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{MiniSat\_JNI}, \,\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 $\mathit{The} = \undef{}
313 (\!\begin{aligned}[t]%
314 & \{a_1, a_2, a_3\} := a_3,\> \{a_1, a_2\} := a_3,\> \{a_1, a_3\} := a_3, \\[-2pt] %% TYPESETTING
315 & \{a_1\} := a_1,\> \{a_2, a_3\} := a_1,\> \{a_2\} := a_2, \\[-2pt]
316 & \{a_3\} := a_3,\> \{\} := a_3)\end{aligned}$
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 $\textit{y} = a_1$
556 To see why the counterexample is genuine, we enable \textit{show\_consts}
557 and \textit{show\_\allowbreak datatypes}:
560 {\slshape Datatype:} \\
561 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\
562 {\slshape Constants:} \\
563 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_1, a_1])$ \\
564 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$
567 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
570 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
571 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_1,
572 a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not
573 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
574 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
575 appending $[a_1, a_1]$ to itself gives $\unk$.
577 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
578 considers the following subsets:
580 \kern-.5\smallskipamount %% TYPESETTING
584 $\{[],\, [a_1],\, [a_2]\}$; \\
585 $\{[],\, [a_1],\, [a_3]\}$; \\
586 $\{[],\, [a_2],\, [a_3]\}$; \\
587 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\
588 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\
589 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\
590 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\
591 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\
592 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\
593 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\
594 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\
595 $\{[],\, [a_3],\, [a_3, a_3]\}$.
599 \kern-2\smallskipamount %% TYPESETTING
601 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
602 are listed and only those. As an example of a non-subterm-closed subset,
603 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe
604 that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin
605 \mathcal{S}$ as a subterm.
607 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
610 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
611 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
613 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
614 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
615 \hbox{}\qquad Free variables: \nopagebreak \\
616 \hbox{}\qquad\qquad $\textit{xs} = [a_1]$ \\
617 \hbox{}\qquad\qquad $\textit{ys} = [a_2]$ \\
618 \hbox{}\qquad Datatypes: \\
619 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
620 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$
623 Because datatypes are approximated using a three-valued logic, there is usually
624 no need to systematically enumerate cardinalities: If Nitpick cannot find a
625 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
626 unlikely that one could be found for smaller cardinalities.
628 \subsection{Typedefs, 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 $P = \{\Abs{0},\, \Abs{1}\}$ \\
646 \hbox{}\qquad\qquad $x = \Abs{2}$ \\
647 \hbox{}\qquad Datatypes: \\
648 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
649 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$
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 $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
668 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
669 \hbox{}\qquad Datatypes: \\
670 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
671 \hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
672 & \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
673 & \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
676 Finally, Nitpick provides rudimentary support for rationals and reals using a
680 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
681 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
682 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
683 \hbox{}\qquad Free variables: \nopagebreak \\
684 \hbox{}\qquad\qquad $x = 1/2$ \\
685 \hbox{}\qquad\qquad $y = -1/2$ \\
686 \hbox{}\qquad Datatypes: \\
687 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
688 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
689 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
692 \subsection{Inductive and Coinductive Predicates}
693 \label{inductive-and-coinductive-predicates}
695 Inductively defined predicates (and sets) are particularly problematic for
696 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
697 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
698 the problem is that they are defined using a least fixed point construction.
700 Nitpick's philosophy is that not all inductive predicates are equal. Consider
701 the \textit{even} predicate below:
704 \textbf{inductive}~\textit{even}~\textbf{where} \\
705 ``\textit{even}~0'' $\,\mid$ \\
706 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
709 This predicate enjoys the desirable property of being well-founded, which means
710 that the introduction rules don't give rise to infinite chains of the form
713 $\cdots\,\Longrightarrow\, \textit{even}~k''
714 \,\Longrightarrow\, \textit{even}~k'
715 \,\Longrightarrow\, \textit{even}~k.$
718 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
722 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
723 \,\Longrightarrow\, \textit{even}~(k - 2)
724 \,\Longrightarrow\, \textit{even}~k.$
727 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
728 fixed point computation.%
729 \footnote{If an inductive predicate is
730 well-founded, then it has exactly one fixed point, which is simultaneously the
731 least and the greatest fixed point. In these circumstances, the computation of
732 the least fixed point amounts to the computation of an arbitrary fixed point,
733 which can be performed using a straightforward recursive equation.}
734 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
735 just as Isabelle's \textbf{function} package usually discharges termination
736 proof obligations automatically.
738 Let's try an example:
741 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
742 \textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
743 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
744 Nitpick can compute it efficiently. \\[2\smallskipamount]
746 \hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
747 Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
748 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
749 Nitpick could not find a better counterexample. \\[2\smallskipamount]
753 No genuine counterexample is possible because Nitpick cannot rule out the
754 existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
755 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
756 existential quantifier:
759 \textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
760 \textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}] \\[2\smallskipamount]
761 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
762 \hbox{}\qquad Empty assignment
765 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
766 we use the following definition instead?
769 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
770 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
771 ``$\textit{even}'~2$'' $\,\mid$ \\
772 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
775 This definition is not well-founded: From $\textit{even}'~0$ and
776 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
777 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
779 Let's check a property involving $\textit{even}'$. To make up for the
780 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
781 \textit{nat}'s cardinality to a mere 10:
784 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
785 \lnot\;\textit{even}'~n$'' \\
786 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
788 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
789 Nitpick might need to unroll it. \\[2\smallskipamount]
791 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
792 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
793 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
794 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
795 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
796 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
797 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
798 \hbox{}\qquad Constant: \nopagebreak \\
799 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
800 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
801 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
802 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
806 Nitpick's output is very instructive. First, it tells us that the predicate is
807 unrolled, meaning that it is computed iteratively from the empty set. Then it
808 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
811 The output also shows how each iteration contributes to $\textit{even}'$. The
812 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
813 predicate depends on an iteration counter. Iteration 0 provides the basis
814 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
815 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
816 iterations would not contribute any new elements.
818 Some values are marked with superscripted question
819 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
820 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
821 \textit{True} or $\unk$, never \textit{False}.
823 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
824 iterations. However, these numbers are bounded by the cardinality of the
825 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
826 ever needed to compute the value of a \textit{nat} predicate. You can specify
827 the number of iterations using the \textit{iter} option, as explained in
828 \S\ref{scope-of-search}.
830 In the next formula, $\textit{even}'$ occurs both positively and negatively:
833 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
834 \textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
835 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
836 \hbox{}\qquad Free variable: \nopagebreak \\
837 \hbox{}\qquad\qquad $n = 1$ \\
838 \hbox{}\qquad Constants: \nopagebreak \\
839 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
840 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
841 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
844 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
845 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
846 fixed point (not necessarily the least one). It is used to falsify
847 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
848 $\textit{even}'~(n - 2)$.
850 Coinductive predicates are handled dually. For example:
853 \textbf{coinductive} \textit{nats} \textbf{where} \\
854 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
855 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
856 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
857 \slshape Nitpick found a counterexample:
858 \\[2\smallskipamount]
859 \hbox{}\qquad Constants: \nopagebreak \\
860 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
861 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
862 & \unr\})\end{aligned}$ \\
863 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
866 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
867 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
868 inductive predicates for which each the predicate occurs in at most one
869 assumption of each introduction rule. For example:
872 \textbf{inductive} \textit{odd} \textbf{where} \\
873 ``$\textit{odd}~1$'' $\,\mid$ \\
874 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
875 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
876 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
877 \slshape Nitpick found a counterexample:
878 \\[2\smallskipamount]
879 \hbox{}\qquad Free variable: \nopagebreak \\
880 \hbox{}\qquad\qquad $n = 1$ \\
881 \hbox{}\qquad Constants: \nopagebreak \\
882 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
883 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
884 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
886 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
887 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
889 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
890 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
891 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
895 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
896 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
897 elements from known ones. The set $\textit{odd}$ consists of all the values
898 reachable through the reflexive transitive closure of
899 $\textit{odd}_{\textrm{step}}$ starting with any element from
900 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
901 transitive closure to encode linear predicates is normally either more thorough
902 or more efficient than unrolling (depending on the value of \textit{iter}), but
903 for those cases where it isn't you can disable it by passing the
904 \textit{dont\_star\_linear\_preds} option.
906 \subsection{Coinductive Datatypes}
907 \label{coinductive-datatypes}
909 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
910 datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
911 list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
912 these lazy lists seamlessly and provides a hook, described in
913 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
916 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
917 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
918 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
919 1, 2, 3, \ldots]$ can be defined as lazy lists using the
920 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
921 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
922 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
924 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
925 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
929 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
930 \textbf{nitpick} \\[2\smallskipamount]
931 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
932 \hbox{}\qquad Free variables: \nopagebreak \\
933 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
934 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
937 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
938 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
939 infinite list $[a_1, a_1, a_1, \ldots]$.
941 The next example is more interesting:
944 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
945 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
946 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
947 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
948 some scopes. \\[2\smallskipamount]
950 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 1,
951 and \textit{bisim\_depth}~= 0. \\
952 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
953 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 8,
954 and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
955 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
956 \textit{card}~``\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
958 \\[2\smallskipamount]
959 \hbox{}\qquad Free variables: \nopagebreak \\
960 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
961 \hbox{}\qquad\qquad $\textit{b} = a_2$ \\
962 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
963 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
967 The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
968 $\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
969 $[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment
970 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
971 the segment leading to the binder is the stem.
973 A salient property of coinductive datatypes is that two objects are considered
974 equal if and only if they lead to the same observations. For example, the lazy
975 lists $\textrm{THE}~\omega.\; \omega =
976 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
977 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
978 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
979 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
980 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
981 concept of equality for coinductive datatypes is called bisimulation and is
982 defined coinductively.
984 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
985 the Kodkod problem to ensure that distinct objects lead to different
986 observations. This precaution is somewhat expensive and often unnecessary, so it
987 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
988 bisimilarity check is then performed \textsl{after} the counterexample has been
989 found to ensure correctness. If this after-the-fact check fails, the
990 counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
991 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
992 check for the previous example saves approximately 150~milli\-seconds; the speed
993 gains can be more significant for larger scopes.
995 The next formula illustrates the need for bisimilarity (either as a Kodkod
996 predicate or as an after-the-fact check) to prevent spurious counterexamples:
999 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
1000 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1001 \textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
1002 \slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
1003 \hbox{}\qquad Free variables: \nopagebreak \\
1004 \hbox{}\qquad\qquad $a = a_1$ \\
1005 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
1006 \textit{LCons}~a_1~\omega$ \\
1007 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1008 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
1009 \hbox{}\qquad\qquad $'a~\textit{llist} =
1010 \{\!\begin{aligned}[t]
1011 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
1012 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
1013 \\[2\smallskipamount]
1014 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1015 that the counterexample is genuine. \\[2\smallskipamount]
1016 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1017 \slshape Nitpick found no counterexample.
1020 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1021 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1023 A compromise between leaving out the bisimilarity predicate from the Kodkod
1024 problem and performing the after-the-fact check is to specify a lower
1025 nonnegative \textit{bisim\_depth} value than the default one provided by
1026 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1027 be distinguished from each other by their prefixes of length $K$. Be aware that
1028 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1029 finding any counterexamples.
1034 Nitpick normally maps function and product types directly to the corresponding
1035 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1036 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1037 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1038 off to treat these types in the same way as plain datatypes, by approximating
1039 them by a subset of a given cardinality. This technique is called ``boxing'' and
1040 is particularly useful for functions passed as arguments to other functions, for
1041 high-arity functions, and for large tuples. Under the hood, boxing involves
1042 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1043 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1045 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1046 using de Bruijn's notation:
1049 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1052 The $\textit{lift}~t~k$ function increments all variables with indices greater
1053 than or equal to $k$ by one:
1056 \textbf{primrec} \textit{lift} \textbf{where} \\
1057 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1058 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1059 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1062 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1063 term $t$ has a loose variable with index $k$ or more:
1066 \textbf{primrec}~\textit{loose} \textbf{where} \\
1067 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1068 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1069 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1072 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1076 \textbf{primrec}~\textit{subst} \textbf{where} \\
1077 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1078 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1079 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1080 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1083 A substitution is a function that maps variable indices to terms. Observe that
1084 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1085 away, because the recursive call for the \textit{Lam} case involves an altered
1086 version. Also notice the \textit{lift} call, which increments the variable
1087 indices when moving under a \textit{Lam}.
1089 A reasonable property to expect of substitution is that it should leave closed
1090 terms unchanged. Alas, even this simple property does not hold:
1093 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1094 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1096 Trying 8 scopes: \nopagebreak \\
1097 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1098 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1099 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1100 \hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
1101 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1102 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1103 \hbox{}\qquad Free variables: \nopagebreak \\
1104 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1105 & 0 := \textit{Var}~0,\>
1106 1 := \textit{Var}~0,\>
1107 2 := \textit{Var}~0, \\[-2pt]
1108 & 3 := \textit{Var}~0,\>
1109 4 := \textit{Var}~0,\>
1110 5 := \textit{Var}~0)\end{aligned}$ \\
1111 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1112 Total time: $4679$ ms.
1115 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1116 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1117 $\lambda$-term notation, $t$~is
1118 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1119 The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
1120 replaced with $\textit{lift}~(\sigma~m)~0$.
1122 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1123 cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1124 For the formula of interest, knowing 6 values of that type was enough to find
1125 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1126 considered, a hopeless undertaking:
1129 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1130 {\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
1134 Boxing can be enabled or disabled globally or on a per-type basis using the
1135 \textit{box} option. Moreover, setting the cardinality of a function or
1136 product type implicitly enables boxing for that type. Nitpick usually performs
1137 reasonable choices about which types should be boxed, but option tweaking
1142 \subsection{Scope Monotonicity}
1143 \label{scope-monotonicity}
1145 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1146 and \textit{max}) controls which scopes are actually tested. In general, to
1147 exhaust all models below a certain cardinality bound, the number of scopes that
1148 Nitpick must consider increases exponentially with the number of type variables
1149 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1150 cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
1151 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1153 Fortunately, many formulas exhibit a property called \textsl{scope
1154 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1155 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1157 Consider the formula
1160 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1163 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1164 $'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
1165 exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
1166 that any counterexample found with a small scope would still be a counterexample
1167 in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
1168 by the larger scope. Nitpick comes to the same conclusion after a careful
1169 inspection of the formula and the relevant definitions:
1172 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1174 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1175 Nitpick might be able to skip some scopes.
1176 \\[2\smallskipamount]
1178 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1179 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1180 \textit{list}''~= 1, \\
1181 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
1182 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
1183 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1184 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1185 \textit{list}''~= 2, \\
1186 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
1187 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
1188 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1189 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
1190 \textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
1191 \textit{list}''~= 8, \\
1192 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
1193 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
1194 \\[2\smallskipamount]
1195 Nitpick found a counterexample for
1196 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1197 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1198 \textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
1199 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
1200 \\[2\smallskipamount]
1201 \hbox{}\qquad Free variables: \nopagebreak \\
1202 \hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
1203 \hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
1204 Total time: 1636 ms.
1207 In theory, it should be sufficient to test a single scope:
1210 \textbf{nitpick}~[\textit{card}~= 8]
1213 However, this is often less efficient in practice and may lead to overly complex
1216 If the monotonicity check fails but we believe that the formula is monotonic (or
1217 we don't mind missing some counterexamples), we can pass the
1218 \textit{mono} option. To convince yourself that this option is risky,
1219 simply consider this example from \S\ref{skolemization}:
1222 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1223 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1224 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1225 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1226 \textbf{nitpick} \\[2\smallskipamount]
1228 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1229 \hbox{}\qquad $\vdots$
1232 (It turns out the formula holds if and only if $\textit{card}~'a \le
1233 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1234 monotonicity checks can be disabled by passing \textit{non\_mono}
1235 (\S\ref{optimizations}).
1237 As insinuated in \S\ref{natural-numbers-and-integers} and
1238 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1239 are normally monotonic and treated as such. The same is true for record types,
1240 \textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
1241 cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
1242 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1243 consider only 8~scopes instead of $32\,768$.
1245 \subsection{Inductive Properties}
1246 \label{inductive-properties}
1248 Inductive properties are a particular pain to prove, because the failure to
1249 establish an induction step can mean several things:
1252 \item The property is invalid.
1253 \item The property is valid but is too weak to support the induction step.
1254 \item The property is valid and strong enough; it's just that we haven't found
1258 Depending on which scenario applies, we would take the appropriate course of
1262 \item Repair the statement of the property so that it becomes valid.
1263 \item Generalize the property and/or prove auxiliary properties.
1264 \item Work harder on a proof.
1267 How can we distinguish between the three scenarios? Nitpick's normal mode of
1268 operation can often detect scenario 1, and Isabelle's automatic tactics help with
1269 scenario 3. Using appropriate techniques, it is also often possible to use
1270 Nitpick to identify scenario 2. Consider the following transition system,
1271 in which natural numbers represent states:
1274 \textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
1275 ``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
1276 ``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
1277 ``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
1280 We will try to prove that only even numbers are reachable:
1283 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
1286 Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
1287 so let's attempt a proof by induction:
1290 \textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
1291 \textbf{apply}~\textit{auto}
1294 This leaves us in the following proof state:
1297 {\slshape goal (2 subgoals): \\
1298 \phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
1299 \phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
1303 If we run Nitpick on the first subgoal, it still won't find any
1304 counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
1305 is helpless. However, notice the $n \in \textit{reach}$ assumption, which
1306 strengthens the induction hypothesis but is not immediately usable in the proof.
1307 If we remove it and invoke Nitpick, this time we get a counterexample:
1310 \textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
1311 \textbf{nitpick} \\[2\smallskipamount]
1312 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1313 \hbox{}\qquad Skolem constant: \nopagebreak \\
1314 \hbox{}\qquad\qquad $n = 0$
1317 Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
1318 to strength the lemma:
1321 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
1324 Unfortunately, the proof by induction still gets stuck, except that Nitpick now
1325 finds the counterexample $n = 2$. We generalize the lemma further to
1328 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
1331 and this time \textit{arith} can finish off the subgoals.
1333 A similar technique can be employed for structural induction. The
1334 following mini formalization of full binary trees will serve as illustration:
1337 \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]
1338 \textbf{primrec}~\textit{labels}~\textbf{where} \\
1339 ``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
1340 ``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
1341 \textbf{primrec}~\textit{swap}~\textbf{where} \\
1342 ``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
1343 \phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
1344 ``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
1347 The \textit{labels} function returns the set of labels occurring on leaves of a
1348 tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
1349 labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
1350 obtained by swapping $a$ and $b$:
1353 \textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
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 this command:
1372 \textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
1375 If we follow the hint, we get a ``nonstandard'' counterexample for the step:
1378 \slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
1379 \hbox{}\qquad Free variables: \nopagebreak \\
1380 \hbox{}\qquad\qquad $a = a_1$ \\
1381 \hbox{}\qquad\qquad $b = a_2$ \\
1382 \hbox{}\qquad\qquad $t = \xi_1$ \\
1383 \hbox{}\qquad\qquad $u = \xi_2$ \\
1384 \hbox{}\qquad Datatype: \nopagebreak \\
1385 \hbox{}\qquad\qquad $\alpha~\textit{btree} = \{\xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \textit{Branch}~\xi_1~\xi_2\}$ \\
1386 \hbox{}\qquad {\slshape Constants:} \nopagebreak \\
1387 \hbox{}\qquad\qquad $\textit{labels} = \undef
1388 (\!\begin{aligned}[t]%
1389 & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
1390 & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
1391 \hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \undef
1392 (\!\begin{aligned}[t]%
1393 & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
1394 & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[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} option told the tool to look for
1401 nonstandard models of binary trees, which means that new ``nonstandard'' trees
1402 $\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees
1403 generated by the \textit{Leaf} and \textit{Branch} constructors.%
1404 \footnote{Notice the similarity between allowing nonstandard trees here and
1405 allowing unreachable states in the preceding example (by removing the ``$n \in
1406 \textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
1407 set of objects over which the induction is performed while doing the step
1408 in order to test the induction hypothesis's strength.}
1409 Unlike standard trees, these new trees contain cycles. We will see later that
1410 every property of acyclic trees that can be proved without using induction also
1411 holds for cyclic trees. Hence,
1414 \textsl{If the induction
1415 hypothesis is strong enough, the induction step will hold even for nonstandard
1416 objects, and Nitpick won't find any nonstandard counterexample.}
1419 But here the tool find some nonstandard trees $t = \xi_1$
1420 and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in
1421 \textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$.
1422 Because neither tree contains both $a$ and $b$, the induction hypothesis tells
1423 us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
1424 and as a result we know nothing about the labels of the tree
1425 $\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
1426 $\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
1427 labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
1428 \textit{labels}$ $(\textit{swap}~u~a~b)$.
1430 The solution is to ensure that we always know what the labels of the subtrees
1431 are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
1432 $t$ in the statement of the lemma:
1435 \textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
1436 \phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
1437 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
1438 \phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
1439 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
1442 This time, Nitpick won't find any nonstandard counterexample, and we can perform
1443 the induction step using \textit{auto}.
1445 \section{Case Studies}
1446 \label{case-studies}
1448 As a didactic device, the previous section focused mostly on toy formulas whose
1449 validity can easily be assessed just by looking at the formula. We will now
1450 review two somewhat more realistic case studies that are within Nitpick's
1451 reach:\ a context-free grammar modeled by mutually inductive sets and a
1452 functional implementation of AA trees. The results presented in this
1453 section were produced with the following settings:
1456 \textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
1459 \subsection{A Context-Free Grammar}
1460 \label{a-context-free-grammar}
1462 Our first case study is taken from section 7.4 in the Isabelle tutorial
1463 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1464 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1467 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1468 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1469 $A$ & $::=$ & $aS \mid bAA$ \\
1470 $B$ & $::=$ & $bS \mid aBB$
1474 The intuition behind the grammar is that $A$ generates all string with one more
1475 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1477 The alphabet consists exclusively of $a$'s and $b$'s:
1480 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1483 Strings over the alphabet are represented by \textit{alphabet list}s.
1484 Nonterminals in the grammar become sets of strings. The production rules
1485 presented above can be expressed as a mutually inductive definition:
1488 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1489 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1490 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1491 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1492 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1493 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1494 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1497 The conversion of the grammar into the inductive definition was done manually by
1498 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1501 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1502 d'\^etre}. A good approach is to state desirable properties of the specification
1503 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1504 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1505 counterexamples will point to bugs in the specification. For our grammar
1506 example, we will proceed in two steps, separating the soundness and the
1507 completeness of the set $S$. First, soundness:
1510 \textbf{theorem}~\textit{S\_sound}: \\
1511 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1512 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1513 \textbf{nitpick} \\[2\smallskipamount]
1514 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1515 \hbox{}\qquad Free variable: \nopagebreak \\
1516 \hbox{}\qquad\qquad $w = [b]$
1519 It would seem that $[b] \in S$. How could this be? An inspection of the
1520 introduction rules reveals that the only rule with a right-hand side of the form
1521 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1525 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1528 On closer inspection, we can see that this rule is wrong. To match the
1529 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1533 \textbf{nitpick} \\[2\smallskipamount]
1534 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1535 \hbox{}\qquad Free variable: \nopagebreak \\
1536 \hbox{}\qquad\qquad $w = [a, a, b]$
1539 Some detective work is necessary to find out what went wrong here. To get $[a,
1540 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1544 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1547 Now, this formula must be wrong: The same assumption occurs twice, and the
1548 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1549 the assumptions should have been a $w$.
1551 With the correction made, we don't get any counterexample from Nitpick. Let's
1552 move on and check completeness:
1555 \textbf{theorem}~\textit{S\_complete}: \\
1556 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1557 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1558 \longrightarrow w \in S$'' \\
1559 \textbf{nitpick} \\[2\smallskipamount]
1560 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1561 \hbox{}\qquad Free variable: \nopagebreak \\
1562 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1565 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1566 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1567 the introduction rules we have are probably correct. Perhaps we simply lack an
1568 introduction rule. Comparing the grammar with the inductive definition, our
1569 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1570 without which the grammar cannot generate two or more $b$'s in a row. So we add
1574 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1577 With this last change, we don't get any counterexamples from Nitpick for either
1578 soundness or completeness. We can even generalize our result to cover $A$ and
1582 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1583 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1584 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1585 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1586 \textbf{nitpick} \\[2\smallskipamount]
1587 \slshape Nitpick found no counterexample.
1590 \subsection{AA Trees}
1593 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1594 similar performance to red-black trees, but with a simpler implementation
1595 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1596 a total order $<$. We start by defining the datatype and some basic extractor
1600 \textbf{datatype} $'a$~\textit{aa\_tree} = \\
1601 \hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}'' \\[2\smallskipamount]
1602 \textbf{primrec} \textit{data} \textbf{where} \\
1603 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1604 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1605 \textbf{primrec} \textit{dataset} \textbf{where} \\
1606 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1607 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1608 \textbf{primrec} \textit{level} \textbf{where} \\
1609 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1610 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1611 \textbf{primrec} \textit{left} \textbf{where} \\
1612 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1613 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1614 \textbf{primrec} \textit{right} \textbf{where} \\
1615 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1616 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1619 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1620 as follows \cite{wikipedia-2009-aa-trees}:
1622 \kern.2\parskip %% TYPESETTING
1625 Each node has a level field, and the following invariants must remain true for
1626 the tree to be valid:
1630 \kern-.4\parskip %% TYPESETTING
1635 \item[1.] The level of a leaf node is one.
1636 \item[2.] The level of a left child is strictly less than that of its parent.
1637 \item[3.] The level of a right child is less than or equal to that of its parent.
1638 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1639 \item[5.] Every node of level greater than one must have two children.
1644 \kern.4\parskip %% TYPESETTING
1646 The \textit{wf} predicate formalizes this description:
1649 \textbf{primrec} \textit{wf} \textbf{where} \\
1650 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1651 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1652 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1653 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1654 \phantom{``$($}$\textrm{else}$ \\
1655 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1656 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1657 \mathrel{\land} \textit{level}~u \le k$ \\
1658 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1661 Rebalancing the tree upon insertion and removal of elements is performed by two
1662 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1665 \textbf{primrec} \textit{skew} \textbf{where} \\
1666 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1667 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1668 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1669 \textit{level}~t~\textrm{then}$ \\
1670 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1671 (\textit{right}~t)~u)$ \\
1672 \phantom{``(}$\textrm{else}$ \\
1673 \phantom{``(\quad}$N~x~k~t~u)$''
1677 \textbf{primrec} \textit{split} \textbf{where} \\
1678 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1679 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1680 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1681 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1682 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1683 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1684 \phantom{``(}$\textrm{else}$ \\
1685 \phantom{``(\quad}$N~x~k~t~u)$''
1688 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1689 of elements stored in the tree:
1692 \textbf{theorem}~\textit{dataset\_skew\_split}:\\
1693 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1694 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1695 \textbf{nitpick} \\[2\smallskipamount]
1696 {\slshape Nitpick found no counterexample.}
1699 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1700 should not alter the tree:
1703 \textbf{theorem}~\textit{wf\_skew\_split}:\\
1704 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1705 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1706 \textbf{nitpick} \\[2\smallskipamount]
1707 {\slshape Nitpick found no counterexample.}
1710 Insertion is implemented recursively. It preserves the sort order:
1713 \textbf{primrec}~\textit{insort} \textbf{where} \\
1714 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1715 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1716 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1717 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1720 Notice that we deliberately commented out the application of \textit{skew} and
1721 \textit{split}. Let's see if this causes any problems:
1724 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1725 \textbf{nitpick} \\[2\smallskipamount]
1726 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1727 \hbox{}\qquad Free variables: \nopagebreak \\
1728 \hbox{}\qquad\qquad $t = N~a_1~1~\Lambda~\Lambda$ \\
1729 \hbox{}\qquad\qquad $x = a_2$
1732 It's hard to see why this is a counterexample. To improve readability, we will
1733 restrict the theorem to \textit{nat}, so that we don't need to look up the value
1734 of the $\textit{op}~{<}$ constant to find out which element is smaller than the
1735 other. In addition, we will tell Nitpick to display the value of
1736 $\textit{insort}~t~x$ using the \textit{eval} option. This gives
1739 \textbf{theorem} \textit{wf\_insort\_nat}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1740 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1741 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1742 \hbox{}\qquad Free variables: \nopagebreak \\
1743 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1744 \hbox{}\qquad\qquad $x = 0$ \\
1745 \hbox{}\qquad Evaluated term: \\
1746 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1749 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1750 where both have a level of 1. This violates the second AA tree invariant, which
1751 states that a left child's level must be less than its parent's. This shouldn't
1752 come as a surprise, considering that we commented out the tree rebalancing code.
1753 Reintroducing the code seems to solve the problem:
1756 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1757 \textbf{nitpick} \\[2\smallskipamount]
1758 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1761 Insertion should transform the set of elements represented by the tree in the
1765 \textbf{theorem} \textit{dataset\_insort}:\kern.4em
1766 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1767 \textbf{nitpick} \\[2\smallskipamount]
1768 {\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
1771 We could continue like this and sketch a complete theory of AA trees. Once the
1772 definitions and main theorems are in place and have been thoroughly tested using
1773 Nitpick, we could start working on the proofs. Developing theories this way
1774 usually saves time, because faulty theorems and definitions are discovered much
1775 earlier in the process.
1777 \section{Option Reference}
1778 \label{option-reference}
1780 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1781 \def\qty#1{$\left<\textit{#1}\right>$}
1782 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1783 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1784 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1785 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1786 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1787 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1788 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1789 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1790 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1792 Nitpick's behavior can be influenced by various options, which can be specified
1793 in brackets after the \textbf{nitpick} command. Default values can be set
1794 using \textbf{nitpick\_\allowbreak params}. For example:
1797 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1800 The options are categorized as follows:\ mode of operation
1801 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1802 format (\S\ref{output-format}), automatic counterexample checks
1803 (\S\ref{authentication}), optimizations
1804 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1806 You can instruct Nitpick to run automatically on newly entered theorems by
1807 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1808 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
1809 and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1810 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1811 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1812 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1813 \textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
1814 Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
1817 The number of options can be overwhelming at first glance. Do not let that worry
1818 you: Nitpick's defaults have been chosen so that it almost always does the right
1819 thing, and the most important options have been covered in context in
1820 \S\ref{first-steps}.
1822 The descriptions below refer to the following syntactic quantities:
1825 \item[$\bullet$] \qtybf{string}: A string.
1826 \item[$\bullet$] \qtybf{bool}: \textit{true} or \textit{false}.
1827 \item[$\bullet$] \qtybf{bool\_or\_smart}: \textit{true}, \textit{false}, or \textit{smart}.
1828 \item[$\bullet$] \qtybf{int}: An integer. Negative integers are prefixed with a hyphen.
1829 \item[$\bullet$] \qtybf{int\_or\_smart}: An integer or \textit{smart}.
1830 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1831 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1833 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1834 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1835 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1836 \item[$\bullet$] \qtybf{const}: The name of a HOL constant.
1837 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1838 \item[$\bullet$] \qtybf{term\_list}: A space-separated list of HOL terms (e.g.,
1839 ``$f~x$''~``$g~y$'').
1840 \item[$\bullet$] \qtybf{type}: A HOL type.
1843 Default values are indicated in square brackets. Boolean options have a negated
1844 counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
1845 Boolean options, ``= \textit{true}'' may be omitted.
1847 \subsection{Mode of Operation}
1848 \label{mode-of-operation}
1851 \optrue{blocking}{non\_blocking}
1852 Specifies whether the \textbf{nitpick} command should operate synchronously.
1853 The asynchronous (non-blocking) mode lets the user start proving the putative
1854 theorem while Nitpick looks for a counterexample, but it can also be more
1855 confusing. For technical reasons, automatic runs currently always block.
1857 \optrue{falsify}{satisfy}
1858 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1859 satisfying examples (models). This manual assumes throughout that
1860 \textit{falsify} is enabled.
1862 \opsmart{user\_axioms}{no\_user\_axioms}
1863 Specifies whether the user-defined axioms (specified using
1864 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1865 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1866 the constants that occur in the formula to falsify. The option is implicitly set
1867 to \textit{true} for automatic runs.
1869 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1870 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1871 these conditions are tagged as ``likely genuine.'' The \textit{debug}
1872 (\S\ref{output-format}) option can be used to find out which axioms were
1876 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1877 (\S\ref{output-format}).}
1879 \optrue{assms}{no\_assms}
1880 Specifies whether the relevant assumptions in structured proof should be
1881 considered. The option is implicitly enabled for automatic runs.
1884 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1886 \opfalse{overlord}{no\_overlord}
1887 Specifies whether Nitpick should put its temporary files in
1888 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1889 debugging Nitpick but also unsafe if several instances of the tool are run
1890 simultaneously. The files are identified by the extensions
1891 \texttt{.kki}, \texttt{.cnf}, \texttt{.out}, and
1892 \texttt{.err}; you may safely remove them after Nitpick has run.
1895 {\small See also \textit{debug} (\S\ref{output-format}).}
1898 \subsection{Scope of Search}
1899 \label{scope-of-search}
1902 \oparg{card}{type}{int\_seq}
1903 Specifies the sequence of cardinalities to use for a given type.
1904 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1905 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1906 Although function and product types are normally mapped directly to the
1907 corresponding Kodkod concepts, setting
1908 the cardinality of such types is also allowed and implicitly enables ``boxing''
1909 for them, as explained in the description of the \textit{box}~\qty{type}
1910 and \textit{box} (\S\ref{scope-of-search}) options.
1913 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1915 \opdefault{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
1916 Specifies the default sequence of cardinalities to use. This can be overridden
1917 on a per-type basis using the \textit{card}~\qty{type} option described above.
1919 \oparg{max}{const}{int\_seq}
1920 Specifies the sequence of maximum multiplicities to use for a given
1921 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1922 number of distinct values that it can construct. Nonsensical values (e.g.,
1923 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1924 datatypes equipped with several constructors.
1926 \opnodefault{max}{int\_seq}
1927 Specifies the default sequence of maximum multiplicities to use for
1928 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
1929 basis using the \textit{max}~\qty{const} option described above.
1931 \opsmart{binary\_ints}{unary\_ints}
1932 Specifies whether natural numbers and integers should be encoded using a unary
1933 or binary notation. In unary mode, the cardinality fully specifies the subset
1934 used to approximate the type. For example:
1936 $$\hbox{\begin{tabular}{@{}rll@{}}%
1937 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
1938 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
1939 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
1944 $$\hbox{\begin{tabular}{@{}rll@{}}%
1945 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
1946 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
1949 In binary mode, the cardinality specifies the number of distinct values that can
1950 be constructed. Each of these value is represented by a bit pattern whose length
1951 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
1952 Nitpick attempts to choose the more appropriate encoding by inspecting the
1953 formula at hand, preferring the binary notation for problems involving
1954 multiplicative operators or large constants.
1956 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
1957 problems that refer to the types \textit{rat} or \textit{real} or the constants
1958 \textit{Suc}, \textit{gcd}, or \textit{lcm}.
1960 {\small See also \textit{bits} (\S\ref{scope-of-search}) and
1961 \textit{show\_datatypes} (\S\ref{output-format}).}
1963 \opdefault{bits}{int\_seq}{$\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{8},\mathbf{10},\mathbf{12}$}
1964 Specifies the number of bits to use to represent natural numbers and integers in
1965 binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
1967 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
1969 \opargboolorsmart{wf}{const}{non\_wf}
1970 Specifies whether the specified (co)in\-duc\-tively defined predicate is
1971 well-founded. The option can take the following values:
1974 \item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
1975 predicate as if it were well-founded. Since this is generally not sound when the
1976 predicate is not well-founded, the counterexamples are tagged as ``likely
1979 \item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
1980 as if it were not well-founded. The predicate is then unrolled as prescribed by
1981 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
1984 \item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
1985 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
1986 \textit{size\_change} tactics. If this succeeds (or the predicate occurs with an
1987 appropriate polarity in the formula to falsify), use an efficient fixed point
1988 equation as specification of the predicate; otherwise, unroll the predicates
1989 according to the \textit{iter}~\qty{const} and \textit{iter} options.
1993 {\small See also \textit{iter} (\S\ref{scope-of-search}),
1994 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
1995 (\S\ref{timeouts}).}
1997 \opsmart{wf}{non\_wf}
1998 Specifies the default wellfoundedness setting to use. This can be overridden on
1999 a per-predicate basis using the \textit{wf}~\qty{const} option above.
2001 \oparg{iter}{const}{int\_seq}
2002 Specifies the sequence of iteration counts to use when unrolling a given
2003 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
2004 predicates that occur negatively and coinductive predicates that occur
2005 positively in the formula to falsify and that cannot be proved to be
2006 well-founded, but this behavior is influenced by the \textit{wf} option. The
2007 iteration counts are automatically bounded by the cardinality of the predicate's
2010 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
2011 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
2013 \opdefault{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
2014 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
2015 predicates. This can be overridden on a per-predicate basis using the
2016 \textit{iter} \qty{const} option above.
2018 \opdefault{bisim\_depth}{int\_seq}{$\mathbf{7}$}
2019 Specifies the sequence of iteration counts to use when unrolling the
2020 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
2021 of $-1$ means that no predicate is generated, in which case Nitpick performs an
2022 after-the-fact check to see if the known coinductive datatype values are
2023 bidissimilar. If two values are found to be bisimilar, the counterexample is
2024 tagged as ``likely genuine.'' The iteration counts are automatically bounded by
2025 the sum of the cardinalities of the coinductive datatypes occurring in the
2028 \opargboolorsmart{box}{type}{dont\_box}
2029 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
2030 product type in an isomorphic datatype internally. Boxing is an effective mean
2031 to reduce the search space and speed up Nitpick, because the isomorphic datatype
2032 is approximated by a subset of the possible function or pair values;
2033 like other drastic optimizations, it can also prevent the discovery of
2034 counterexamples. The option can take the following values:
2037 \item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
2039 \item[$\bullet$] \textbf{\textit{false}}: Never box the type.
2040 \item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
2041 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
2042 higher-order functions are good candidates for boxing.
2045 Setting the \textit{card}~\qty{type} option for a function or product type
2046 implicitly enables boxing for that type.
2049 {\small See also \textit{verbose} (\S\ref{output-format})
2050 and \textit{debug} (\S\ref{output-format}).}
2052 \opsmart{box}{dont\_box}
2053 Specifies the default boxing setting to use. This can be overridden on a
2054 per-type basis using the \textit{box}~\qty{type} option described above.
2056 \opargboolorsmart{mono}{type}{non\_mono}
2057 Specifies whether the given type should be considered monotonic when
2058 enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a
2059 monotonicity check on the type. Setting this option to \textit{true} can reduce
2060 the number of scopes tried, but it also diminishes the theoretical chance of
2061 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
2064 {\small See also \textit{card} (\S\ref{scope-of-search}),
2065 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
2066 (\S\ref{output-format}).}
2068 \opsmart{mono}{non\_box}
2069 Specifies the default monotonicity setting to use. This can be overridden on a
2070 per-type basis using the \textit{mono}~\qty{type} option described above.
2072 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
2073 Specifies whether type variables with the same sort constraints should be
2074 merged. Setting this option to \textit{true} can reduce the number of scopes
2075 tried and the size of the generated Kodkod formulas, but it also diminishes the
2076 theoretical chance of finding a counterexample.
2078 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
2080 \opargbool{std}{type}{non\_std}
2081 Specifies whether the given type should be given standard models.
2082 Nonstandard models are unsound but can help debug inductive arguments,
2083 as explained in \S\ref{inductive-properties}.
2085 \optrue{std}{non\_std}
2086 Specifies the default standardness to use. This can be overridden on a per-type
2087 basis using the \textit{std}~\qty{type} option described above.
2090 \subsection{Output Format}
2091 \label{output-format}
2094 \opfalse{verbose}{quiet}
2095 Specifies whether the \textbf{nitpick} command should explain what it does. This
2096 option is useful to determine which scopes are tried or which SAT solver is
2097 used. This option is implicitly disabled for automatic runs.
2099 \opfalse{debug}{no\_debug}
2100 Specifies whether Nitpick should display additional debugging information beyond
2101 what \textit{verbose} already displays. Enabling \textit{debug} also enables
2102 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
2103 option is implicitly disabled for automatic runs.
2106 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
2107 \textit{batch\_size} (\S\ref{optimizations}).}
2109 \optrue{show\_skolems}{hide\_skolem}
2110 Specifies whether the values of Skolem constants should be displayed as part of
2111 counterexamples. Skolem constants correspond to bound variables in the original
2112 formula and usually help us to understand why the counterexample falsifies the
2116 {\small See also \textit{skolemize} (\S\ref{optimizations}).}
2118 \opfalse{show\_datatypes}{hide\_datatypes}
2119 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
2120 be displayed as part of counterexamples. Such subsets are sometimes helpful when
2121 investigating whether a potential counterexample is genuine or spurious, but
2122 their potential for clutter is real.
2124 \opfalse{show\_consts}{hide\_consts}
2125 Specifies whether the values of constants occurring in the formula (including
2126 its axioms) should be displayed along with any counterexample. These values are
2127 sometimes helpful when investigating why a counterexample is
2128 genuine, but they can clutter the output.
2130 \opfalse{show\_all}{dont\_show\_all}
2131 Enabling this option effectively enables \textit{show\_skolems},
2132 \textit{show\_datatypes}, and \textit{show\_consts}.
2134 \opdefault{max\_potential}{int}{$\mathbf{1}$}
2135 Specifies the maximum number of potential counterexamples to display. Setting
2136 this option to 0 speeds up the search for a genuine counterexample. This option
2137 is implicitly set to 0 for automatic runs. If you set this option to a value
2138 greater than 1, you will need an incremental SAT solver: For efficiency, it is
2139 recommended to install the JNI version of MiniSat and set \textit{sat\_solver} =
2140 \textit{MiniSat\_JNI}. Also be aware that many of the counterexamples may look
2141 identical, unless the \textit{show\_all} (\S\ref{output-format}) option is
2145 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
2146 \textit{sat\_solver} (\S\ref{optimizations}).}
2148 \opdefault{max\_genuine}{int}{$\mathbf{1}$}
2149 Specifies the maximum number of genuine counterexamples to display. If you set
2150 this option to a value greater than 1, you will need an incremental SAT solver:
2151 For efficiency, it is recommended to install the JNI version of MiniSat and set
2152 \textit{sat\_solver} = \textit{MiniSat\_JNI}. Also be aware that many of the
2153 counterexamples may look identical, unless the \textit{show\_all}
2154 (\S\ref{output-format}) option is enabled.
2157 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
2158 \textit{sat\_solver} (\S\ref{optimizations}).}
2160 \opnodefault{eval}{term\_list}
2161 Specifies the list of terms whose values should be displayed along with
2162 counterexamples. This option suffers from an ``observer effect'': Nitpick might
2163 find different counterexamples for different values of this option.
2165 \oparg{format}{term}{int\_seq}
2166 Specifies how to uncurry the value displayed for a variable or constant.
2167 Uncurrying sometimes increases the readability of the output for high-arity
2168 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
2169 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
2170 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
2171 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
2172 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
2173 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
2174 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
2175 arguments that are not accounted for are left alone, as if the specification had
2176 been $1,\ldots,1,n_1,\ldots,n_k$.
2179 {\small See also \textit{uncurry} (\S\ref{optimizations}).}
2181 \opdefault{format}{int\_seq}{$\mathbf{1}$}
2182 Specifies the default format to use. Irrespective of the default format, the
2183 extra arguments to a Skolem constant corresponding to the outer bound variables
2184 are kept separated from the remaining arguments, the \textbf{for} arguments of
2185 an inductive definitions are kept separated from the remaining arguments, and
2186 the iteration counter of an unrolled inductive definition is shown alone. The
2187 default format can be overridden on a per-variable or per-constant basis using
2188 the \textit{format}~\qty{term} option described above.
2191 %% MARK: Authentication
2192 \subsection{Authentication}
2193 \label{authentication}
2196 \opfalse{check\_potential}{trust\_potential}
2197 Specifies whether potential counterexamples should be given to Isabelle's
2198 \textit{auto} tactic to assess their validity. If a potential counterexample is
2199 shown to be genuine, Nitpick displays a message to this effect and terminates.
2202 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
2204 \opfalse{check\_genuine}{trust\_genuine}
2205 Specifies whether genuine and likely genuine counterexamples should be given to
2206 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
2207 counterexample is shown to be spurious, the user is kindly asked to send a bug
2208 report to the author at
2209 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
2212 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
2214 \opnodefault{expect}{string}
2215 Specifies the expected outcome, which must be one of the following:
2218 \item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
2219 \item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a ``likely
2220 genuine'' counterexample (i.e., a counterexample that is genuine unless
2221 it contradicts a missing axiom or a dangerous option was used inappropriately).
2222 \item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
2223 \item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
2224 \item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
2225 Kodkod ran out of memory).
2228 Nitpick emits an error if the actual outcome differs from the expected outcome.
2229 This option is useful for regression testing.
2232 \subsection{Optimizations}
2233 \label{optimizations}
2235 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
2240 \opdefault{sat\_solver}{string}{smart}
2241 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2242 to be faster than their Java counterparts, but they can be more difficult to
2243 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2244 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2245 you will need an incremental SAT solver, such as \textit{MiniSat\_JNI}
2246 (recommended) or \textit{SAT4J}.
2248 The supported solvers are listed below:
2252 \item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
2253 written in \cpp{}. To use MiniSat, set the environment variable
2254 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2255 executable. The \cpp{} sources and executables for MiniSat are available at
2256 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2257 and 2.0 beta (2007-07-21).
2259 \item[$\bullet$] \textbf{\textit{MiniSat\_JNI}}: The JNI (Java Native Interface)
2260 version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which
2261 you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2262 version of MiniSat, the JNI version can be used incrementally.
2265 %%% "It is bundled with Kodkodi and requires no further installation or
2266 %%% configuration steps. Alternatively,"
2267 \item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
2268 written in C. You can install a standard version of
2269 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2270 that contains the \texttt{picosat} executable. The C sources for PicoSAT are
2271 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2272 Nitpick has been tested with version 913.
2274 \item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
2275 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2276 the directory that contains the \texttt{zchaff} executable. The \cpp{} sources
2277 and executables for zChaff are available at
2278 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2279 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2281 \item[$\bullet$] \textbf{\textit{zChaff\_JNI}}: The JNI version of zChaff is
2282 bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on
2283 Kodkod's web site \cite{kodkod-2009}.
2285 \item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
2286 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2287 directory that contains the \texttt{rsat} executable. The \cpp{} sources for
2288 RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been
2289 tested with version 2.01.
2291 \item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
2292 written in C. To use BerkMin, set the environment variable
2293 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2294 executable. The BerkMin executables are available at
2295 \url{http://eigold.tripod.com/BerkMin.html}.
2297 \item[$\bullet$] \textbf{\textit{BerkMin\_Alloy}}: Variant of BerkMin that is
2298 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2299 version of BerkMin, set the environment variable
2300 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2303 \item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
2304 written in C. To use Jerusat, set the environment variable
2305 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2306 executable. The C sources for Jerusat are available at
2307 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2309 \item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
2310 written in Java that can be used incrementally. It is bundled with Kodkodi and
2311 requires no further installation or configuration steps. Do not attempt to
2312 install the official SAT4J packages, because their API is incompatible with
2315 \item[$\bullet$] \textbf{\textit{SAT4J\_Light}}: Variant of SAT4J that is
2316 optimized for small problems. It can also be used incrementally.
2318 \item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
2319 experimental solver written in \cpp. To use HaifaSat, set the environment
2320 variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
2321 \texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at
2322 \url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
2324 \item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
2325 \textit{smart}, Nitpick selects the first solver among MiniSat,
2326 PicoSAT, zChaff, RSat, BerkMin, BerkMin\_Alloy, Jerusat, MiniSat\_JNI, and zChaff\_JNI
2327 that is recognized by Isabelle. If none is found, it falls back on SAT4J, which
2328 should always be available. If \textit{verbose} (\S\ref{output-format}) is
2329 enabled, Nitpick displays which SAT solver was chosen.
2333 \opdefault{batch\_size}{int\_or\_smart}{smart}
2334 Specifies the maximum number of Kodkod problems that should be lumped together
2335 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2336 together ensures that Kodkodi is launched less often, but it makes the verbose
2337 output less readable and is sometimes detrimental to performance. If
2338 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2339 \textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
2341 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2342 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2343 be rewritten to use (automatically generated) discriminators and destructors.
2344 This optimization can drastically reduce the size of the Boolean formulas given
2348 {\small See also \textit{debug} (\S\ref{output-format}).}
2350 \optrue{specialize}{dont\_specialize}
2351 Specifies whether functions invoked with static arguments should be specialized.
2352 This optimization can drastically reduce the search space, especially for
2353 higher-order functions.
2356 {\small See also \textit{debug} (\S\ref{output-format}) and
2357 \textit{show\_consts} (\S\ref{output-format}).}
2359 \optrue{skolemize}{dont\_skolemize}
2360 Specifies whether the formula should be skolemized. For performance reasons,
2361 (positive) $\forall$-quanti\-fiers that occur in the scope of a higher-order
2362 (positive) $\exists$-quanti\-fier are left unchanged.
2365 {\small See also \textit{debug} (\S\ref{output-format}) and
2366 \textit{show\_skolems} (\S\ref{output-format}).}
2368 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2369 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2370 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2371 predicates for which each the predicate occurs in at most one assumption of each
2372 introduction rule. Using the reflexive transitive closure is in principle
2373 equivalent to setting \textit{iter} to the cardinality of the predicate's
2374 domain, but it is usually more efficient.
2376 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2377 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2379 \optrue{uncurry}{dont\_uncurry}
2380 Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no
2381 tangible effect on efficiency, but it creates opportunities for the boxing
2385 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug}
2386 (\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).}
2388 \optrue{fast\_descrs}{full\_descrs}
2389 Specifies whether Nitpick should optimize the definite and indefinite
2390 description operators (THE and SOME). The optimized versions usually help
2391 Nitpick generate more counterexamples or at least find them faster, but only the
2392 unoptimized versions are complete when all types occurring in the formula are
2395 {\small See also \textit{debug} (\S\ref{output-format}).}
2397 \optrue{peephole\_optim}{no\_peephole\_optim}
2398 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2399 peephole optimizer. These optimizations can make a significant difference.
2400 Unless you are tracking down a bug in Nitpick or distrust the peephole
2401 optimizer, you should leave this option enabled.
2403 \opdefault{sym\_break}{int}{20}
2404 Specifies an upper bound on the number of relations for which Kodkod generates
2405 symmetry breaking predicates. According to the Kodkod documentation
2406 \cite{kodkod-2009-options}, ``in general, the higher this value, the more
2407 symmetries will be broken, and the faster the formula will be solved. But,
2408 setting the value too high may have the opposite effect and slow down the
2411 \opdefault{sharing\_depth}{int}{3}
2412 Specifies the depth to which Kodkod should check circuits for equivalence during
2413 the translation to SAT. The default of 3 is the same as in Alloy. The minimum
2414 allowed depth is 1. Increasing the sharing may result in a smaller SAT problem,
2415 but can also slow down Kodkod.
2417 \opfalse{flatten\_props}{dont\_flatten\_props}
2418 Specifies whether Kodkod should try to eliminate intermediate Boolean variables.
2419 Although this might sound like a good idea, in practice it can drastically slow
2422 \opdefault{max\_threads}{int}{0}
2423 Specifies the maximum number of threads to use in Kodkod. If this option is set
2424 to 0, Kodkod will compute an appropriate value based on the number of processor
2428 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2429 \textit{timeout} (\S\ref{timeouts}).}
2432 \subsection{Timeouts}
2436 \opdefault{timeout}{time}{$\mathbf{30}$ s}
2437 Specifies the maximum amount of time that the \textbf{nitpick} command should
2438 spend looking for a counterexample. Nitpick tries to honor this constraint as
2439 well as it can but offers no guarantees. For automatic runs,
2440 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2441 a time slot whose length is specified by the ``Auto Counterexample Time
2442 Limit'' option in Proof General.
2445 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2447 \opdefault{tac\_timeout}{time}{$\mathbf{500}$\,ms}
2448 Specifies the maximum amount of time that the \textit{auto} tactic should use
2449 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2450 and \textit{size\_change} should use when checking whether a (co)in\-duc\-tive
2451 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2452 can but offers no guarantees.
2455 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2456 \textit{check\_potential} (\S\ref{authentication}),
2457 and \textit{check\_genuine} (\S\ref{authentication}).}
2460 \section{Attribute Reference}
2461 \label{attribute-reference}
2463 Nitpick needs to consider the definitions of all constants occurring in a
2464 formula in order to falsify it. For constants introduced using the
2465 \textbf{definition} command, the definition is simply the associated
2466 \textit{\_def} axiom. In contrast, instead of using the internal representation
2467 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2468 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2469 equational specification entered by the user.
2471 Behind the scenes, Isabelle's built-in packages and theories rely on the
2472 following attributes to affect Nitpick's behavior:
2475 \flushitem{\textit{nitpick\_def}}
2478 This attribute specifies an alternative definition of a constant. The
2479 alternative definition should be logically equivalent to the constant's actual
2480 axiomatic definition and should be of the form
2482 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2484 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2487 \flushitem{\textit{nitpick\_simp}}
2490 This attribute specifies the equations that constitute the specification of a
2491 constant. For functions defined using the \textbf{primrec}, \textbf{function},
2492 and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
2493 \textit{simps} rules. The equations must be of the form
2495 \qquad $c~t_1~\ldots\ t_n \,=\, u.$
2497 \flushitem{\textit{nitpick\_psimp}}
2500 This attribute specifies the equations that constitute the partial specification
2501 of a constant. For functions defined using the \textbf{function} package, this
2502 corresponds to the \textit{psimps} rules. The conditional equations must be of
2505 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
2507 \flushitem{\textit{nitpick\_intro}}
2510 This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
2511 For predicates defined using the \textbf{inductive} or \textbf{coinductive}
2512 command, this corresponds to the \textit{intros} rules. The introduction rules
2515 \qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
2516 \ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\
2519 where the $P_i$'s are side conditions that do not involve $c$ and $M$ is an
2520 optional monotonic operator. The order of the assumptions is irrelevant.
2524 When faced with a constant, Nitpick proceeds as follows:
2527 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2528 is not empty, Nitpick uses these rules as the specification of the constant.
2530 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2531 the constant is not empty, it uses these rules as the specification of the
2534 \item[3.] Otherwise, it looks up the definition of the constant:
2537 \item[1.] If the \textit{nitpick\_def} set associated with the constant
2538 is not empty, it uses the latest rule added to the set as the definition of the
2539 constant; otherwise it uses the actual definition axiom.
2540 \item[2.] If the definition is of the form
2542 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
2544 then Nitpick assumes that the definition was made using an inductive package and
2545 based on the introduction rules marked with \textit{nitpick\_\allowbreak
2546 ind\_\allowbreak intros} tries to determine whether the definition is
2551 As an illustration, consider the inductive definition
2554 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2555 ``\textit{odd}~1'' $\,\mid$ \\
2556 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2559 Isabelle automatically attaches the \textit{nitpick\_intro} attribute to
2560 the above rules. Nitpick then uses the \textit{lfp}-based definition in
2561 conjunction with these rules. To override this, we can specify an alternative
2562 definition as follows:
2565 \textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]: ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2568 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2569 = 1$. Alternatively, we can specify an equational specification of the constant:
2572 \textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]: ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2575 Such tweaks should be done with great care, because Nitpick will assume that the
2576 constant is completely defined by its equational specification. For example, if
2577 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
2578 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2579 (\S\ref{output-format}) option is extremely useful to understand what is going
2580 on when experimenting with \textit{nitpick\_} attributes.
2582 \section{Standard ML Interface}
2583 \label{standard-ml-interface}
2585 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2586 and debugging. Among the most interesting functions exported by Nitpick are
2587 those that let you invoke the tool programmatically and those that let you
2588 register and unregister custom coinductive datatypes.
2590 \subsection{Invocation of Nitpick}
2591 \label{invocation-of-nitpick}
2593 The \textit{Nitpick} structure offers the following functions for invoking your
2594 favorite counterexample generator:
2597 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2598 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2599 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2600 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2601 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2604 The return value is a new proof state paired with an outcome string
2605 (``genuine'', ``likely\_genuine'', ``potential'', ``none'', or ``unknown''). The
2606 \textit{params} type is a large record that lets you set Nitpick's options. The
2607 current default options can be retrieved by calling the following function
2608 defined in the \textit{Nitpick\_Isar} structure:
2611 $\textbf{val}\,~\textit{default\_params} :\,
2612 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2615 The second argument lets you override option values before they are parsed and
2616 put into a \textit{params} record. Here is an example:
2619 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2620 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2621 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2622 & \textit{subgoal}\end{aligned}$
2627 \subsection{Registration of Coinductive Datatypes}
2628 \label{registration-of-coinductive-datatypes}
2630 If you have defined a custom coinductive datatype, you can tell Nitpick about
2631 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2632 uses for lazy lists. The interface for registering and unregistering coinductive
2633 datatypes consists of the following pair of functions defined in the
2634 \textit{Nitpick} structure:
2637 $\textbf{val}\,~\textit{register\_codatatype} :\,
2638 \textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2639 $\textbf{val}\,~\textit{unregister\_codatatype} :\,
2640 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2643 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2644 not been, you could have told Nitpick about it by adding the following line
2645 to your theory file:
2648 $\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
2649 & \textit{Nitpick.register\_codatatype} \\[-2pt]
2650 & \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
2651 & \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
2654 The \textit{register\_codatatype} function takes a coinductive type, its case
2655 function, and the list of its constructors. The case function must take its
2656 arguments in the order that the constructors are listed. If no case function
2657 with the correct signature is available, simply pass the empty string.
2659 On the other hand, if your goal is to cripple Nitpick, add the following line to
2660 your theory file and try to check a few conjectures about lazy lists:
2663 $\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
2664 \kern1pt'a~\textit{list}\textrm{''}\}\ \,{*}\}$
2667 Inductive datatypes can be registered as coinductive datatypes, given
2668 appropriate coinductive constructors. However, doing so precludes
2669 the use of the inductive constructors---Nitpick will generate an error if they
2672 \section{Known Bugs and Limitations}
2673 \label{known-bugs-and-limitations}
2675 Here are the known bugs and limitations in Nitpick at the time of writing:
2678 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2679 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2680 Nitpick to generate spurious counterexamples for theorems that refer to values
2681 for which the function is not defined. For example:
2684 \textbf{primrec} \textit{prec} \textbf{where} \\
2685 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2686 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2687 \textbf{nitpick} \\[2\smallskipamount]
2688 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2690 \\[2\smallskipamount]
2691 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2692 \textbf{by}~(\textit{auto simp}:~\textit{prec\_def})
2695 Such theorems are considered bad style because they rely on the internal
2696 representation of functions synthesized by Isabelle, which is an implementation
2699 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2700 which can become invalid if you change the definition of an inductive predicate
2701 that is registered in the cache. To clear the cache,
2702 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2703 501$\,\textit{ms}$).
2705 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2706 \textbf{guess} command in a structured proof.
2708 \item[$\bullet$] The \textit{nitpick\_} attributes and the
2709 \textit{Nitpick.register\_} functions can cause havoc if used improperly.
2711 \item[$\bullet$] Although this has never been observed, arbitrary theorem
2712 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
2714 \item[$\bullet$] Local definitions are not supported and result in an error.
2716 %\item[$\bullet$] All constants and types whose names start with
2717 %\textit{Nitpick}{.} are reserved for internal use.
2721 \bibliography{../manual}{}
2722 \bibliographystyle{abbrv}