, consider the inductive definition
2553 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2554 ``\textit{odd}~1'' $\,\mid$ \\
2555 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2558 Isabelle automatically attaches the \textit{nitpick\_intro} attribute to
2559 the above rules. Nitpick then uses the \textit{lfp}-based definition in
2560 conjunction with these rules. To override this, we can specify an alternative
2561 definition as follows:
2564 \textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]: ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2567 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2568 = 1$. Alternatively, we can specify an equational specification of the constant:
2571 \textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]: ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2574 Such tweaks should be done with great care, because Nitpick will assume that the
2575 constant is completely defined by its equational specification. For example, if
2576 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
2577 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2578 (\S\ref{output-format}) option is extremely useful to understand what is going
2579 on when experimenting with \textit{nitpick\_} attributes.
2581 \section{Standard ML Interface}
2582 \label{standard-ml-interface}
2584 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2585 and debugging. Among the most interesting functions exported by Nitpick are
2586 those that let you invoke the tool programmatically and those that let you
2587 register and unregister custom coinductive datatypes.
2589 \subsection{Invocation of Nitpick}
2590 \label{invocation-of-nitpick}
2592 The \textit{Nitpick} structure offers the following functions for invoking your
2593 favorite counterexample generator:
2596 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2597 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2598 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2599 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2600 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2603 The return value is a new proof state paired with an outcome string
2604 (``genuine'', ``likely\_genuine'', ``potential'', ``none'', or ``unknown''). The
2605 \textit{params} type is a large record that lets you set Nitpick's options. The
2606 current default options can be retrieved by calling the following function
2607 defined in the \textit{Nitpick\_Isar} structure:
2610 $\textbf{val}\,~\textit{default\_params} :\,
2611 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2614 The second argument lets you override option values before they are parsed and
2615 put into a \textit{params} record. Here is an example:
2618 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2619 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2620 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2621 & \textit{subgoal}\end{aligned}$
2626 \subsection{Registration of Coinductive Datatypes}
2627 \label{registration-of-coinductive-datatypes}
2629 If you have defined a custom coinductive datatype, you can tell Nitpick about
2630 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2631 uses for lazy lists. The interface for registering and unregistering coinductive
2632 datatypes consists of the following pair of functions defined in the
2633 \textit{Nitpick} structure:
2636 $\textbf{val}\,~\textit{register\_codatatype} :\,
2637 \textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2638 $\textbf{val}\,~\textit{unregister\_codatatype} :\,
2639 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2642 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2643 not been, you could have told Nitpick about it by adding the following line
2644 to your theory file:
2647 $\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
2648 & \textit{Nitpick.register\_codatatype} \\[-2pt]
2649 & \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
2650 & \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
2653 The \textit{register\_codatatype} function takes a coinductive type, its case
2654 function, and the list of its constructors. The case function must take its
2655 arguments in the order that the constructors are listed. If no case function
2656 with the correct signature is available, simply pass the empty string.
2658 On the other hand, if your goal is to cripple Nitpick, add the following line to
2659 your theory file and try to check a few conjectures about lazy lists:
2662 $\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
2663 \kern1pt'a~\textit{list}\textrm{''}\}\ \,{*}\}$
2666 Inductive datatypes can be registered as coinductive datatypes, given
2667 appropriate coinductive constructors. However, doing so precludes
2668 the use of the inductive constructors---Nitpick will generate an error if they
2671 \section{Known Bugs and Limitations}
2672 \label{known-bugs-and-limitations}
2674 Here are the known bugs and limitations in Nitpick at the time of writing:
2677 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2678 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2679 Nitpick to generate spurious counterexamples for theorems that refer to values
2680 for which the function is not defined. For example:
2683 \textbf{primrec} \textit{prec} \textbf{where} \\
2684 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2685 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2686 \textbf{nitpick} \\[2\smallskipamount]
2687 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2689 \\[2\smallskipamount]
2690 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2691 \textbf{by}~(\textit{auto simp}:~\textit{prec\_def})
2694 Such theorems are considered bad style because they rely on the internal
2695 representation of functions synthesized by Isabelle, which is an implementation
2698 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2699 which can become invalid if you change the definition of an inductive predicate
2700 that is registered in the cache. To clear the cache,
2701 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2702 501$\,\textit{ms}$).
2704 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2705 \textbf{guess} command in a structured proof.
2707 \item[$\bullet$] The \textit{nitpick\_} attributes and the
2708 \textit{Nitpick.register\_} functions can cause havoc if used improperly.
2710 \item[$\bullet$] Although this has never been observed, arbitrary theorem
2711 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
2713 \item[$\bullet$] Local definitions are not supported and result in an error.
2715 %\item[$\bullet$] All constants and types whose names start with
2716 %\textit{Nitpick}{.} are reserved for internal use.
2720 \bibliography{../manual}{}
2721 \bibliographystyle{abbrv}