added support for nonstandard "nat"s to Nitpick and fixed bugs in binary "nat"s and "int"s
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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 Warning: The conjecture either trivially holds for the given scopes or (more likely) lies outside Nitpick's supported
476 fragment. Only potential counterexamples may be found. \\[2\smallskipamount]
477 Nitpick found a potential counterexample: \\[2\smallskipamount]
478 \hbox{}\qquad Free variable: \nopagebreak \\
479 \hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
480 Confirmation by ``\textit{auto}'': The above counterexample is genuine.
483 You might wonder why the counterexample is first reported as potential. The root
484 of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
485 \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
486 that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
487 \textit{False}; but otherwise, it does not know anything about values of $n \ge
488 \textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
489 \textit{True}. Since the assumption can never be satisfied, the putative lemma
490 can never be falsified.
492 Incidentally, if you distrust the so-called genuine counterexamples, you can
493 enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
494 aware that \textit{auto} will usually fail to prove that the counterexample is
497 Some conjectures involving elementary number theory make Nitpick look like a
498 giant with feet of clay:
501 \textbf{lemma} ``$P~\textit{Suc}$'' \\
502 \textbf{nitpick} [\textit{card} = 1--6] \\[2\smallskipamount]
504 Nitpick found no counterexample.
507 On any finite set $N$, \textit{Suc} is a partial function; for example, if $N =
508 \{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\,
509 \ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when passed as
510 argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$. The next
514 \textbf{lemma} ``$P~(\textit{op}~{+}\Colon
515 \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
516 \textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
517 {\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
518 \hbox{}\qquad Free variable: \nopagebreak \\
519 \hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
520 \textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
521 {\slshape Nitpick found no counterexample.}
524 The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
525 $\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
528 Because numbers are infinite and are approximated using a three-valued logic,
529 there is usually no need to systematically enumerate domain sizes. If Nitpick
530 cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
531 unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
532 example above is an exception to this principle.) Nitpick nonetheless enumerates
533 all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
534 cardinalities are fast to handle and give rise to simpler counterexamples. This
535 is explained in more detail in \S\ref{scope-monotonicity}.
537 \subsection{Inductive Datatypes}
538 \label{inductive-datatypes}
540 Like natural numbers and integers, inductive datatypes with recursive
541 constructors admit no finite models and must be approximated by a subterm-closed
542 subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
543 Nitpick looks for all counterexamples that can be built using at most 10
546 Let's see with an example involving \textit{hd} (which returns the first element
547 of a list) and $@$ (which concatenates two lists):
550 \textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
551 \textbf{nitpick} \\[2\smallskipamount]
552 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
553 \hbox{}\qquad Free variables: \nopagebreak \\
554 \hbox{}\qquad\qquad $\textit{xs} = []$ \\
555 \hbox{}\qquad\qquad $\textit{y} = a_1$
558 To see why the counterexample is genuine, we enable \textit{show\_consts}
559 and \textit{show\_\allowbreak datatypes}:
562 {\slshape Datatype:} \\
563 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\
564 {\slshape Constants:} \\
565 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_1, a_1])$ \\
566 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$
569 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
572 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
573 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_1,
574 a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not
575 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
576 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
577 appending $[a_1, a_1]$ to itself gives $\unk$.
579 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
580 considers the following subsets:
582 \kern-.5\smallskipamount %% TYPESETTING
586 $\{[],\, [a_1],\, [a_2]\}$; \\
587 $\{[],\, [a_1],\, [a_3]\}$; \\
588 $\{[],\, [a_2],\, [a_3]\}$; \\
589 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\
590 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\
591 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\
592 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\
593 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\
594 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\
595 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\
596 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\
597 $\{[],\, [a_3],\, [a_3, a_3]\}$.
601 \kern-2\smallskipamount %% TYPESETTING
603 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
604 are listed and only those. As an example of a non-subterm-closed subset,
605 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe
606 that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin
607 \mathcal{S}$ as a subterm.
609 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
612 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
613 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
615 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
616 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
617 \hbox{}\qquad Free variables: \nopagebreak \\
618 \hbox{}\qquad\qquad $\textit{xs} = [a_1]$ \\
619 \hbox{}\qquad\qquad $\textit{ys} = [a_2]$ \\
620 \hbox{}\qquad Datatypes: \\
621 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
622 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$
625 Because datatypes are approximated using a three-valued logic, there is usually
626 no need to systematically enumerate cardinalities: If Nitpick cannot find a
627 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
628 unlikely that one could be found for smaller cardinalities.
630 \subsection{Typedefs, Records, Rationals, and Reals}
631 \label{typedefs-records-rationals-and-reals}
633 Nitpick generally treats types declared using \textbf{typedef} as datatypes
634 whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
638 \textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
639 \textbf{by}~\textit{blast} \\[2\smallskipamount]
640 \textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
641 \textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
642 \textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
643 \textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
644 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
645 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
646 \hbox{}\qquad Free variables: \nopagebreak \\
647 \hbox{}\qquad\qquad $P = \{\Abs{0},\, \Abs{1}\}$ \\
648 \hbox{}\qquad\qquad $x = \Abs{2}$ \\
649 \hbox{}\qquad Datatypes: \\
650 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
651 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$
655 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
658 Records, which are implemented as \textbf{typedef}s behind the scenes, are
659 handled in much the same way:
662 \textbf{record} \textit{point} = \\
663 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
664 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
665 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
666 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
667 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
668 \hbox{}\qquad Free variables: \nopagebreak \\
669 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
670 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
671 \hbox{}\qquad Datatypes: \\
672 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
673 \hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
674 & \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
675 & \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
678 Finally, Nitpick provides rudimentary support for rationals and reals using a
682 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
683 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
684 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
685 \hbox{}\qquad Free variables: \nopagebreak \\
686 \hbox{}\qquad\qquad $x = 1/2$ \\
687 \hbox{}\qquad\qquad $y = -1/2$ \\
688 \hbox{}\qquad Datatypes: \\
689 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
690 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
691 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
694 \subsection{Inductive and Coinductive Predicates}
695 \label{inductive-and-coinductive-predicates}
697 Inductively defined predicates (and sets) are particularly problematic for
698 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
699 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
700 the problem is that they are defined using a least fixed point construction.
702 Nitpick's philosophy is that not all inductive predicates are equal. Consider
703 the \textit{even} predicate below:
706 \textbf{inductive}~\textit{even}~\textbf{where} \\
707 ``\textit{even}~0'' $\,\mid$ \\
708 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
711 This predicate enjoys the desirable property of being well-founded, which means
712 that the introduction rules don't give rise to infinite chains of the form
715 $\cdots\,\Longrightarrow\, \textit{even}~k''
716 \,\Longrightarrow\, \textit{even}~k'
717 \,\Longrightarrow\, \textit{even}~k.$
720 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
724 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
725 \,\Longrightarrow\, \textit{even}~(k - 2)
726 \,\Longrightarrow\, \textit{even}~k.$
729 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
730 fixed point computation.%
731 \footnote{If an inductive predicate is
732 well-founded, then it has exactly one fixed point, which is simultaneously the
733 least and the greatest fixed point. In these circumstances, the computation of
734 the least fixed point amounts to the computation of an arbitrary fixed point,
735 which can be performed using a straightforward recursive equation.}
736 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
737 just as Isabelle's \textbf{function} package usually discharges termination
738 proof obligations automatically.
740 Let's try an example:
743 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
744 \textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
745 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
746 Nitpick can compute it efficiently. \\[2\smallskipamount]
748 \hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
749 Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
750 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
751 Nitpick could not find a better counterexample. \\[2\smallskipamount]
755 No genuine counterexample is possible because Nitpick cannot rule out the
756 existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
757 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
758 existential quantifier:
761 \textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
762 \textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}] \\[2\smallskipamount]
763 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
764 \hbox{}\qquad Empty assignment
767 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
768 we use the following definition instead?
771 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
772 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
773 ``$\textit{even}'~2$'' $\,\mid$ \\
774 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
777 This definition is not well-founded: From $\textit{even}'~0$ and
778 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
779 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
781 Let's check a property involving $\textit{even}'$. To make up for the
782 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
783 \textit{nat}'s cardinality to a mere 10:
786 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
787 \lnot\;\textit{even}'~n$'' \\
788 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
790 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
791 Nitpick might need to unroll it. \\[2\smallskipamount]
793 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
794 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
795 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
796 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
797 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
798 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
799 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
800 \hbox{}\qquad Constant: \nopagebreak \\
801 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
802 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
803 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
804 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
808 Nitpick's output is very instructive. First, it tells us that the predicate is
809 unrolled, meaning that it is computed iteratively from the empty set. Then it
810 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
813 The output also shows how each iteration contributes to $\textit{even}'$. The
814 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
815 predicate depends on an iteration counter. Iteration 0 provides the basis
816 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
817 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
818 iterations would not contribute any new elements.
820 Some values are marked with superscripted question
821 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
822 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
823 \textit{True} or $\unk$, never \textit{False}.
825 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
826 iterations. However, these numbers are bounded by the cardinality of the
827 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
828 ever needed to compute the value of a \textit{nat} predicate. You can specify
829 the number of iterations using the \textit{iter} option, as explained in
830 \S\ref{scope-of-search}.
832 In the next formula, $\textit{even}'$ occurs both positively and negatively:
835 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
836 \textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
837 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
838 \hbox{}\qquad Free variable: \nopagebreak \\
839 \hbox{}\qquad\qquad $n = 1$ \\
840 \hbox{}\qquad Constants: \nopagebreak \\
841 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
842 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
843 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
846 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
847 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
848 fixed point (not necessarily the least one). It is used to falsify
849 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
850 $\textit{even}'~(n - 2)$.
852 Coinductive predicates are handled dually. For example:
855 \textbf{coinductive} \textit{nats} \textbf{where} \\
856 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
857 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
858 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
859 \slshape Nitpick found a counterexample:
860 \\[2\smallskipamount]
861 \hbox{}\qquad Constants: \nopagebreak \\
862 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
863 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
864 & \unr\})\end{aligned}$ \\
865 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
868 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
869 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
870 inductive predicates for which each the predicate occurs in at most one
871 assumption of each introduction rule. For example:
874 \textbf{inductive} \textit{odd} \textbf{where} \\
875 ``$\textit{odd}~1$'' $\,\mid$ \\
876 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
877 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
878 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
879 \slshape Nitpick found a counterexample:
880 \\[2\smallskipamount]
881 \hbox{}\qquad Free variable: \nopagebreak \\
882 \hbox{}\qquad\qquad $n = 1$ \\
883 \hbox{}\qquad Constants: \nopagebreak \\
884 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
885 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
886 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
888 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
889 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
891 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
892 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
893 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
897 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
898 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
899 elements from known ones. The set $\textit{odd}$ consists of all the values
900 reachable through the reflexive transitive closure of
901 $\textit{odd}_{\textrm{step}}$ starting with any element from
902 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
903 transitive closure to encode linear predicates is normally either more thorough
904 or more efficient than unrolling (depending on the value of \textit{iter}), but
905 for those cases where it isn't you can disable it by passing the
906 \textit{dont\_star\_linear\_preds} option.
908 \subsection{Coinductive Datatypes}
909 \label{coinductive-datatypes}
911 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
912 datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
913 list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
914 these lazy lists seamlessly and provides a hook, described in
915 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
918 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
919 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
920 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
921 1, 2, 3, \ldots]$ can be defined as lazy lists using the
922 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
923 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
924 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
926 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
927 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
931 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
932 \textbf{nitpick} \\[2\smallskipamount]
933 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
934 \hbox{}\qquad Free variables: \nopagebreak \\
935 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
936 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
939 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
940 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
941 infinite list $[a_1, a_1, a_1, \ldots]$.
943 The next example is more interesting:
946 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
947 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
948 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
949 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
950 some scopes. \\[2\smallskipamount]
952 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 1,
953 and \textit{bisim\_depth}~= 0. \\
954 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
955 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 8,
956 and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
957 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
958 \textit{card}~``\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
960 \\[2\smallskipamount]
961 \hbox{}\qquad Free variables: \nopagebreak \\
962 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
963 \hbox{}\qquad\qquad $\textit{b} = a_2$ \\
964 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
965 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
969 The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
970 $\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
971 $[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment
972 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
973 the segment leading to the binder is the stem.
975 A salient property of coinductive datatypes is that two objects are considered
976 equal if and only if they lead to the same observations. For example, the lazy
977 lists $\textrm{THE}~\omega.\; \omega =
978 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
979 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
980 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
981 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
982 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
983 concept of equality for coinductive datatypes is called bisimulation and is
984 defined coinductively.
986 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
987 the Kodkod problem to ensure that distinct objects lead to different
988 observations. This precaution is somewhat expensive and often unnecessary, so it
989 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
990 bisimilarity check is then performed \textsl{after} the counterexample has been
991 found to ensure correctness. If this after-the-fact check fails, the
992 counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
993 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
994 check for the previous example saves approximately 150~milli\-seconds; the speed
995 gains can be more significant for larger scopes.
997 The next formula illustrates the need for bisimilarity (either as a Kodkod
998 predicate or as an after-the-fact check) to prevent spurious counterexamples:
1001 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
1002 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1003 \textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
1004 \slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
1005 \hbox{}\qquad Free variables: \nopagebreak \\
1006 \hbox{}\qquad\qquad $a = a_1$ \\
1007 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
1008 \textit{LCons}~a_1~\omega$ \\
1009 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1010 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
1011 \hbox{}\qquad\qquad $'a~\textit{llist} =
1012 \{\!\begin{aligned}[t]
1013 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
1014 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
1015 \\[2\smallskipamount]
1016 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1017 that the counterexample is genuine. \\[2\smallskipamount]
1018 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1019 \slshape Nitpick found no counterexample.
1022 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1023 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1025 A compromise between leaving out the bisimilarity predicate from the Kodkod
1026 problem and performing the after-the-fact check is to specify a lower
1027 nonnegative \textit{bisim\_depth} value than the default one provided by
1028 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1029 be distinguished from each other by their prefixes of length $K$. Be aware that
1030 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1031 finding any counterexamples.
1036 Nitpick normally maps function and product types directly to the corresponding
1037 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1038 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1039 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1040 off to treat these types in the same way as plain datatypes, by approximating
1041 them by a subset of a given cardinality. This technique is called ``boxing'' and
1042 is particularly useful for functions passed as arguments to other functions, for
1043 high-arity functions, and for large tuples. Under the hood, boxing involves
1044 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1045 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1047 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1048 using de Bruijn's notation:
1051 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1054 The $\textit{lift}~t~k$ function increments all variables with indices greater
1055 than or equal to $k$ by one:
1058 \textbf{primrec} \textit{lift} \textbf{where} \\
1059 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1060 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1061 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1064 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1065 term $t$ has a loose variable with index $k$ or more:
1068 \textbf{primrec}~\textit{loose} \textbf{where} \\
1069 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1070 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1071 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1074 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1078 \textbf{primrec}~\textit{subst} \textbf{where} \\
1079 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1080 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1081 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1082 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1085 A substitution is a function that maps variable indices to terms. Observe that
1086 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1087 away, because the recursive call for the \textit{Lam} case involves an altered
1088 version. Also notice the \textit{lift} call, which increments the variable
1089 indices when moving under a \textit{Lam}.
1091 A reasonable property to expect of substitution is that it should leave closed
1092 terms unchanged. Alas, even this simple property does not hold:
1095 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1096 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1098 Trying 8 scopes: \nopagebreak \\
1099 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1100 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1101 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1102 \hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
1103 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1104 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1105 \hbox{}\qquad Free variables: \nopagebreak \\
1106 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1107 & 0 := \textit{Var}~0,\>
1108 1 := \textit{Var}~0,\>
1109 2 := \textit{Var}~0, \\[-2pt]
1110 & 3 := \textit{Var}~0,\>
1111 4 := \textit{Var}~0,\>
1112 5 := \textit{Var}~0)\end{aligned}$ \\
1113 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1114 Total time: $4679$ ms.
1117 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1118 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1119 $\lambda$-term notation, $t$~is
1120 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1121 The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
1122 replaced with $\textit{lift}~(\sigma~m)~0$.
1124 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1125 cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1126 For the formula of interest, knowing 6 values of that type was enough to find
1127 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1128 considered, a hopeless undertaking:
1131 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1132 {\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
1136 Boxing can be enabled or disabled globally or on a per-type basis using the
1137 \textit{box} option. Moreover, setting the cardinality of a function or
1138 product type implicitly enables boxing for that type. Nitpick usually performs
1139 reasonable choices about which types should be boxed, but option tweaking
1144 \subsection{Scope Monotonicity}
1145 \label{scope-monotonicity}
1147 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1148 and \textit{max}) controls which scopes are actually tested. In general, to
1149 exhaust all models below a certain cardinality bound, the number of scopes that
1150 Nitpick must consider increases exponentially with the number of type variables
1151 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1152 cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
1153 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1155 Fortunately, many formulas exhibit a property called \textsl{scope
1156 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1157 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1159 Consider the formula
1162 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1165 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1166 $'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
1167 exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
1168 that any counterexample found with a small scope would still be a counterexample
1169 in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
1170 by the larger scope. Nitpick comes to the same conclusion after a careful
1171 inspection of the formula and the relevant definitions:
1174 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1176 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1177 Nitpick might be able to skip some scopes.
1178 \\[2\smallskipamount]
1180 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1181 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1182 \textit{list}''~= 1, \\
1183 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
1184 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
1185 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1186 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1187 \textit{list}''~= 2, \\
1188 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
1189 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
1190 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1191 \hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
1192 \textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
1193 \textit{list}''~= 8, \\
1194 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
1195 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
1196 \\[2\smallskipamount]
1197 Nitpick found a counterexample for
1198 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1199 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1200 \textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
1201 \textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
1202 \\[2\smallskipamount]
1203 \hbox{}\qquad Free variables: \nopagebreak \\
1204 \hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
1205 \hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
1206 Total time: 1636 ms.
1209 In theory, it should be sufficient to test a single scope:
1212 \textbf{nitpick}~[\textit{card}~= 8]
1215 However, this is often less efficient in practice and may lead to overly complex
1218 If the monotonicity check fails but we believe that the formula is monotonic (or
1219 we don't mind missing some counterexamples), we can pass the
1220 \textit{mono} option. To convince yourself that this option is risky,
1221 simply consider this example from \S\ref{skolemization}:
1224 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1225 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1226 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1227 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1228 \textbf{nitpick} \\[2\smallskipamount]
1230 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1231 \hbox{}\qquad $\vdots$
1234 (It turns out the formula holds if and only if $\textit{card}~'a \le
1235 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1236 monotonicity checks can be disabled by passing \textit{non\_mono}
1237 (\S\ref{optimizations}).
1239 As insinuated in \S\ref{natural-numbers-and-integers} and
1240 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1241 are normally monotonic and treated as such. The same is true for record types,
1242 \textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
1243 cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
1244 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1245 consider only 8~scopes instead of $32\,768$.
1247 \subsection{Inductive Properties}
1248 \label{inductive-properties}
1250 Inductive properties are a particular pain to prove, because the failure to
1251 establish an induction step can mean several things:
1254 \item The property is invalid.
1255 \item The property is valid but is too weak to support the induction step.
1256 \item The property is valid and strong enough; it's just that we haven't found
1260 Depending on which scenario applies, we would take the appropriate course of
1264 \item Repair the statement of the property so that it becomes valid.
1265 \item Generalize the property and/or prove auxiliary properties.
1266 \item Work harder on a proof.
1269 How can we distinguish between the three scenarios? Nitpick's normal mode of
1270 operation can often detect scenario 1, and Isabelle's automatic tactics help with
1271 scenario 3. Using appropriate techniques, it is also often possible to use
1272 Nitpick to identify scenario 2. Consider the following transition system,
1273 in which natural numbers represent states:
1276 \textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
1277 ``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
1278 ``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
1279 ``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
1282 We will try to prove that only even numbers are reachable:
1285 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
1288 Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
1289 so let's attempt a proof by induction:
1292 \textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
1293 \textbf{apply}~\textit{auto}
1296 This leaves us in the following proof state:
1299 {\slshape goal (2 subgoals): \\
1300 \phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
1301 \phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
1305 If we run Nitpick on the first subgoal, it still won't find any
1306 counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
1307 is helpless. However, notice the $n \in \textit{reach}$ assumption, which
1308 strengthens the induction hypothesis but is not immediately usable in the proof.
1309 If we remove it and invoke Nitpick, this time we get a counterexample:
1312 \textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
1313 \textbf{nitpick} \\[2\smallskipamount]
1314 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1315 \hbox{}\qquad Skolem constant: \nopagebreak \\
1316 \hbox{}\qquad\qquad $n = 0$
1319 Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
1320 to strength the lemma:
1323 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
1326 Unfortunately, the proof by induction still gets stuck, except that Nitpick now
1327 finds the counterexample $n = 2$. We generalize the lemma further to
1330 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
1333 and this time \textit{arith} can finish off the subgoals.
1335 A similar technique can be employed for structural induction. The
1336 following mini formalization of full binary trees will serve as illustration:
1339 \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]
1340 \textbf{primrec}~\textit{labels}~\textbf{where} \\
1341 ``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
1342 ``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
1343 \textbf{primrec}~\textit{swap}~\textbf{where} \\
1344 ``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
1345 \phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
1346 ``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
1349 The \textit{labels} function returns the set of labels occurring on leaves of a
1350 tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
1351 labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
1352 obtained by swapping $a$ and $b$:
1355 \textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
1358 Nitpick can't find any counterexample, so we proceed with induction
1359 (this time favoring a more structured style):
1362 \textbf{proof}~(\textit{induct}~$t$) \\
1363 \hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
1365 \hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
1368 Nitpick can't find any counterexample at this point either, but it makes the
1369 following suggestion:
1373 Hint: To check that the induction hypothesis is general enough, try this command:
1374 \textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
1377 If we follow the hint, we get a ``nonstandard'' counterexample for the step:
1380 \slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
1381 \hbox{}\qquad Free variables: \nopagebreak \\
1382 \hbox{}\qquad\qquad $a = a_1$ \\
1383 \hbox{}\qquad\qquad $b = a_2$ \\
1384 \hbox{}\qquad\qquad $t = \xi_1$ \\
1385 \hbox{}\qquad\qquad $u = \xi_2$ \\
1386 \hbox{}\qquad Datatype: \nopagebreak \\
1387 \hbox{}\qquad\qquad $\alpha~\textit{btree} = \{\xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \textit{Branch}~\xi_1~\xi_2\}$ \\
1388 \hbox{}\qquad {\slshape Constants:} \nopagebreak \\
1389 \hbox{}\qquad\qquad $\textit{labels} = \undef
1390 (\!\begin{aligned}[t]%
1391 & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
1392 & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
1393 \hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \undef
1394 (\!\begin{aligned}[t]%
1395 & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
1396 & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount]
1397 The existence of a nonstandard model suggests that the induction hypothesis is not general enough or perhaps
1398 even wrong. See the ``Inductive Properties'' section of the Nitpick manual for details (``\textit{isabelle doc nitpick}'').
1401 Reading the Nitpick manual is a most excellent idea.
1402 But what's going on? The \textit{non\_std} option told the tool to look for
1403 nonstandard models of binary trees, which means that new ``nonstandard'' trees
1404 $\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees
1405 generated by the \textit{Leaf} and \textit{Branch} constructors.%
1406 \footnote{Notice the similarity between allowing nonstandard trees here and
1407 allowing unreachable states in the preceding example (by removing the ``$n \in
1408 \textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
1409 set of objects over which the induction is performed while doing the step
1410 in order to test the induction hypothesis's strength.}
1411 Unlike standard trees, these new trees contain cycles. We will see later that
1412 every property of acyclic trees that can be proved without using induction also
1413 holds for cyclic trees. Hence,
1416 \textsl{If the induction
1417 hypothesis is strong enough, the induction step will hold even for nonstandard
1418 objects, and Nitpick won't find any nonstandard counterexample.}
1421 But here the tool find some nonstandard trees $t = \xi_1$
1422 and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in
1423 \textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$.
1424 Because neither tree contains both $a$ and $b$, the induction hypothesis tells
1425 us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
1426 and as a result we know nothing about the labels of the tree
1427 $\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
1428 $\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
1429 labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
1430 \textit{labels}$ $(\textit{swap}~u~a~b)$.
1432 The solution is to ensure that we always know what the labels of the subtrees
1433 are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
1434 $t$ in the statement of the lemma:
1437 \textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
1438 \phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
1439 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
1440 \phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
1441 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
1444 This time, Nitpick won't find any nonstandard counterexample, and we can perform
1445 the induction step using \textit{auto}.
1447 \section{Case Studies}
1448 \label{case-studies}
1450 As a didactic device, the previous section focused mostly on toy formulas whose
1451 validity can easily be assessed just by looking at the formula. We will now
1452 review two somewhat more realistic case studies that are within Nitpick's
1453 reach:\ a context-free grammar modeled by mutually inductive sets and a
1454 functional implementation of AA trees. The results presented in this
1455 section were produced with the following settings:
1458 \textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
1461 \subsection{A Context-Free Grammar}
1462 \label{a-context-free-grammar}
1464 Our first case study is taken from section 7.4 in the Isabelle tutorial
1465 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1466 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1469 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1470 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1471 $A$ & $::=$ & $aS \mid bAA$ \\
1472 $B$ & $::=$ & $bS \mid aBB$
1476 The intuition behind the grammar is that $A$ generates all string with one more
1477 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1479 The alphabet consists exclusively of $a$'s and $b$'s:
1482 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1485 Strings over the alphabet are represented by \textit{alphabet list}s.
1486 Nonterminals in the grammar become sets of strings. The production rules
1487 presented above can be expressed as a mutually inductive definition:
1490 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1491 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1492 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1493 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1494 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1495 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1496 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1499 The conversion of the grammar into the inductive definition was done manually by
1500 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1503 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1504 d'\^etre}. A good approach is to state desirable properties of the specification
1505 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1506 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1507 counterexamples will point to bugs in the specification. For our grammar
1508 example, we will proceed in two steps, separating the soundness and the
1509 completeness of the set $S$. First, soundness:
1512 \textbf{theorem}~\textit{S\_sound}: \\
1513 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1514 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1515 \textbf{nitpick} \\[2\smallskipamount]
1516 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1517 \hbox{}\qquad Free variable: \nopagebreak \\
1518 \hbox{}\qquad\qquad $w = [b]$
1521 It would seem that $[b] \in S$. How could this be? An inspection of the
1522 introduction rules reveals that the only rule with a right-hand side of the form
1523 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1527 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1530 On closer inspection, we can see that this rule is wrong. To match the
1531 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1535 \textbf{nitpick} \\[2\smallskipamount]
1536 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1537 \hbox{}\qquad Free variable: \nopagebreak \\
1538 \hbox{}\qquad\qquad $w = [a, a, b]$
1541 Some detective work is necessary to find out what went wrong here. To get $[a,
1542 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1546 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1549 Now, this formula must be wrong: The same assumption occurs twice, and the
1550 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1551 the assumptions should have been a $w$.
1553 With the correction made, we don't get any counterexample from Nitpick. Let's
1554 move on and check completeness:
1557 \textbf{theorem}~\textit{S\_complete}: \\
1558 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1559 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1560 \longrightarrow w \in S$'' \\
1561 \textbf{nitpick} \\[2\smallskipamount]
1562 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1563 \hbox{}\qquad Free variable: \nopagebreak \\
1564 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1567 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1568 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1569 the introduction rules we have are probably correct. Perhaps we simply lack an
1570 introduction rule. Comparing the grammar with the inductive definition, our
1571 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1572 without which the grammar cannot generate two or more $b$'s in a row. So we add
1576 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1579 With this last change, we don't get any counterexamples from Nitpick for either
1580 soundness or completeness. We can even generalize our result to cover $A$ and
1584 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1585 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1586 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1587 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1588 \textbf{nitpick} \\[2\smallskipamount]
1589 \slshape Nitpick found no counterexample.
1592 \subsection{AA Trees}
1595 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1596 similar performance to red-black trees, but with a simpler implementation
1597 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1598 a total order $<$. We start by defining the datatype and some basic extractor
1602 \textbf{datatype} $'a$~\textit{aa\_tree} = \\
1603 \hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}'' \\[2\smallskipamount]
1604 \textbf{primrec} \textit{data} \textbf{where} \\
1605 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1606 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1607 \textbf{primrec} \textit{dataset} \textbf{where} \\
1608 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1609 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1610 \textbf{primrec} \textit{level} \textbf{where} \\
1611 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1612 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1613 \textbf{primrec} \textit{left} \textbf{where} \\
1614 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1615 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1616 \textbf{primrec} \textit{right} \textbf{where} \\
1617 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1618 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1621 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1622 as follows \cite{wikipedia-2009-aa-trees}:
1624 \kern.2\parskip %% TYPESETTING
1627 Each node has a level field, and the following invariants must remain true for
1628 the tree to be valid:
1632 \kern-.4\parskip %% TYPESETTING
1637 \item[1.] The level of a leaf node is one.
1638 \item[2.] The level of a left child is strictly less than that of its parent.
1639 \item[3.] The level of a right child is less than or equal to that of its parent.
1640 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1641 \item[5.] Every node of level greater than one must have two children.
1646 \kern.4\parskip %% TYPESETTING
1648 The \textit{wf} predicate formalizes this description:
1651 \textbf{primrec} \textit{wf} \textbf{where} \\
1652 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1653 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1654 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1655 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1656 \phantom{``$($}$\textrm{else}$ \\
1657 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1658 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1659 \mathrel{\land} \textit{level}~u \le k$ \\
1660 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1663 Rebalancing the tree upon insertion and removal of elements is performed by two
1664 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1667 \textbf{primrec} \textit{skew} \textbf{where} \\
1668 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1669 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1670 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1671 \textit{level}~t~\textrm{then}$ \\
1672 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1673 (\textit{right}~t)~u)$ \\
1674 \phantom{``(}$\textrm{else}$ \\
1675 \phantom{``(\quad}$N~x~k~t~u)$''
1679 \textbf{primrec} \textit{split} \textbf{where} \\
1680 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1681 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1682 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1683 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1684 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1685 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1686 \phantom{``(}$\textrm{else}$ \\
1687 \phantom{``(\quad}$N~x~k~t~u)$''
1690 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1691 of elements stored in the tree:
1694 \textbf{theorem}~\textit{dataset\_skew\_split}:\\
1695 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1696 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1697 \textbf{nitpick} \\[2\smallskipamount]
1698 {\slshape Nitpick found no counterexample.}
1701 Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
1702 should not alter the tree:
1705 \textbf{theorem}~\textit{wf\_skew\_split}:\\
1706 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1707 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1708 \textbf{nitpick} \\[2\smallskipamount]
1709 {\slshape Nitpick found no counterexample.}
1712 Insertion is implemented recursively. It preserves the sort order:
1715 \textbf{primrec}~\textit{insort} \textbf{where} \\
1716 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1717 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1718 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1719 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1722 Notice that we deliberately commented out the application of \textit{skew} and
1723 \textit{split}. Let's see if this causes any problems:
1726 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1727 \textbf{nitpick} \\[2\smallskipamount]
1728 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1729 \hbox{}\qquad Free variables: \nopagebreak \\
1730 \hbox{}\qquad\qquad $t = N~a_1~1~\Lambda~\Lambda$ \\
1731 \hbox{}\qquad\qquad $x = a_2$
1734 It's hard to see why this is a counterexample. To improve readability, we will
1735 restrict the theorem to \textit{nat}, so that we don't need to look up the value
1736 of the $\textit{op}~{<}$ constant to find out which element is smaller than the
1737 other. In addition, we will tell Nitpick to display the value of
1738 $\textit{insort}~t~x$ using the \textit{eval} option. This gives
1741 \textbf{theorem} \textit{wf\_insort\_nat}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1742 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1743 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1744 \hbox{}\qquad Free variables: \nopagebreak \\
1745 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1746 \hbox{}\qquad\qquad $x = 0$ \\
1747 \hbox{}\qquad Evaluated term: \\
1748 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1751 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1752 where both have a level of 1. This violates the second AA tree invariant, which
1753 states that a left child's level must be less than its parent's. This shouldn't
1754 come as a surprise, considering that we commented out the tree rebalancing code.
1755 Reintroducing the code seems to solve the problem:
1758 \textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1759 \textbf{nitpick} \\[2\smallskipamount]
1760 {\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
1763 Insertion should transform the set of elements represented by the tree in the
1767 \textbf{theorem} \textit{dataset\_insort}:\kern.4em
1768 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1769 \textbf{nitpick} \\[2\smallskipamount]
1770 {\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
1773 We could continue like this and sketch a complete theory of AA trees. Once the
1774 definitions and main theorems are in place and have been thoroughly tested using
1775 Nitpick, we could start working on the proofs. Developing theories this way
1776 usually saves time, because faulty theorems and definitions are discovered much
1777 earlier in the process.
1779 \section{Option Reference}
1780 \label{option-reference}
1782 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1783 \def\qty#1{$\left<\textit{#1}\right>$}
1784 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1785 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1786 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1787 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1788 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1789 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1790 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1791 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1792 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1794 Nitpick's behavior can be influenced by various options, which can be specified
1795 in brackets after the \textbf{nitpick} command. Default values can be set
1796 using \textbf{nitpick\_\allowbreak params}. For example:
1799 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
1802 The options are categorized as follows:\ mode of operation
1803 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1804 format (\S\ref{output-format}), automatic counterexample checks
1805 (\S\ref{authentication}), optimizations
1806 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1808 You can instruct Nitpick to run automatically on newly entered theorems by
1809 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1810 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
1811 and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
1812 \textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
1813 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
1814 disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
1815 \textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
1816 Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
1819 The number of options can be overwhelming at first glance. Do not let that worry
1820 you: Nitpick's defaults have been chosen so that it almost always does the right
1821 thing, and the most important options have been covered in context in
1822 \S\ref{first-steps}.
1824 The descriptions below refer to the following syntactic quantities:
1827 \item[$\bullet$] \qtybf{string}: A string.
1828 \item[$\bullet$] \qtybf{bool}: \textit{true} or \textit{false}.
1829 \item[$\bullet$] \qtybf{bool\_or\_smart}: \textit{true}, \textit{false}, or \textit{smart}.
1830 \item[$\bullet$] \qtybf{int}: An integer. Negative integers are prefixed with a hyphen.
1831 \item[$\bullet$] \qtybf{int\_or\_smart}: An integer or \textit{smart}.
1832 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1833 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1835 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1836 \item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
1837 (milliseconds), or the keyword \textit{none} ($\infty$ years).
1838 \item[$\bullet$] \qtybf{const}: The name of a HOL constant.
1839 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1840 \item[$\bullet$] \qtybf{term\_list}: A space-separated list of HOL terms (e.g.,
1841 ``$f~x$''~``$g~y$'').
1842 \item[$\bullet$] \qtybf{type}: A HOL type.
1845 Default values are indicated in square brackets. Boolean options have a negated
1846 counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
1847 Boolean options, ``= \textit{true}'' may be omitted.
1849 \subsection{Mode of Operation}
1850 \label{mode-of-operation}
1853 \optrue{blocking}{non\_blocking}
1854 Specifies whether the \textbf{nitpick} command should operate synchronously.
1855 The asynchronous (non-blocking) mode lets the user start proving the putative
1856 theorem while Nitpick looks for a counterexample, but it can also be more
1857 confusing. For technical reasons, automatic runs currently always block.
1859 \optrue{falsify}{satisfy}
1860 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1861 satisfying examples (models). This manual assumes throughout that
1862 \textit{falsify} is enabled.
1864 \opsmart{user\_axioms}{no\_user\_axioms}
1865 Specifies whether the user-defined axioms (specified using
1866 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1867 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1868 the constants that occur in the formula to falsify. The option is implicitly set
1869 to \textit{true} for automatic runs.
1871 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1872 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1873 these conditions are tagged as ``likely genuine.'' The \textit{debug}
1874 (\S\ref{output-format}) option can be used to find out which axioms were
1878 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1879 (\S\ref{output-format}).}
1881 \optrue{assms}{no\_assms}
1882 Specifies whether the relevant assumptions in structured proof should be
1883 considered. The option is implicitly enabled for automatic runs.
1886 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1888 \opfalse{overlord}{no\_overlord}
1889 Specifies whether Nitpick should put its temporary files in
1890 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1891 debugging Nitpick but also unsafe if several instances of the tool are run
1892 simultaneously. The files are identified by the extensions
1893 \texttt{.kki}, \texttt{.cnf}, \texttt{.out}, and
1894 \texttt{.err}; you may safely remove them after Nitpick has run.
1897 {\small See also \textit{debug} (\S\ref{output-format}).}
1900 \subsection{Scope of Search}
1901 \label{scope-of-search}
1904 \oparg{card}{type}{int\_seq}
1905 Specifies the sequence of cardinalities to use for a given type.
1906 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1907 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1908 Although function and product types are normally mapped directly to the
1909 corresponding Kodkod concepts, setting
1910 the cardinality of such types is also allowed and implicitly enables ``boxing''
1911 for them, as explained in the description of the \textit{box}~\qty{type}
1912 and \textit{box} (\S\ref{scope-of-search}) options.
1915 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
1917 \opdefault{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
1918 Specifies the default sequence of cardinalities to use. This can be overridden
1919 on a per-type basis using the \textit{card}~\qty{type} option described above.
1921 \oparg{max}{const}{int\_seq}
1922 Specifies the sequence of maximum multiplicities to use for a given
1923 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1924 number of distinct values that it can construct. Nonsensical values (e.g.,
1925 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1926 datatypes equipped with several constructors.
1928 \opnodefault{max}{int\_seq}
1929 Specifies the default sequence of maximum multiplicities to use for
1930 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
1931 basis using the \textit{max}~\qty{const} option described above.
1933 \opsmart{binary\_ints}{unary\_ints}
1934 Specifies whether natural numbers and integers should be encoded using a unary
1935 or binary notation. In unary mode, the cardinality fully specifies the subset
1936 used to approximate the type. For example:
1938 $$\hbox{\begin{tabular}{@{}rll@{}}%
1939 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
1940 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
1941 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
1946 $$\hbox{\begin{tabular}{@{}rll@{}}%
1947 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
1948 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
1951 In binary mode, the cardinality specifies the number of distinct values that can
1952 be constructed. Each of these value is represented by a bit pattern whose length
1953 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
1954 Nitpick attempts to choose the more appropriate encoding by inspecting the
1955 formula at hand, preferring the binary notation for problems involving
1956 multiplicative operators or large constants.
1958 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
1959 problems that refer to the types \textit{rat} or \textit{real} or the constants
1960 \textit{Suc}, \textit{gcd}, or \textit{lcm}.
1962 {\small See also \textit{bits} (\S\ref{scope-of-search}) and
1963 \textit{show\_datatypes} (\S\ref{output-format}).}
1965 \opdefault{bits}{int\_seq}{$\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{8},\mathbf{10},\mathbf{12}$}
1966 Specifies the number of bits to use to represent natural numbers and integers in
1967 binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
1969 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
1971 \opargboolorsmart{wf}{const}{non\_wf}
1972 Specifies whether the specified (co)in\-duc\-tively defined predicate is
1973 well-founded. The option can take the following values:
1976 \item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
1977 predicate as if it were well-founded. Since this is generally not sound when the
1978 predicate is not well-founded, the counterexamples are tagged as ``likely
1981 \item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
1982 as if it were not well-founded. The predicate is then unrolled as prescribed by
1983 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
1986 \item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
1987 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
1988 \textit{size\_change} tactics. If this succeeds (or the predicate occurs with an
1989 appropriate polarity in the formula to falsify), use an efficient fixed point
1990 equation as specification of the predicate; otherwise, unroll the predicates
1991 according to the \textit{iter}~\qty{const} and \textit{iter} options.
1995 {\small See also \textit{iter} (\S\ref{scope-of-search}),
1996 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
1997 (\S\ref{timeouts}).}
1999 \opsmart{wf}{non\_wf}
2000 Specifies the default wellfoundedness setting to use. This can be overridden on
2001 a per-predicate basis using the \textit{wf}~\qty{const} option above.
2003 \oparg{iter}{const}{int\_seq}
2004 Specifies the sequence of iteration counts to use when unrolling a given
2005 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
2006 predicates that occur negatively and coinductive predicates that occur
2007 positively in the formula to falsify and that cannot be proved to be
2008 well-founded, but this behavior is influenced by the \textit{wf} option. The
2009 iteration counts are automatically bounded by the cardinality of the predicate's
2012 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
2013 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
2015 \opdefault{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
2016 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
2017 predicates. This can be overridden on a per-predicate basis using the
2018 \textit{iter} \qty{const} option above.
2020 \opdefault{bisim\_depth}{int\_seq}{$\mathbf{7}$}
2021 Specifies the sequence of iteration counts to use when unrolling the
2022 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
2023 of $-1$ means that no predicate is generated, in which case Nitpick performs an
2024 after-the-fact check to see if the known coinductive datatype values are
2025 bidissimilar. If two values are found to be bisimilar, the counterexample is
2026 tagged as ``likely genuine.'' The iteration counts are automatically bounded by
2027 the sum of the cardinalities of the coinductive datatypes occurring in the
2030 \opargboolorsmart{box}{type}{dont\_box}
2031 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
2032 product type in an isomorphic datatype internally. Boxing is an effective mean
2033 to reduce the search space and speed up Nitpick, because the isomorphic datatype
2034 is approximated by a subset of the possible function or pair values;
2035 like other drastic optimizations, it can also prevent the discovery of
2036 counterexamples. The option can take the following values:
2039 \item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
2041 \item[$\bullet$] \textbf{\textit{false}}: Never box the type.
2042 \item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
2043 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
2044 higher-order functions are good candidates for boxing.
2047 Setting the \textit{card}~\qty{type} option for a function or product type
2048 implicitly enables boxing for that type.
2051 {\small See also \textit{verbose} (\S\ref{output-format})
2052 and \textit{debug} (\S\ref{output-format}).}
2054 \opsmart{box}{dont\_box}
2055 Specifies the default boxing setting to use. This can be overridden on a
2056 per-type basis using the \textit{box}~\qty{type} option described above.
2058 \opargboolorsmart{mono}{type}{non\_mono}
2059 Specifies whether the given type should be considered monotonic when
2060 enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a
2061 monotonicity check on the type. Setting this option to \textit{true} can reduce
2062 the number of scopes tried, but it also diminishes the theoretical chance of
2063 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
2066 {\small See also \textit{card} (\S\ref{scope-of-search}),
2067 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
2068 (\S\ref{output-format}).}
2070 \opsmart{mono}{non\_box}
2071 Specifies the default monotonicity setting to use. This can be overridden on a
2072 per-type basis using the \textit{mono}~\qty{type} option described above.
2074 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
2075 Specifies whether type variables with the same sort constraints should be
2076 merged. Setting this option to \textit{true} can reduce the number of scopes
2077 tried and the size of the generated Kodkod formulas, but it also diminishes the
2078 theoretical chance of finding a counterexample.
2080 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
2082 \opargbool{std}{type}{non\_std}
2083 Specifies whether the given (recursive) datatype should be given standard
2084 models. Nonstandard models are unsound but can help debug structural induction
2085 proofs, as explained in \S\ref{inductive-properties}.
2087 \optrue{std}{non\_std}
2088 Specifies the default standardness to use. This can be overridden on a per-type
2089 basis using the \textit{std}~\qty{type} option described above.
2092 \subsection{Output Format}
2093 \label{output-format}
2096 \opfalse{verbose}{quiet}
2097 Specifies whether the \textbf{nitpick} command should explain what it does. This
2098 option is useful to determine which scopes are tried or which SAT solver is
2099 used. This option is implicitly disabled for automatic runs.
2101 \opfalse{debug}{no\_debug}
2102 Specifies whether Nitpick should display additional debugging information beyond
2103 what \textit{verbose} already displays. Enabling \textit{debug} also enables
2104 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
2105 option is implicitly disabled for automatic runs.
2108 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
2109 \textit{batch\_size} (\S\ref{optimizations}).}
2111 \optrue{show\_skolems}{hide\_skolem}
2112 Specifies whether the values of Skolem constants should be displayed as part of
2113 counterexamples. Skolem constants correspond to bound variables in the original
2114 formula and usually help us to understand why the counterexample falsifies the
2118 {\small See also \textit{skolemize} (\S\ref{optimizations}).}
2120 \opfalse{show\_datatypes}{hide\_datatypes}
2121 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
2122 be displayed as part of counterexamples. Such subsets are sometimes helpful when
2123 investigating whether a potential counterexample is genuine or spurious, but
2124 their potential for clutter is real.
2126 \opfalse{show\_consts}{hide\_consts}
2127 Specifies whether the values of constants occurring in the formula (including
2128 its axioms) should be displayed along with any counterexample. These values are
2129 sometimes helpful when investigating why a counterexample is
2130 genuine, but they can clutter the output.
2132 \opfalse{show\_all}{dont\_show\_all}
2133 Enabling this option effectively enables \textit{show\_skolems},
2134 \textit{show\_datatypes}, and \textit{show\_consts}.
2136 \opdefault{max\_potential}{int}{$\mathbf{1}$}
2137 Specifies the maximum number of potential counterexamples to display. Setting
2138 this option to 0 speeds up the search for a genuine counterexample. This option
2139 is implicitly set to 0 for automatic runs. If you set this option to a value
2140 greater than 1, you will need an incremental SAT solver: For efficiency, it is
2141 recommended to install the JNI version of MiniSat and set \textit{sat\_solver} =
2142 \textit{MiniSat\_JNI}. Also be aware that many of the counterexamples may look
2143 identical, unless the \textit{show\_all} (\S\ref{output-format}) option is
2147 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
2148 \textit{sat\_solver} (\S\ref{optimizations}).}
2150 \opdefault{max\_genuine}{int}{$\mathbf{1}$}
2151 Specifies the maximum number of genuine counterexamples to display. If you set
2152 this option to a value greater than 1, you will need an incremental SAT solver:
2153 For efficiency, it is recommended to install the JNI version of MiniSat and set
2154 \textit{sat\_solver} = \textit{MiniSat\_JNI}. Also be aware that many of the
2155 counterexamples may look identical, unless the \textit{show\_all}
2156 (\S\ref{output-format}) option is enabled.
2159 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
2160 \textit{sat\_solver} (\S\ref{optimizations}).}
2162 \opnodefault{eval}{term\_list}
2163 Specifies the list of terms whose values should be displayed along with
2164 counterexamples. This option suffers from an ``observer effect'': Nitpick might
2165 find different counterexamples for different values of this option.
2167 \oparg{format}{term}{int\_seq}
2168 Specifies how to uncurry the value displayed for a variable or constant.
2169 Uncurrying sometimes increases the readability of the output for high-arity
2170 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
2171 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
2172 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
2173 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
2174 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
2175 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
2176 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
2177 arguments that are not accounted for are left alone, as if the specification had
2178 been $1,\ldots,1,n_1,\ldots,n_k$.
2181 {\small See also \textit{uncurry} (\S\ref{optimizations}).}
2183 \opdefault{format}{int\_seq}{$\mathbf{1}$}
2184 Specifies the default format to use. Irrespective of the default format, the
2185 extra arguments to a Skolem constant corresponding to the outer bound variables
2186 are kept separated from the remaining arguments, the \textbf{for} arguments of
2187 an inductive definitions are kept separated from the remaining arguments, and
2188 the iteration counter of an unrolled inductive definition is shown alone. The
2189 default format can be overridden on a per-variable or per-constant basis using
2190 the \textit{format}~\qty{term} option described above.
2193 %% MARK: Authentication
2194 \subsection{Authentication}
2195 \label{authentication}
2198 \opfalse{check\_potential}{trust\_potential}
2199 Specifies whether potential counterexamples should be given to Isabelle's
2200 \textit{auto} tactic to assess their validity. If a potential counterexample is
2201 shown to be genuine, Nitpick displays a message to this effect and terminates.
2204 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
2206 \opfalse{check\_genuine}{trust\_genuine}
2207 Specifies whether genuine and likely genuine counterexamples should be given to
2208 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
2209 counterexample is shown to be spurious, the user is kindly asked to send a bug
2210 report to the author at
2211 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
2214 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
2216 \opnodefault{expect}{string}
2217 Specifies the expected outcome, which must be one of the following:
2220 \item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
2221 \item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a ``likely
2222 genuine'' counterexample (i.e., a counterexample that is genuine unless
2223 it contradicts a missing axiom or a dangerous option was used inappropriately).
2224 \item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
2225 \item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
2226 \item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
2227 Kodkod ran out of memory).
2230 Nitpick emits an error if the actual outcome differs from the expected outcome.
2231 This option is useful for regression testing.
2234 \subsection{Optimizations}
2235 \label{optimizations}
2237 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
2242 \opdefault{sat\_solver}{string}{smart}
2243 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2244 to be faster than their Java counterparts, but they can be more difficult to
2245 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2246 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2247 you will need an incremental SAT solver, such as \textit{MiniSat\_JNI}
2248 (recommended) or \textit{SAT4J}.
2250 The supported solvers are listed below:
2254 \item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
2255 written in \cpp{}. To use MiniSat, set the environment variable
2256 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2257 executable. The \cpp{} sources and executables for MiniSat are available at
2258 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2259 and 2.0 beta (2007-07-21).
2261 \item[$\bullet$] \textbf{\textit{MiniSat\_JNI}}: The JNI (Java Native Interface)
2262 version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which
2263 you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2264 version of MiniSat, the JNI version can be used incrementally.
2267 %%% "It is bundled with Kodkodi and requires no further installation or
2268 %%% configuration steps. Alternatively,"
2269 \item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
2270 written in C. You can install a standard version of
2271 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2272 that contains the \texttt{picosat} executable. The C sources for PicoSAT are
2273 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2274 Nitpick has been tested with version 913.
2276 \item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
2277 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2278 the directory that contains the \texttt{zchaff} executable. The \cpp{} sources
2279 and executables for zChaff are available at
2280 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2281 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2283 \item[$\bullet$] \textbf{\textit{zChaff\_JNI}}: The JNI version of zChaff is
2284 bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on
2285 Kodkod's web site \cite{kodkod-2009}.
2287 \item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
2288 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2289 directory that contains the \texttt{rsat} executable. The \cpp{} sources for
2290 RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been
2291 tested with version 2.01.
2293 \item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
2294 written in C. To use BerkMin, set the environment variable
2295 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2296 executable. The BerkMin executables are available at
2297 \url{http://eigold.tripod.com/BerkMin.html}.
2299 \item[$\bullet$] \textbf{\textit{BerkMin\_Alloy}}: Variant of BerkMin that is
2300 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2301 version of BerkMin, set the environment variable
2302 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2305 \item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
2306 written in C. To use Jerusat, set the environment variable
2307 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2308 executable. The C sources for Jerusat are available at
2309 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2311 \item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
2312 written in Java that can be used incrementally. It is bundled with Kodkodi and
2313 requires no further installation or configuration steps. Do not attempt to
2314 install the official SAT4J packages, because their API is incompatible with
2317 \item[$\bullet$] \textbf{\textit{SAT4J\_Light}}: Variant of SAT4J that is
2318 optimized for small problems. It can also be used incrementally.
2320 \item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
2321 experimental solver written in \cpp. To use HaifaSat, set the environment
2322 variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
2323 \texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at
2324 \url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
2326 \item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
2327 \textit{smart}, Nitpick selects the first solver among MiniSat,
2328 PicoSAT, zChaff, RSat, BerkMin, BerkMin\_Alloy, Jerusat, MiniSat\_JNI, and zChaff\_JNI
2329 that is recognized by Isabelle. If none is found, it falls back on SAT4J, which
2330 should always be available. If \textit{verbose} (\S\ref{output-format}) is
2331 enabled, Nitpick displays which SAT solver was chosen.
2335 \opdefault{batch\_size}{int\_or\_smart}{smart}
2336 Specifies the maximum number of Kodkod problems that should be lumped together
2337 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2338 together ensures that Kodkodi is launched less often, but it makes the verbose
2339 output less readable and is sometimes detrimental to performance. If
2340 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2341 \textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
2343 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2344 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2345 be rewritten to use (automatically generated) discriminators and destructors.
2346 This optimization can drastically reduce the size of the Boolean formulas given
2350 {\small See also \textit{debug} (\S\ref{output-format}).}
2352 \optrue{specialize}{dont\_specialize}
2353 Specifies whether functions invoked with static arguments should be specialized.
2354 This optimization can drastically reduce the search space, especially for
2355 higher-order functions.
2358 {\small See also \textit{debug} (\S\ref{output-format}) and
2359 \textit{show\_consts} (\S\ref{output-format}).}
2361 \optrue{skolemize}{dont\_skolemize}
2362 Specifies whether the formula should be skolemized. For performance reasons,
2363 (positive) $\forall$-quanti\-fiers that occur in the scope of a higher-order
2364 (positive) $\exists$-quanti\-fier are left unchanged.
2367 {\small See also \textit{debug} (\S\ref{output-format}) and
2368 \textit{show\_skolems} (\S\ref{output-format}).}
2370 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2371 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2372 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2373 predicates for which each the predicate occurs in at most one assumption of each
2374 introduction rule. Using the reflexive transitive closure is in principle
2375 equivalent to setting \textit{iter} to the cardinality of the predicate's
2376 domain, but it is usually more efficient.
2378 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2379 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2381 \optrue{uncurry}{dont\_uncurry}
2382 Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no
2383 tangible effect on efficiency, but it creates opportunities for the boxing
2387 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug}
2388 (\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).}
2390 \optrue{fast\_descrs}{full\_descrs}
2391 Specifies whether Nitpick should optimize the definite and indefinite
2392 description operators (THE and SOME). The optimized versions usually help
2393 Nitpick generate more counterexamples or at least find them faster, but only the
2394 unoptimized versions are complete when all types occurring in the formula are
2397 {\small See also \textit{debug} (\S\ref{output-format}).}
2399 \optrue{peephole\_optim}{no\_peephole\_optim}
2400 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2401 peephole optimizer. These optimizations can make a significant difference.
2402 Unless you are tracking down a bug in Nitpick or distrust the peephole
2403 optimizer, you should leave this option enabled.
2405 \opdefault{sym\_break}{int}{20}
2406 Specifies an upper bound on the number of relations for which Kodkod generates
2407 symmetry breaking predicates. According to the Kodkod documentation
2408 \cite{kodkod-2009-options}, ``in general, the higher this value, the more
2409 symmetries will be broken, and the faster the formula will be solved. But,
2410 setting the value too high may have the opposite effect and slow down the
2413 \opdefault{sharing\_depth}{int}{3}
2414 Specifies the depth to which Kodkod should check circuits for equivalence during
2415 the translation to SAT. The default of 3 is the same as in Alloy. The minimum
2416 allowed depth is 1. Increasing the sharing may result in a smaller SAT problem,
2417 but can also slow down Kodkod.
2419 \opfalse{flatten\_props}{dont\_flatten\_props}
2420 Specifies whether Kodkod should try to eliminate intermediate Boolean variables.
2421 Although this might sound like a good idea, in practice it can drastically slow
2424 \opdefault{max\_threads}{int}{0}
2425 Specifies the maximum number of threads to use in Kodkod. If this option is set
2426 to 0, Kodkod will compute an appropriate value based on the number of processor
2430 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2431 \textit{timeout} (\S\ref{timeouts}).}
2434 \subsection{Timeouts}
2438 \opdefault{timeout}{time}{$\mathbf{30}$ s}
2439 Specifies the maximum amount of time that the \textbf{nitpick} command should
2440 spend looking for a counterexample. Nitpick tries to honor this constraint as
2441 well as it can but offers no guarantees. For automatic runs,
2442 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2443 a time slot whose length is specified by the ``Auto Counterexample Time
2444 Limit'' option in Proof General.
2447 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2449 \opdefault{tac\_timeout}{time}{$\mathbf{500}$\,ms}
2450 Specifies the maximum amount of time that the \textit{auto} tactic should use
2451 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2452 and \textit{size\_change} should use when checking whether a (co)in\-duc\-tive
2453 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2454 can but offers no guarantees.
2457 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2458 \textit{check\_potential} (\S\ref{authentication}),
2459 and \textit{check\_genuine} (\S\ref{authentication}).}
2462 \section{Attribute Reference}
2463 \label{attribute-reference}
2465 Nitpick needs to consider the definitions of all constants occurring in a
2466 formula in order to falsify it. For constants introduced using the
2467 \textbf{definition} command, the definition is simply the associated
2468 \textit{\_def} axiom. In contrast, instead of using the internal representation
2469 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2470 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2471 equational specification entered by the user.
2473 Behind the scenes, Isabelle's built-in packages and theories rely on the
2474 following attributes to affect Nitpick's behavior:
2477 \flushitem{\textit{nitpick\_def}}
2480 This attribute specifies an alternative definition of a constant. The
2481 alternative definition should be logically equivalent to the constant's actual
2482 axiomatic definition and should be of the form
2484 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2486 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2489 \flushitem{\textit{nitpick\_simp}}
2492 This attribute specifies the equations that constitute the specification of a
2493 constant. For functions defined using the \textbf{primrec}, \textbf{function},
2494 and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
2495 \textit{simps} rules. The equations must be of the form
2497 \qquad $c~t_1~\ldots\ t_n \,=\, u.$
2499 \flushitem{\textit{nitpick\_psimp}}
2502 This attribute specifies the equations that constitute the partial specification
2503 of a constant. For functions defined using the \textbf{function} package, this
2504 corresponds to the \textit{psimps} rules. The conditional equations must be of
2507 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
2509 \flushitem{\textit{nitpick\_intro}}
2512 This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
2513 For predicates defined using the \textbf{inductive} or \textbf{coinductive}
2514 command, this corresponds to the \textit{intros} rules. The introduction rules
2517 \qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
2518 \ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\
2521 where the $P_i$'s are side conditions that do not involve $c$ and $M$ is an
2522 optional monotonic operator. The order of the assumptions is irrelevant.
2526 When faced with a constant, Nitpick proceeds as follows:
2529 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2530 is not empty, Nitpick uses these rules as the specification of the constant.
2532 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2533 the constant is not empty, it uses these rules as the specification of the
2536 \item[3.] Otherwise, it looks up the definition of the constant:
2539 \item[1.] If the \textit{nitpick\_def} set associated with the constant
2540 is not empty, it uses the latest rule added to the set as the definition of the
2541 constant; otherwise it uses the actual definition axiom.
2542 \item[2.] If the definition is of the form
2544 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
2546 then Nitpick assumes that the definition was made using an inductive package and
2547 based on the introduction rules marked with \textit{nitpick\_\allowbreak
2548 ind\_\allowbreak intros} tries to determine whether the definition is
2553 As an illustration, consider the inductive definition
2556 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2557 ``\textit{odd}~1'' $\,\mid$ \\
2558 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2561 Isabelle automatically attaches the \textit{nitpick\_intro} attribute to
2562 the above rules. Nitpick then uses the \textit{lfp}-based definition in
2563 conjunction with these rules. To override this, we can specify an alternative
2564 definition as follows:
2567 \textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]: ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2570 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2571 = 1$. Alternatively, we can specify an equational specification of the constant:
2574 \textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]: ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2577 Such tweaks should be done with great care, because Nitpick will assume that the
2578 constant is completely defined by its equational specification. For example, if
2579 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
2580 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2581 (\S\ref{output-format}) option is extremely useful to understand what is going
2582 on when experimenting with \textit{nitpick\_} attributes.
2584 \section{Standard ML Interface}
2585 \label{standard-ml-interface}
2587 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2588 and debugging. Among the most interesting functions exported by Nitpick are
2589 those that let you invoke the tool programmatically and those that let you
2590 register and unregister custom coinductive datatypes.
2592 \subsection{Invocation of Nitpick}
2593 \label{invocation-of-nitpick}
2595 The \textit{Nitpick} structure offers the following functions for invoking your
2596 favorite counterexample generator:
2599 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2600 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2601 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2602 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2603 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2606 The return value is a new proof state paired with an outcome string
2607 (``genuine'', ``likely\_genuine'', ``potential'', ``none'', or ``unknown''). The
2608 \textit{params} type is a large record that lets you set Nitpick's options. The
2609 current default options can be retrieved by calling the following function
2610 defined in the \textit{Nitpick\_Isar} structure:
2613 $\textbf{val}\,~\textit{default\_params} :\,
2614 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2617 The second argument lets you override option values before they are parsed and
2618 put into a \textit{params} record. Here is an example:
2621 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2622 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2623 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2624 & \textit{subgoal}\end{aligned}$
2629 \subsection{Registration of Coinductive Datatypes}
2630 \label{registration-of-coinductive-datatypes}
2632 If you have defined a custom coinductive datatype, you can tell Nitpick about
2633 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2634 uses for lazy lists. The interface for registering and unregistering coinductive
2635 datatypes consists of the following pair of functions defined in the
2636 \textit{Nitpick} structure:
2639 $\textbf{val}\,~\textit{register\_codatatype} :\,
2640 \textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
2641 $\textbf{val}\,~\textit{unregister\_codatatype} :\,
2642 \textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
2645 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2646 not been, you could have told Nitpick about it by adding the following line
2647 to your theory file:
2650 $\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
2651 & \textit{Nitpick.register\_codatatype} \\[-2pt]
2652 & \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
2653 & \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
2656 The \textit{register\_codatatype} function takes a coinductive type, its case
2657 function, and the list of its constructors. The case function must take its
2658 arguments in the order that the constructors are listed. If no case function
2659 with the correct signature is available, simply pass the empty string.
2661 On the other hand, if your goal is to cripple Nitpick, add the following line to
2662 your theory file and try to check a few conjectures about lazy lists:
2665 $\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
2666 \kern1pt'a~\textit{list}\textrm{''}\}\ \,{*}\}$
2669 Inductive datatypes can be registered as coinductive datatypes, given
2670 appropriate coinductive constructors. However, doing so precludes
2671 the use of the inductive constructors---Nitpick will generate an error if they
2674 \section{Known Bugs and Limitations}
2675 \label{known-bugs-and-limitations}
2677 Here are the known bugs and limitations in Nitpick at the time of writing:
2680 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2681 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2682 Nitpick to generate spurious counterexamples for theorems that refer to values
2683 for which the function is not defined. For example:
2686 \textbf{primrec} \textit{prec} \textbf{where} \\
2687 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2688 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2689 \textbf{nitpick} \\[2\smallskipamount]
2690 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2692 \\[2\smallskipamount]
2693 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2694 \textbf{by}~(\textit{auto simp}:~\textit{prec\_def})
2697 Such theorems are considered bad style because they rely on the internal
2698 representation of functions synthesized by Isabelle, which is an implementation
2701 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2702 which can become invalid if you change the definition of an inductive predicate
2703 that is registered in the cache. To clear the cache,
2704 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2705 501$\,\textit{ms}$).
2707 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2708 \textbf{guess} command in a structured proof.
2710 \item[$\bullet$] The \textit{nitpick\_} attributes and the
2711 \textit{Nitpick.register\_} functions can cause havoc if used improperly.
2713 \item[$\bullet$] Although this has never been observed, arbitrary theorem
2714 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
2716 \item[$\bullet$] Local definitions are not supported and result in an error.
2718 %\item[$\bullet$] All constants and types whose names start with
2719 %\textit{Nitpick}{.} are reserved for internal use.
2723 \bibliography{../manual}{}
2724 \bibliographystyle{abbrv}