1 \documentclass[a4paper,12pt]{article}
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23 \def\Colon{\mathord{:\mkern-1.5mu:}}
24 %\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}}
25 %\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}}
26 \def\lparr{\mathopen{(\mkern-4mu\mid}}
27 \def\rparr{\mathclose{\mid\mkern-4mu)}}
30 \def\undef{(\lambda x.\; \unk)}
31 %\def\unr{\textit{others}}
33 \def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
34 \def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
36 \hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
37 counter-example counter-examples data-type data-types co-data-type
38 co-data-types in-duc-tive co-in-duc-tive}
44 \selectlanguage{english}
46 \title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
47 Picking Nits \\[\smallskipamount]
48 \Large A User's Guide to Nitpick for Isabelle/HOL}
50 Jasmin Christian Blanchette \\
51 {\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\
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81 \section{Introduction}
84 Nitpick \cite{blanchette-nipkow-2010} is a counterexample generator for
85 Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
86 combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
87 quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
88 first-order relational model finder developed by the Software Design Group at
89 MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
90 borrows many ideas and code fragments, but it benefits from Kodkod's
91 optimizations and a new encoding scheme. The name Nitpick is shamelessly
92 appropriated from a now retired Alloy precursor.
94 Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
95 theorem and wait a few seconds. Nonetheless, there are situations where knowing
96 how it works under the hood and how it reacts to various options helps
97 increase the test coverage. This manual also explains how to install the tool on
98 your workstation. Should the motivation fail you, think of the many hours of
99 hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.
101 Another common use of Nitpick is to find out whether the axioms of a locale are
102 satisfiable, while the locale is being developed. To check this, it suffices to
106 \textbf{lemma}~``$\textit{False}$'' \\
107 \textbf{nitpick}~[\textit{show\_all}]
110 after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
111 must find a model for the axioms. If it finds no model, we have an indication
112 that the axioms might be unsatisfiable.
114 You can also invoke Nitpick from the ``Commands'' submenu of the
115 ``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c C-a
116 C-n. This is equivalent to entering the \textbf{nitpick} command with no
117 arguments in the theory text.
119 Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
120 machine called \texttt{java}. The examples presented in this manual can be found
121 in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.
123 Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
124 Nitpick also provides an automatic mode that can be enabled via the ``Auto
125 Nitpick'' option from the ``Isabelle'' menu in Proof General. In this mode,
126 Nitpick is run on every newly entered theorem. The time limit for Auto Nitpick
127 and other automatic tools can be set using the ``Auto Tools Time Limit'' option.
130 \setbox\boxA=\hbox{\texttt{nospam}}
132 The known bugs and limitations at the time of writing are listed in
133 \S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
134 or this manual should be directed to
135 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
136 in.\allowbreak tum.\allowbreak de}.
138 \vskip2.5\smallskipamount
140 \textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
141 suggesting several textual improvements.
142 % and Perry James for reporting a typo.
144 %\section{Installation}
145 %\label{installation}
149 % * Nitpick is part of Isabelle/HOL
150 % * but it relies on an external tool called Kodkodi (Kodkod wrapper)
152 % * if you use a prebuilt Isabelle package, Kodkodi is automatically there
153 % * if you work from sources, the latest Kodkodi can be obtained from ...
154 % download it, install it in some directory of your choice (e.g.,
155 % $ISABELLE_HOME/contrib/kodkodi), and add the absolute path to Kodkodi
156 % in your .isabelle/etc/components file
158 % * If you're not sure, just try the example in the next section
160 \section{First Steps}
163 This section introduces Nitpick by presenting small examples. If possible, you
164 should try out the examples on your workstation. Your theory file should start
168 \textbf{theory}~\textit{Scratch} \\
169 \textbf{imports}~\textit{Main~Quotient\_Product~RealDef} \\
173 The results presented here were obtained using the JNI (Java Native Interface)
174 version of MiniSat and with multithreading disabled to reduce nondeterminism.
175 This was done by adding the line
178 \textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSat\_JNI}, \,\textit{max\_threads}~= 1]
181 after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
182 Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
183 be installed, as explained in \S\ref{optimizations}. If you have already
184 configured SAT solvers in Isabelle (e.g., for Refute), these will also be
185 available to Nitpick.
187 \subsection{Propositional Logic}
188 \label{propositional-logic}
190 Let's start with a trivial example from propositional logic:
193 \textbf{lemma}~``$P \longleftrightarrow Q$'' \\
197 You should get the following output:
201 Nitpick found a counterexample: \\[2\smallskipamount]
202 \hbox{}\qquad Free variables: \nopagebreak \\
203 \hbox{}\qquad\qquad $P = \textit{True}$ \\
204 \hbox{}\qquad\qquad $Q = \textit{False}$
207 %FIXME: If you get the output:...
208 %Then do such-and-such.
210 Nitpick can also be invoked on individual subgoals, as in the example below:
213 \textbf{apply}~\textit{auto} \\[2\smallskipamount]
214 {\slshape goal (2 subgoals): \\
215 \phantom{0}1. $P\,\Longrightarrow\, Q$ \\
216 \phantom{0}2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
217 \textbf{nitpick}~1 \\[2\smallskipamount]
218 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
219 \hbox{}\qquad Free variables: \nopagebreak \\
220 \hbox{}\qquad\qquad $P = \textit{True}$ \\
221 \hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
222 \textbf{nitpick}~2 \\[2\smallskipamount]
223 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
224 \hbox{}\qquad Free variables: \nopagebreak \\
225 \hbox{}\qquad\qquad $P = \textit{False}$ \\
226 \hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
230 \subsection{Type Variables}
231 \label{type-variables}
233 If you are left unimpressed by the previous example, don't worry. The next
234 one is more mind- and computer-boggling:
237 \textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
239 \pagebreak[2] %% TYPESETTING
241 The putative lemma involves the definite description operator, {THE}, presented
242 in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
243 operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
244 lemma is merely asserting the indefinite description operator axiom with {THE}
245 substituted for {SOME}.
247 The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
248 containing type variables, Nitpick enumerates the possible domains for each type
249 variable, up to a given cardinality (10 by default), looking for a finite
253 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
255 Trying 10 scopes: \nopagebreak \\
256 \hbox{}\qquad \textit{card}~$'a$~= 1; \\
257 \hbox{}\qquad \textit{card}~$'a$~= 2; \\
258 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
259 \hbox{}\qquad \textit{card}~$'a$~= 10. \\[2\smallskipamount]
260 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
261 \hbox{}\qquad Free variables: \nopagebreak \\
262 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
263 \hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
267 Nitpick found a counterexample in which $'a$ has cardinality 3. (For
268 cardinalities 1 and 2, the formula holds.) In the counterexample, the three
269 values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
271 The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
272 \textit{verbose} is enabled. You can specify \textit{verbose} each time you
273 invoke \textbf{nitpick}, or you can set it globally using the command
276 \textbf{nitpick\_params} [\textit{verbose}]
279 This command also displays the current default values for all of the options
280 supported by Nitpick. The options are listed in \S\ref{option-reference}.
282 \subsection{Constants}
285 By just looking at Nitpick's output, it might not be clear why the
286 counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
287 this time telling it to show the values of the constants that occur in the
291 \textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
292 \textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
294 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
295 \hbox{}\qquad Free variables: \nopagebreak \\
296 \hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
297 \hbox{}\qquad\qquad $x = a_3$ \\
298 \hbox{}\qquad Constant: \nopagebreak \\
299 \hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
302 As the result of an optimization, Nitpick directly assigned a value to the
303 subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
304 disable this optimization by using the command
307 \textbf{nitpick}~[\textit{dont\_specialize},\, \textit{show\_consts}]
313 \slshape Constant: \nopagebreak \\
314 \hbox{}\qquad $\mathit{The} = \undef{}
315 (\!\begin{aligned}[t]%
316 & \{a_1, a_2, a_3\} := a_3,\> \{a_1, a_2\} := a_3,\> \{a_1, a_3\} := a_3, \\[-2pt] %% TYPESETTING
317 & \{a_1\} := a_1,\> \{a_2, a_3\} := a_1,\> \{a_2\} := a_2, \\[-2pt]
318 & \{a_3\} := a_3,\> \{\} := a_3)\end{aligned}$
321 Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
322 just like before.\footnote{The Isabelle/HOL notation $f(x :=
323 y)$ denotes the function that maps $x$ to $y$ and that otherwise behaves like
326 Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
327 unique $x$ such that'') at the front of our putative lemma's assumption:
330 \textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
333 The fix appears to work:
336 \textbf{nitpick} \\[2\smallskipamount]
337 \slshape Nitpick found no counterexample.
340 We can further increase our confidence in the formula by exhausting all
341 cardinalities up to 50:
344 \textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
345 can be entered as \texttt{-} (hyphen) or
346 \texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
347 \slshape Nitpick found no counterexample.
350 Let's see if Sledgehammer can find a proof:
353 \textbf{sledgehammer} \\[2\smallskipamount]
354 {\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
355 $\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
356 Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
357 \textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
358 {\slshape No subgoals!}% \\[2\smallskipamount]
362 This must be our lucky day.
364 \subsection{Skolemization}
365 \label{skolemization}
367 Are all invertible functions onto? Let's find out:
370 \textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
371 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
372 \textbf{nitpick} \\[2\smallskipamount]
374 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
375 \hbox{}\qquad Free variable: \nopagebreak \\
376 \hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
377 \hbox{}\qquad Skolem constants: \nopagebreak \\
378 \hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
379 \hbox{}\qquad\qquad $y = a_2$
382 Although $f$ is the only free variable occurring in the formula, Nitpick also
383 displays values for the bound variables $g$ and $y$. These values are available
384 to Nitpick because it performs skolemization as a preprocessing step.
386 In the previous example, skolemization only affected the outermost quantifiers.
387 This is not always the case, as illustrated below:
390 \textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
391 \textbf{nitpick} \\[2\smallskipamount]
393 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
394 \hbox{}\qquad Skolem constant: \nopagebreak \\
395 \hbox{}\qquad\qquad $\lambda x.\; f =
396 \undef{}(\!\begin{aligned}[t]
397 & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
398 & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
401 The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
402 $x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
403 function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
404 maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
406 The source of the Skolem constants is sometimes more obscure:
409 \textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
410 \textbf{nitpick} \\[2\smallskipamount]
412 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
413 \hbox{}\qquad Free variable: \nopagebreak \\
414 \hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
415 \hbox{}\qquad Skolem constants: \nopagebreak \\
416 \hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
417 \hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
420 What happened here is that Nitpick expanded the \textit{sym} constant to its
424 $\mathit{sym}~r \,\equiv\,
425 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
428 As their names suggest, the Skolem constants $\mathit{sym}.x$ and
429 $\mathit{sym}.y$ are simply the bound variables $x$ and $y$
430 from \textit{sym}'s definition.
432 \subsection{Natural Numbers and Integers}
433 \label{natural-numbers-and-integers}
435 Because of the axiom of infinity, the type \textit{nat} does not admit any
436 finite models. To deal with this, Nitpick's approach is to consider finite
437 subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined
438 value (displayed as `$\unk$'). The type \textit{int} is handled similarly.
439 Internally, undefined values lead to a three-valued logic.
441 Here is an example involving \textit{int\/}:
444 \textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
445 \textbf{nitpick} \\[2\smallskipamount]
446 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
447 \hbox{}\qquad Free variables: \nopagebreak \\
448 \hbox{}\qquad\qquad $i = 0$ \\
449 \hbox{}\qquad\qquad $j = 1$ \\
450 \hbox{}\qquad\qquad $m = 1$ \\
451 \hbox{}\qquad\qquad $n = 0$
454 Internally, Nitpick uses either a unary or a binary representation of numbers.
455 The unary representation is more efficient but only suitable for numbers very
456 close to zero. By default, Nitpick attempts to choose the more appropriate
457 encoding by inspecting the formula at hand. This behavior can be overridden by
458 passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For
459 binary notation, the number of bits to use can be specified using
460 the \textit{bits} option. For example:
463 \textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$]
466 With infinite types, we don't always have the luxury of a genuine counterexample
467 and must often content ourselves with a potential one. The tedious task of
468 finding out whether the potential counterexample is in fact genuine can be
469 outsourced to \textit{auto} by passing \textit{check\_potential}. For example:
472 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
473 \textbf{nitpick} [\textit{card~nat}~= 50, \textit{check\_potential}] \\[2\smallskipamount]
474 \slshape Warning: The conjecture either trivially holds for the given scopes or lies outside Nitpick's supported
475 fragment. Only potential counterexamples may be found. \\[2\smallskipamount]
476 Nitpick found a potential counterexample: \\[2\smallskipamount]
477 \hbox{}\qquad Free variable: \nopagebreak \\
478 \hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
479 Confirmation by ``\textit{auto}'': The above counterexample is genuine.
482 You might wonder why the counterexample is first reported as potential. The root
483 of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
484 \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
485 that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
486 \textit{False}; but otherwise, it does not know anything about values of $n \ge
487 \textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
488 \textit{True}. Since the assumption can never be satisfied, the putative lemma
489 can never be falsified.
491 Incidentally, if you distrust the so-called genuine counterexamples, you can
492 enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
493 aware that \textit{auto} will usually fail to prove that the counterexample is
496 Some conjectures involving elementary number theory make Nitpick look like a
497 giant with feet of clay:
500 \textbf{lemma} ``$P~\textit{Suc}$'' \\
501 \textbf{nitpick} \\[2\smallskipamount]
503 Nitpick found no counterexample.
506 On any finite set $N$, \textit{Suc} is a partial function; for example, if $N =
507 \{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\,
508 \ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when passed as
509 argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$. The next
513 \textbf{lemma} ``$P~(\textit{op}~{+}\Colon
514 \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
515 \textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
516 {\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
517 \hbox{}\qquad Free variable: \nopagebreak \\
518 \hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
519 \textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
520 {\slshape Nitpick found no counterexample.}
523 The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
524 $\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
527 Because numbers are infinite and are approximated using a three-valued logic,
528 there is usually no need to systematically enumerate domain sizes. If Nitpick
529 cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
530 unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
531 example above is an exception to this principle.) Nitpick nonetheless enumerates
532 all cardinalities from 1 to 10 for \textit{nat}, mainly because smaller
533 cardinalities are fast to handle and give rise to simpler counterexamples. This
534 is explained in more detail in \S\ref{scope-monotonicity}.
536 \subsection{Inductive Datatypes}
537 \label{inductive-datatypes}
539 Like natural numbers and integers, inductive datatypes with recursive
540 constructors admit no finite models and must be approximated by a subterm-closed
541 subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
542 Nitpick looks for all counterexamples that can be built using at most 10
545 Let's see with an example involving \textit{hd} (which returns the first element
546 of a list) and $@$ (which concatenates two lists):
549 \textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
550 \textbf{nitpick} \\[2\smallskipamount]
551 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
552 \hbox{}\qquad Free variables: \nopagebreak \\
553 \hbox{}\qquad\qquad $\textit{xs} = []$ \\
554 \hbox{}\qquad\qquad $\textit{y} = a_1$
557 To see why the counterexample is genuine, we enable \textit{show\_consts}
558 and \textit{show\_\allowbreak datatypes}:
561 {\slshape Datatype:} \\
562 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\
563 {\slshape Constants:} \\
564 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_1, a_1])$ \\
565 \hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$
568 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
571 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
572 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_1,
573 a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not
574 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
575 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
576 appending $[a_1, a_1]$ to itself gives $\unk$.
578 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
579 considers the following subsets:
581 \kern-.5\smallskipamount %% TYPESETTING
585 $\{[],\, [a_1],\, [a_2]\}$; \\
586 $\{[],\, [a_1],\, [a_3]\}$; \\
587 $\{[],\, [a_2],\, [a_3]\}$; \\
588 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\
589 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\
590 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\
591 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\
592 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\
593 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\
594 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\
595 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\
596 $\{[],\, [a_3],\, [a_3, a_3]\}$.
600 \kern-2\smallskipamount %% TYPESETTING
602 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
603 are listed and only those. As an example of a non-subterm-closed subset,
604 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe
605 that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin
606 \mathcal{S}$ as a subterm.
608 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
611 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
612 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
614 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
615 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
616 \hbox{}\qquad Free variables: \nopagebreak \\
617 \hbox{}\qquad\qquad $\textit{xs} = [a_1]$ \\
618 \hbox{}\qquad\qquad $\textit{ys} = [a_2]$ \\
619 \hbox{}\qquad Datatypes: \\
620 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
621 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$
624 Because datatypes are approximated using a three-valued logic, there is usually
625 no need to systematically enumerate cardinalities: If Nitpick cannot find a
626 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
627 unlikely that one could be found for smaller cardinalities.
629 \subsection{Typedefs, Quotient Types, Records, Rationals, and Reals}
630 \label{typedefs-quotient-types-records-rationals-and-reals}
632 Nitpick generally treats types declared using \textbf{typedef} as datatypes
633 whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
637 \textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
638 \textbf{by}~\textit{blast} \\[2\smallskipamount]
639 \textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
640 \textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
641 \textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
642 \textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
643 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
644 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
645 \hbox{}\qquad Free variables: \nopagebreak \\
646 \hbox{}\qquad\qquad $P = \{\Abs{0},\, \Abs{1}\}$ \\
647 \hbox{}\qquad\qquad $x = \Abs{2}$ \\
648 \hbox{}\qquad Datatypes: \\
649 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
650 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$
653 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
655 Quotient types are handled in much the same way. The following fragment defines
656 the integer type \textit{my\_int} by encoding the integer $x$ by a pair of
657 natural numbers $(m, n)$ such that $x + n = m$:
660 \textbf{fun} \textit{my\_int\_rel} \textbf{where} \\
661 ``$\textit{my\_int\_rel}~(x,\, y)~(u,\, v) = (x + v = u + y)$'' \\[2\smallskipamount]
663 \textbf{quotient\_type}~\textit{my\_int} = ``$\textit{nat} \times \textit{nat\/}$''$\;{/}\;$\textit{my\_int\_rel} \\
664 \textbf{by}~(\textit{auto simp add\/}:\ \textit{equivp\_def expand\_fun\_eq}) \\[2\smallskipamount]
666 \textbf{definition}~\textit{add\_raw}~\textbf{where} \\
667 ``$\textit{add\_raw} \,\equiv\, \lambda(x,\, y)~(u,\, v).\; (x + (u\Colon\textit{nat}), y + (v\Colon\textit{nat}))$'' \\[2\smallskipamount]
669 \textbf{quotient\_definition} ``$\textit{add\/}\Colon\textit{my\_int} \Rightarrow \textit{my\_int} \Rightarrow \textit{my\_int\/}$'' \textbf{is} \textit{add\_raw} \\[2\smallskipamount]
671 \textbf{lemma} ``$\textit{add}~x~y = \textit{add}~x~x$'' \\
672 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
673 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
674 \hbox{}\qquad Free variables: \nopagebreak \\
675 \hbox{}\qquad\qquad $x = \Abs{(0,\, 0)}$ \\
676 \hbox{}\qquad\qquad $y = \Abs{(1,\, 0)}$ \\
677 \hbox{}\qquad Datatypes: \\
678 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, \unr\}$ \\
679 \hbox{}\qquad\qquad $\textit{nat} \times \textit{nat}~[\textsl{boxed\/}] = \{(0,\, 0),\> (1,\, 0),\> \unr\}$ \\
680 \hbox{}\qquad\qquad $\textit{my\_int} = \{\Abs{(0,\, 0)},\> \Abs{(1,\, 0)},\> \unr\}$
683 In the counterexample, $\Abs{(0,\, 0)}$ and $\Abs{(1,\, 0)}$ represent the
684 integers $0$ and $1$, respectively. Other representants would have been
685 possible---e.g., $\Abs{(5,\, 5)}$ and $\Abs{(12,\, 11)}$. If we are going to
686 use \textit{my\_int} extensively, it pays off to install a term postprocessor
687 that converts the pair notation to the standard mathematical notation:
690 $\textbf{ML}~\,\{{*} \\
692 %& ({*}~\,\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \\[-2pt]
693 %& \phantom{(*}~\,{\rightarrow}\;\textit{term}~\,{*}) \\[-2pt]
694 & \textbf{fun}\,~\textit{my\_int\_postproc}~\_~\_~\_~T~(\textit{Const}~\_~\$~(\textit{Const}~\_~\$~\textit{t1}~\$~\textit{t2\/})) = {} \\[-2pt]
695 & \phantom{fun}\,~\textit{HOLogic.mk\_number}~T~(\textit{snd}~(\textit{HOLogic.dest\_number~t1}) \\[-2pt]
696 & \phantom{fun\,~\textit{HOLogic.mk\_number}~T~(}{-}~\textit{snd}~(\textit{HOLogic.dest\_number~t2\/})) \\[-2pt]
697 & \phantom{fun}\!{\mid}\,~\textit{my\_int\_postproc}~\_~\_~\_~\_~t = t \\[-2pt]
698 {*}\}\end{aligned}$ \\[2\smallskipamount]
699 $\textbf{declaration}~\,\{{*} \\
701 & \textit{Nitpick\_Model.register\_term\_postprocessor}~\!\begin{aligned}[t]
702 & @\{\textrm{typ}~\textit{my\_int}\} \\[-2pt]
703 & \textit{my\_int\_postproc}\end{aligned} \\[-2pt]
707 Records are also handled as datatypes with a single constructor:
710 \textbf{record} \textit{point} = \\
711 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
712 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
713 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
714 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
715 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
716 \hbox{}\qquad Free variables: \nopagebreak \\
717 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
718 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
719 \hbox{}\qquad Datatypes: \\
720 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
721 \hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
722 & \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
723 & \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
726 Finally, Nitpick provides rudimentary support for rationals and reals using a
730 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
731 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
732 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
733 \hbox{}\qquad Free variables: \nopagebreak \\
734 \hbox{}\qquad\qquad $x = 1/2$ \\
735 \hbox{}\qquad\qquad $y = -1/2$ \\
736 \hbox{}\qquad Datatypes: \\
737 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
738 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
739 \hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
742 \subsection{Inductive and Coinductive Predicates}
743 \label{inductive-and-coinductive-predicates}
745 Inductively defined predicates (and sets) are particularly problematic for
746 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
747 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
748 the problem is that they are defined using a least fixed-point construction.
750 Nitpick's philosophy is that not all inductive predicates are equal. Consider
751 the \textit{even} predicate below:
754 \textbf{inductive}~\textit{even}~\textbf{where} \\
755 ``\textit{even}~0'' $\,\mid$ \\
756 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
759 This predicate enjoys the desirable property of being well-founded, which means
760 that the introduction rules don't give rise to infinite chains of the form
763 $\cdots\,\Longrightarrow\, \textit{even}~k''
764 \,\Longrightarrow\, \textit{even}~k'
765 \,\Longrightarrow\, \textit{even}~k.$
768 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
772 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
773 \,\Longrightarrow\, \textit{even}~(k - 2)
774 \,\Longrightarrow\, \textit{even}~k.$
777 Wellfoundedness is desirable because it enables Nitpick to use a very efficient
778 fixed-point computation.%
779 \footnote{If an inductive predicate is
780 well-founded, then it has exactly one fixed point, which is simultaneously the
781 least and the greatest fixed point. In these circumstances, the computation of
782 the least fixed point amounts to the computation of an arbitrary fixed point,
783 which can be performed using a straightforward recursive equation.}
784 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
785 just as Isabelle's \textbf{function} package usually discharges termination
786 proof obligations automatically.
788 Let's try an example:
791 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
792 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
793 \slshape The inductive predicate ``\textit{even}'' was proved well-founded.
794 Nitpick can compute it efficiently. \\[2\smallskipamount]
796 \hbox{}\qquad \textit{card nat}~= 50. \\[2\smallskipamount]
797 Nitpick found a potential counterexample for \textit{card nat}~= 50: \\[2\smallskipamount]
798 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
799 Nitpick could not find a better counterexample. It checked 0 of 1 scope. \\[2\smallskipamount]
803 No genuine counterexample is possible because Nitpick cannot rule out the
804 existence of a natural number $n \ge 50$ such that both $\textit{even}~n$ and
805 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
806 existential quantifier:
809 \textbf{lemma} ``$\exists n \mathbin{\le} 49.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
810 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}] \\[2\smallskipamount]
811 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
812 \hbox{}\qquad Empty assignment
815 So far we were blessed by the wellfoundedness of \textit{even}. What happens if
816 we use the following definition instead?
819 \textbf{inductive} $\textit{even}'$ \textbf{where} \\
820 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
821 ``$\textit{even}'~2$'' $\,\mid$ \\
822 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
825 This definition is not well-founded: From $\textit{even}'~0$ and
826 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
827 predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
829 Let's check a property involving $\textit{even}'$. To make up for the
830 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
831 \textit{nat}'s cardinality to a mere 10:
834 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
835 \lnot\;\textit{even}'~n$'' \\
836 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
838 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
839 Nitpick might need to unroll it. \\[2\smallskipamount]
841 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
842 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
843 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
844 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
845 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
846 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
847 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
848 \hbox{}\qquad Constant: \nopagebreak \\
849 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
850 & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
851 & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
852 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
856 Nitpick's output is very instructive. First, it tells us that the predicate is
857 unrolled, meaning that it is computed iteratively from the empty set. Then it
858 lists six scopes specifying different bounds on the numbers of iterations:\ 0,
861 The output also shows how each iteration contributes to $\textit{even}'$. The
862 notation $\lambda i.\; \textit{even}'$ indicates that the value of the
863 predicate depends on an iteration counter. Iteration 0 provides the basis
864 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
865 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
866 iterations would not contribute any new elements.
868 Some values are marked with superscripted question
869 marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
870 predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
871 \textit{True} or $\unk$, never \textit{False}.
873 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, 20, 24, and 28
874 iterations. However, these numbers are bounded by the cardinality of the
875 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
876 ever needed to compute the value of a \textit{nat} predicate. You can specify
877 the number of iterations using the \textit{iter} option, as explained in
878 \S\ref{scope-of-search}.
880 In the next formula, $\textit{even}'$ occurs both positively and negatively:
883 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
884 \textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
885 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
886 \hbox{}\qquad Free variable: \nopagebreak \\
887 \hbox{}\qquad\qquad $n = 1$ \\
888 \hbox{}\qquad Constants: \nopagebreak \\
889 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
890 & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\
891 \hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
894 Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
895 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
896 fixed point (not necessarily the least one). It is used to falsify
897 $\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
898 $\textit{even}'~(n - 2)$.
900 Coinductive predicates are handled dually. For example:
903 \textbf{coinductive} \textit{nats} \textbf{where} \\
904 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
905 \textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
906 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
907 \slshape Nitpick found a counterexample:
908 \\[2\smallskipamount]
909 \hbox{}\qquad Constants: \nopagebreak \\
910 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
911 & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
912 & \unr\})\end{aligned}$ \\
913 \hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
916 As a special case, Nitpick uses Kodkod's transitive closure operator to encode
917 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
918 inductive predicates for which each the predicate occurs in at most one
919 assumption of each introduction rule. For example:
922 \textbf{inductive} \textit{odd} \textbf{where} \\
923 ``$\textit{odd}~1$'' $\,\mid$ \\
924 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
925 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
926 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
927 \slshape Nitpick found a counterexample:
928 \\[2\smallskipamount]
929 \hbox{}\qquad Free variable: \nopagebreak \\
930 \hbox{}\qquad\qquad $n = 1$ \\
931 \hbox{}\qquad Constants: \nopagebreak \\
932 \hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
933 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
934 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
936 & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
937 & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
939 & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
940 & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
941 \hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
945 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
946 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
947 elements from known ones. The set $\textit{odd}$ consists of all the values
948 reachable through the reflexive transitive closure of
949 $\textit{odd}_{\textrm{step}}$ starting with any element from
950 $\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
951 transitive closure to encode linear predicates is normally either more thorough
952 or more efficient than unrolling (depending on the value of \textit{iter}), but
953 for those cases where it isn't you can disable it by passing the
954 \textit{dont\_star\_linear\_preds} option.
956 \subsection{Coinductive Datatypes}
957 \label{coinductive-datatypes}
959 While Isabelle regrettably lacks a high-level mechanism for defining coinductive
960 datatypes, the \textit{Coinductive\_List} theory from Andreas Lochbihler's
961 \textit{Coinductive} AFP entry \cite{lochbihler-2010} provides a coinductive
962 ``lazy list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick
963 supports these lazy lists seamlessly and provides a hook, described in
964 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
967 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but
968 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
969 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
970 1, 2, 3, \ldots]$ can be defined as lazy lists using the
971 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
972 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
973 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
975 Although it is otherwise no friend of infinity, Nitpick can find counterexamples
976 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
980 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
981 \textbf{nitpick} \\[2\smallskipamount]
982 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
983 \hbox{}\qquad Free variables: \nopagebreak \\
984 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
985 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
988 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
989 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
990 infinite list $[a_1, a_1, a_1, \ldots]$.
992 The next example is more interesting:
995 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
996 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
997 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
998 \slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
999 some scopes. \\[2\smallskipamount]
1000 Trying 10 scopes: \\
1001 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 1,
1002 and \textit{bisim\_depth}~= 0. \\
1003 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1004 \hbox{}\qquad \textit{card} $'a$~= 10, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 10,
1005 and \textit{bisim\_depth}~= 9. \\[2\smallskipamount]
1006 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
1007 \textit{card}~``\kern1pt$'a~\textit{list\/}$''~= 2, and \textit{bisim\_\allowbreak
1009 \\[2\smallskipamount]
1010 \hbox{}\qquad Free variables: \nopagebreak \\
1011 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
1012 \hbox{}\qquad\qquad $\textit{b} = a_2$ \\
1013 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1014 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
1018 The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
1019 $\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
1020 $[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment
1021 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
1022 the segment leading to the binder is the stem.
1024 A salient property of coinductive datatypes is that two objects are considered
1025 equal if and only if they lead to the same observations. For example, the lazy
1026 lists $\textrm{THE}~\omega.\; \omega =
1027 \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
1028 $\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
1029 \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
1030 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
1031 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
1032 concept of equality for coinductive datatypes is called bisimulation and is
1033 defined coinductively.
1035 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
1036 the Kodkod problem to ensure that distinct objects lead to different
1037 observations. This precaution is somewhat expensive and often unnecessary, so it
1038 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
1039 bisimilarity check is then performed \textsl{after} the counterexample has been
1040 found to ensure correctness. If this after-the-fact check fails, the
1041 counterexample is tagged as ``quasi genuine'' and Nitpick recommends to try
1042 again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
1043 check for the previous example saves approximately 150~milli\-seconds; the speed
1044 gains can be more significant for larger scopes.
1046 The next formula illustrates the need for bisimilarity (either as a Kodkod
1047 predicate or as an after-the-fact check) to prevent spurious counterexamples:
1050 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
1051 \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
1052 \textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
1053 \slshape Nitpick found a quasi genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
1054 \hbox{}\qquad Free variables: \nopagebreak \\
1055 \hbox{}\qquad\qquad $a = a_1$ \\
1056 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
1057 \textit{LCons}~a_1~\omega$ \\
1058 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
1059 \hbox{}\qquad Codatatype:\strut \nopagebreak \\
1060 \hbox{}\qquad\qquad $'a~\textit{llist} =
1061 \{\!\begin{aligned}[t]
1062 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
1063 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
1064 \\[2\smallskipamount]
1065 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
1066 that the counterexample is genuine. \\[2\smallskipamount]
1067 {\upshape\textbf{nitpick}} \\[2\smallskipamount]
1068 \slshape Nitpick found no counterexample.
1071 In the first \textbf{nitpick} invocation, the after-the-fact check discovered
1072 that the two known elements of type $'a~\textit{llist}$ are bisimilar.
1074 A compromise between leaving out the bisimilarity predicate from the Kodkod
1075 problem and performing the after-the-fact check is to specify a lower
1076 nonnegative \textit{bisim\_depth} value than the default one provided by
1077 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
1078 be distinguished from each other by their prefixes of length $K$. Be aware that
1079 setting $K$ to a too low value can overconstrain Nitpick, preventing it from
1080 finding any counterexamples.
1085 Nitpick normally maps function and product types directly to the corresponding
1086 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
1087 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
1088 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
1089 off to treat these types in the same way as plain datatypes, by approximating
1090 them by a subset of a given cardinality. This technique is called ``boxing'' and
1091 is particularly useful for functions passed as arguments to other functions, for
1092 high-arity functions, and for large tuples. Under the hood, boxing involves
1093 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
1094 isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
1096 To illustrate boxing, we consider a formalization of $\lambda$-terms represented
1097 using de Bruijn's notation:
1100 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
1103 The $\textit{lift}~t~k$ function increments all variables with indices greater
1104 than or equal to $k$ by one:
1107 \textbf{primrec} \textit{lift} \textbf{where} \\
1108 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
1109 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
1110 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
1113 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
1114 term $t$ has a loose variable with index $k$ or more:
1117 \textbf{primrec}~\textit{loose} \textbf{where} \\
1118 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
1119 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
1120 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
1123 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
1127 \textbf{primrec}~\textit{subst} \textbf{where} \\
1128 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
1129 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
1130 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
1131 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
1134 A substitution is a function that maps variable indices to terms. Observe that
1135 $\sigma$ is a function passed as argument and that Nitpick can't optimize it
1136 away, because the recursive call for the \textit{Lam} case involves an altered
1137 version. Also notice the \textit{lift} call, which increments the variable
1138 indices when moving under a \textit{Lam}.
1140 A reasonable property to expect of substitution is that it should leave closed
1141 terms unchanged. Alas, even this simple property does not hold:
1144 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
1145 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
1147 Trying 10 scopes: \nopagebreak \\
1148 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
1149 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
1150 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1151 \hbox{}\qquad \textit{card~nat}~= 10, \textit{card tm}~= 10, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 10. \\[2\smallskipamount]
1152 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
1153 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
1154 \hbox{}\qquad Free variables: \nopagebreak \\
1155 \hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
1156 & 0 := \textit{Var}~0,\>
1157 1 := \textit{Var}~0,\>
1158 2 := \textit{Var}~0, \\[-2pt]
1159 & 3 := \textit{Var}~0,\>
1160 4 := \textit{Var}~0,\>
1161 5 := \textit{Var}~0)\end{aligned}$ \\
1162 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
1166 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
1167 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
1168 $\lambda$-term notation, $t$~is
1169 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
1170 The bug is in \textit{subst\/}: The $\textit{lift}~(\sigma~m)~1$ call should be
1171 replaced with $\textit{lift}~(\sigma~m)~0$.
1173 An interesting aspect of Nitpick's verbose output is that it assigned inceasing
1174 cardinalities from 1 to 10 to the type $\textit{nat} \Rightarrow \textit{tm}$.
1175 For the formula of interest, knowing 6 values of that type was enough to find
1176 the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
1177 considered, a hopeless undertaking:
1180 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
1181 {\slshape Nitpick ran out of time after checking 3 of 10 scopes.}
1185 Boxing can be enabled or disabled globally or on a per-type basis using the
1186 \textit{box} option. Nitpick usually performs reasonable choices about which
1187 types should be boxed, but option tweaking sometimes helps. A related optimization,
1188 ``finalization,'' attempts to wrap functions that constant at all but finitely
1189 many points (e.g., finite sets); see the documentation for the \textit{finalize}
1190 option in \S\ref{scope-of-search} for details.
1194 \subsection{Scope Monotonicity}
1195 \label{scope-monotonicity}
1197 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
1198 and \textit{max}) controls which scopes are actually tested. In general, to
1199 exhaust all models below a certain cardinality bound, the number of scopes that
1200 Nitpick must consider increases exponentially with the number of type variables
1201 (and \textbf{typedecl}'d types) occurring in the formula. Given the default
1202 cardinality specification of 1--10, no fewer than $10^4 = 10\,000$ scopes must be
1203 considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
1205 Fortunately, many formulas exhibit a property called \textsl{scope
1206 monotonicity}, meaning that if the formula is falsifiable for a given scope,
1207 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
1209 Consider the formula
1212 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
1215 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
1216 $'b~\textit{list}$. A priori, Nitpick would need to consider $1\,000$ scopes to
1217 exhaust the specification \textit{card}~= 1--10 (10 cardinalies for $'a$
1218 $\times$ 10 cardinalities for $'b$ $\times$ 10 cardinalities for the datatypes).
1219 However, our intuition tells us that any counterexample found with a small scope
1220 would still be a counterexample in a larger scope---by simply ignoring the fresh
1221 $'a$ and $'b$ values provided by the larger scope. Nitpick comes to the same
1222 conclusion after a careful inspection of the formula and the relevant
1226 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
1228 The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
1229 Nitpick might be able to skip some scopes.
1230 \\[2\smallskipamount]
1231 Trying 10 scopes: \\
1232 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
1233 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
1234 \textit{list\/}''~= 1, \\
1235 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 1, and
1236 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 1. \\
1237 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
1238 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
1239 \textit{list\/}''~= 2, \\
1240 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 2, and
1241 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 2. \\
1242 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
1243 \hbox{}\qquad \textit{card} $'a$~= 10, \textit{card} $'b$~= 10,
1244 \textit{card} \textit{nat}~= 10, \textit{card} ``$('a \times {'}b)$
1245 \textit{list\/}''~= 10, \\
1246 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 10, and
1247 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 10.
1248 \\[2\smallskipamount]
1249 Nitpick found a counterexample for
1250 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
1251 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
1252 \textit{list\/}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 5, and
1253 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 5:
1254 \\[2\smallskipamount]
1255 \hbox{}\qquad Free variables: \nopagebreak \\
1256 \hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
1257 \hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
1261 In theory, it should be sufficient to test a single scope:
1264 \textbf{nitpick}~[\textit{card}~= 10]
1267 However, this is often less efficient in practice and may lead to overly complex
1270 If the monotonicity check fails but we believe that the formula is monotonic (or
1271 we don't mind missing some counterexamples), we can pass the
1272 \textit{mono} option. To convince yourself that this option is risky,
1273 simply consider this example from \S\ref{skolemization}:
1276 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
1277 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
1278 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
1279 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
1280 \textbf{nitpick} \\[2\smallskipamount]
1282 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
1283 \hbox{}\qquad $\vdots$
1286 (It turns out the formula holds if and only if $\textit{card}~'a \le
1287 \textit{card}~'b$.) Although this is rarely advisable, the automatic
1288 monotonicity checks can be disabled by passing \textit{non\_mono}
1289 (\S\ref{optimizations}).
1291 As insinuated in \S\ref{natural-numbers-and-integers} and
1292 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
1293 are normally monotonic and treated as such. The same is true for record types,
1294 \textit{rat}, and \textit{real}. Thus, given the
1295 cardinality specification 1--10, a formula involving \textit{nat}, \textit{int},
1296 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
1297 consider only 10~scopes instead of $10\,000$. On the other hand,
1298 \textbf{typedef}s and quotient types are generally nonmonotonic.
1300 \subsection{Inductive Properties}
1301 \label{inductive-properties}
1303 Inductive properties are a particular pain to prove, because the failure to
1304 establish an induction step can mean several things:
1307 \item The property is invalid.
1308 \item The property is valid but is too weak to support the induction step.
1309 \item The property is valid and strong enough; it's just that we haven't found
1313 Depending on which scenario applies, we would take the appropriate course of
1317 \item Repair the statement of the property so that it becomes valid.
1318 \item Generalize the property and/or prove auxiliary properties.
1319 \item Work harder on a proof.
1322 How can we distinguish between the three scenarios? Nitpick's normal mode of
1323 operation can often detect scenario 1, and Isabelle's automatic tactics help with
1324 scenario 3. Using appropriate techniques, it is also often possible to use
1325 Nitpick to identify scenario 2. Consider the following transition system,
1326 in which natural numbers represent states:
1329 \textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
1330 ``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
1331 ``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
1332 ``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
1335 We will try to prove that only even numbers are reachable:
1338 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
1341 Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
1342 so let's attempt a proof by induction:
1345 \textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
1346 \textbf{apply}~\textit{auto}
1349 This leaves us in the following proof state:
1352 {\slshape goal (2 subgoals): \\
1353 \phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
1354 \phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
1358 If we run Nitpick on the first subgoal, it still won't find any
1359 counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
1360 is helpless. However, notice the $n \in \textit{reach}$ assumption, which
1361 strengthens the induction hypothesis but is not immediately usable in the proof.
1362 If we remove it and invoke Nitpick, this time we get a counterexample:
1365 \textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
1366 \textbf{nitpick} \\[2\smallskipamount]
1367 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1368 \hbox{}\qquad Skolem constant: \nopagebreak \\
1369 \hbox{}\qquad\qquad $n = 0$
1372 Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
1373 to strength the lemma:
1376 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
1379 Unfortunately, the proof by induction still gets stuck, except that Nitpick now
1380 finds the counterexample $n = 2$. We generalize the lemma further to
1383 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
1386 and this time \textit{arith} can finish off the subgoals.
1388 A similar technique can be employed for structural induction. The
1389 following mini formalization of full binary trees will serve as illustration:
1392 \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]
1393 \textbf{primrec}~\textit{labels}~\textbf{where} \\
1394 ``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
1395 ``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
1396 \textbf{primrec}~\textit{swap}~\textbf{where} \\
1397 ``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
1398 \phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
1399 ``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
1402 The \textit{labels} function returns the set of labels occurring on leaves of a
1403 tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
1404 labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
1405 obtained by swapping $a$ and $b$:
1408 \textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
1411 Nitpick can't find any counterexample, so we proceed with induction
1412 (this time favoring a more structured style):
1415 \textbf{proof}~(\textit{induct}~$t$) \\
1416 \hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
1418 \hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
1421 Nitpick can't find any counterexample at this point either, but it makes the
1422 following suggestion:
1426 Hint: To check that the induction hypothesis is general enough, try this command:
1427 \textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
1430 If we follow the hint, we get a ``nonstandard'' counterexample for the step:
1433 \slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
1434 \hbox{}\qquad Free variables: \nopagebreak \\
1435 \hbox{}\qquad\qquad $a = a_1$ \\
1436 \hbox{}\qquad\qquad $b = a_2$ \\
1437 \hbox{}\qquad\qquad $t = \xi_1$ \\
1438 \hbox{}\qquad\qquad $u = \xi_2$ \\
1439 \hbox{}\qquad Datatype: \nopagebreak \\
1440 \hbox{}\qquad\qquad $\alpha~\textit{btree} = \{\xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \textit{Branch}~\xi_1~\xi_2\}$ \\
1441 \hbox{}\qquad {\slshape Constants:} \nopagebreak \\
1442 \hbox{}\qquad\qquad $\textit{labels} = \undef
1443 (\!\begin{aligned}[t]%
1444 & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
1445 & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
1446 \hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \undef
1447 (\!\begin{aligned}[t]%
1448 & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
1449 & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount]
1450 The existence of a nonstandard model suggests that the induction hypothesis is not general enough or may even
1451 be wrong. See the Nitpick manual's ``Inductive Properties'' section for details (``\textit{isabelle doc nitpick}'').
1454 Reading the Nitpick manual is a most excellent idea.
1455 But what's going on? The \textit{non\_std} option told the tool to look for
1456 nonstandard models of binary trees, which means that new ``nonstandard'' trees
1457 $\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees
1458 generated by the \textit{Leaf} and \textit{Branch} constructors.%
1459 \footnote{Notice the similarity between allowing nonstandard trees here and
1460 allowing unreachable states in the preceding example (by removing the ``$n \in
1461 \textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
1462 set of objects over which the induction is performed while doing the step
1463 in order to test the induction hypothesis's strength.}
1464 Unlike standard trees, these new trees contain cycles. We will see later that
1465 every property of acyclic trees that can be proved without using induction also
1466 holds for cyclic trees. Hence,
1469 \textsl{If the induction
1470 hypothesis is strong enough, the induction step will hold even for nonstandard
1471 objects, and Nitpick won't find any nonstandard counterexample.}
1474 But here the tool find some nonstandard trees $t = \xi_1$
1475 and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in
1476 \textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$.
1477 Because neither tree contains both $a$ and $b$, the induction hypothesis tells
1478 us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
1479 and as a result we know nothing about the labels of the tree
1480 $\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
1481 $\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
1482 labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
1483 \textit{labels}$ $(\textit{swap}~u~a~b)$.
1485 The solution is to ensure that we always know what the labels of the subtrees
1486 are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
1487 $t$ in the statement of the lemma:
1490 \textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
1491 \phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
1492 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
1493 \phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
1494 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
1497 This time, Nitpick won't find any nonstandard counterexample, and we can perform
1498 the induction step using \textit{auto}.
1500 \section{Case Studies}
1501 \label{case-studies}
1503 As a didactic device, the previous section focused mostly on toy formulas whose
1504 validity can easily be assessed just by looking at the formula. We will now
1505 review two somewhat more realistic case studies that are within Nitpick's
1506 reach:\ a context-free grammar modeled by mutually inductive sets and a
1507 functional implementation of AA trees. The results presented in this
1508 section were produced with the following settings:
1511 \textbf{nitpick\_params} [\textit{max\_potential}~= 0]
1514 \subsection{A Context-Free Grammar}
1515 \label{a-context-free-grammar}
1517 Our first case study is taken from section 7.4 in the Isabelle tutorial
1518 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
1519 Ullman, produces all strings with an equal number of $a$'s and $b$'s:
1522 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
1523 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
1524 $A$ & $::=$ & $aS \mid bAA$ \\
1525 $B$ & $::=$ & $bS \mid aBB$
1529 The intuition behind the grammar is that $A$ generates all string with one more
1530 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
1532 The alphabet consists exclusively of $a$'s and $b$'s:
1535 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
1538 Strings over the alphabet are represented by \textit{alphabet list}s.
1539 Nonterminals in the grammar become sets of strings. The production rules
1540 presented above can be expressed as a mutually inductive definition:
1543 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
1544 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
1545 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1546 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
1547 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
1548 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
1549 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1552 The conversion of the grammar into the inductive definition was done manually by
1553 Joe Blow, an underpaid undergraduate student. As a result, some errors might
1556 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
1557 d'\^etre}. A good approach is to state desirable properties of the specification
1558 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
1559 as $b$'s) and check them with Nitpick. If the properties are correctly stated,
1560 counterexamples will point to bugs in the specification. For our grammar
1561 example, we will proceed in two steps, separating the soundness and the
1562 completeness of the set $S$. First, soundness:
1565 \textbf{theorem}~\textit{S\_sound\/}: \\
1566 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1567 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
1568 \textbf{nitpick} \\[2\smallskipamount]
1569 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1570 \hbox{}\qquad Free variable: \nopagebreak \\
1571 \hbox{}\qquad\qquad $w = [b]$
1574 It would seem that $[b] \in S$. How could this be? An inspection of the
1575 introduction rules reveals that the only rule with a right-hand side of the form
1576 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
1580 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
1583 On closer inspection, we can see that this rule is wrong. To match the
1584 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
1588 \textbf{nitpick} \\[2\smallskipamount]
1589 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1590 \hbox{}\qquad Free variable: \nopagebreak \\
1591 \hbox{}\qquad\qquad $w = [a, a, b]$
1594 Some detective work is necessary to find out what went wrong here. To get $[a,
1595 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
1599 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
1602 Now, this formula must be wrong: The same assumption occurs twice, and the
1603 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
1604 the assumptions should have been a $w$.
1606 With the correction made, we don't get any counterexample from Nitpick. Let's
1607 move on and check completeness:
1610 \textbf{theorem}~\textit{S\_complete}: \\
1611 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
1612 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
1613 \longrightarrow w \in S$'' \\
1614 \textbf{nitpick} \\[2\smallskipamount]
1615 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1616 \hbox{}\qquad Free variable: \nopagebreak \\
1617 \hbox{}\qquad\qquad $w = [b, b, a, a]$
1620 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
1621 $a$'s and $b$'s. But since our inductive definition passed the soundness check,
1622 the introduction rules we have are probably correct. Perhaps we simply lack an
1623 introduction rule. Comparing the grammar with the inductive definition, our
1624 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
1625 without which the grammar cannot generate two or more $b$'s in a row. So we add
1629 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
1632 With this last change, we don't get any counterexamples from Nitpick for either
1633 soundness or completeness. We can even generalize our result to cover $A$ and
1637 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
1638 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
1639 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
1640 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
1641 \textbf{nitpick} \\[2\smallskipamount]
1642 \slshape Nitpick found no counterexample.
1645 \subsection{AA Trees}
1648 AA trees are a kind of balanced trees discovered by Arne Andersson that provide
1649 similar performance to red-black trees, but with a simpler implementation
1650 \cite{andersson-1993}. They can be used to store sets of elements equipped with
1651 a total order $<$. We start by defining the datatype and some basic extractor
1655 \textbf{datatype} $'a$~\textit{aa\_tree} = \\
1656 \hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}'' \\[2\smallskipamount]
1657 \textbf{primrec} \textit{data} \textbf{where} \\
1658 ``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
1659 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
1660 \textbf{primrec} \textit{dataset} \textbf{where} \\
1661 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
1662 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
1663 \textbf{primrec} \textit{level} \textbf{where} \\
1664 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
1665 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
1666 \textbf{primrec} \textit{left} \textbf{where} \\
1667 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
1668 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
1669 \textbf{primrec} \textit{right} \textbf{where} \\
1670 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
1671 ``$\textit{right}~(N~\_~\_~\_~u) = u$''
1674 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
1675 as follows \cite{wikipedia-2009-aa-trees}:
1677 \kern.2\parskip %% TYPESETTING
1680 Each node has a level field, and the following invariants must remain true for
1681 the tree to be valid:
1685 \kern-.4\parskip %% TYPESETTING
1690 \item[1.] The level of a leaf node is one.
1691 \item[2.] The level of a left child is strictly less than that of its parent.
1692 \item[3.] The level of a right child is less than or equal to that of its parent.
1693 \item[4.] The level of a right grandchild is strictly less than that of its grandparent.
1694 \item[5.] Every node of level greater than one must have two children.
1699 \kern.4\parskip %% TYPESETTING
1701 The \textit{wf} predicate formalizes this description:
1704 \textbf{primrec} \textit{wf} \textbf{where} \\
1705 ``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
1706 ``$\textit{wf}~(N~\_~k~t~u) =$ \\
1707 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
1708 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
1709 \phantom{``$($}$\textrm{else}$ \\
1710 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
1711 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
1712 \mathrel{\land} \textit{level}~u \le k$ \\
1713 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
1716 Rebalancing the tree upon insertion and removal of elements is performed by two
1717 auxiliary functions called \textit{skew} and \textit{split}, defined below:
1720 \textbf{primrec} \textit{skew} \textbf{where} \\
1721 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
1722 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\
1723 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
1724 \textit{level}~t~\textrm{then}$ \\
1725 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
1726 (\textit{right}~t)~u)$ \\
1727 \phantom{``(}$\textrm{else}$ \\
1728 \phantom{``(\quad}$N~x~k~t~u)$''
1732 \textbf{primrec} \textit{split} \textbf{where} \\
1733 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
1734 ``$\textit{split}~(N~x~k~t~u) = {}$ \\
1735 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
1736 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
1737 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
1738 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
1739 \phantom{``(}$\textrm{else}$ \\
1740 \phantom{``(\quad}$N~x~k~t~u)$''
1743 Performing a \textit{skew} or a \textit{split} should have no impact on the set
1744 of elements stored in the tree:
1747 \textbf{theorem}~\textit{dataset\_skew\_split\/}:\\
1748 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
1749 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
1750 \textbf{nitpick} \\[2\smallskipamount]
1751 {\slshape Nitpick ran out of time after checking 9 of 10 scopes.}
1754 Furthermore, applying \textit{skew} or \textit{split} on a well-formed tree
1755 should not alter the tree:
1758 \textbf{theorem}~\textit{wf\_skew\_split\/}:\\
1759 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
1760 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
1761 \textbf{nitpick} \\[2\smallskipamount]
1762 {\slshape Nitpick found no counterexample.}
1765 Insertion is implemented recursively. It preserves the sort order:
1768 \textbf{primrec}~\textit{insort} \textbf{where} \\
1769 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
1770 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\
1771 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
1772 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
1775 Notice that we deliberately commented out the application of \textit{skew} and
1776 \textit{split}. Let's see if this causes any problems:
1779 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1780 \textbf{nitpick} \\[2\smallskipamount]
1781 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
1782 \hbox{}\qquad Free variables: \nopagebreak \\
1783 \hbox{}\qquad\qquad $t = N~a_1~1~\Lambda~\Lambda$ \\
1784 \hbox{}\qquad\qquad $x = a_2$
1787 It's hard to see why this is a counterexample. To improve readability, we will
1788 restrict the theorem to \textit{nat}, so that we don't need to look up the value
1789 of the $\textit{op}~{<}$ constant to find out which element is smaller than the
1790 other. In addition, we will tell Nitpick to display the value of
1791 $\textit{insort}~t~x$ using the \textit{eval} option. This gives
1794 \textbf{theorem} \textit{wf\_insort\_nat\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
1795 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
1796 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
1797 \hbox{}\qquad Free variables: \nopagebreak \\
1798 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
1799 \hbox{}\qquad\qquad $x = 0$ \\
1800 \hbox{}\qquad Evaluated term: \\
1801 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
1804 Nitpick's output reveals that the element $0$ was added as a left child of $1$,
1805 where both nodes have a level of 1. This violates the second AA tree invariant,
1806 which states that a left child's level must be less than its parent's. This
1807 shouldn't come as a surprise, considering that we commented out the tree
1808 rebalancing code. Reintroducing the code seems to solve the problem:
1811 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
1812 \textbf{nitpick} \\[2\smallskipamount]
1813 {\slshape Nitpick ran out of time after checking 8 of 10 scopes.}
1816 Insertion should transform the set of elements represented by the tree in the
1820 \textbf{theorem} \textit{dataset\_insort\/}:\kern.4em
1821 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
1822 \textbf{nitpick} \\[2\smallskipamount]
1823 {\slshape Nitpick ran out of time after checking 7 of 10 scopes.}
1826 We could continue like this and sketch a complete theory of AA trees. Once the
1827 definitions and main theorems are in place and have been thoroughly tested using
1828 Nitpick, we could start working on the proofs. Developing theories this way
1829 usually saves time, because faulty theorems and definitions are discovered much
1830 earlier in the process.
1832 \section{Option Reference}
1833 \label{option-reference}
1835 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
1836 \def\qty#1{$\left<\textit{#1}\right>$}
1837 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
1838 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1839 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1840 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
1841 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
1842 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
1843 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
1844 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1845 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
1847 Nitpick's behavior can be influenced by various options, which can be specified
1848 in brackets after the \textbf{nitpick} command. Default values can be set
1849 using \textbf{nitpick\_\allowbreak params}. For example:
1852 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60]
1855 The options are categorized as follows:\ mode of operation
1856 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
1857 format (\S\ref{output-format}), automatic counterexample checks
1858 (\S\ref{authentication}), optimizations
1859 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
1861 You can instruct Nitpick to run automatically on newly entered theorems by
1862 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
1863 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation}),
1864 \textit{assms} (\S\ref{mode-of-operation}), and \textit{mono}
1865 (\S\ref{scope-of-search}) are implicitly enabled, \textit{blocking}
1866 (\S\ref{mode-of-operation}), \textit{verbose} (\S\ref{output-format}), and
1867 \textit{debug} (\S\ref{output-format}) are disabled, \textit{max\_threads}
1868 (\S\ref{optimizations}) is taken to be 1, \textit{max\_potential}
1869 (\S\ref{output-format}) is taken to be 0, and \textit{timeout}
1870 (\S\ref{timeouts}) is superseded by the ``Auto Tools Time Limit'' in
1871 Proof General's ``Isabelle'' menu. Nitpick's output is also more concise.
1873 The number of options can be overwhelming at first glance. Do not let that worry
1874 you: Nitpick's defaults have been chosen so that it almost always does the right
1875 thing, and the most important options have been covered in context in
1876 \S\ref{first-steps}.
1878 The descriptions below refer to the following syntactic quantities:
1881 \item[$\bullet$] \qtybf{string}: A string.
1882 \item[$\bullet$] \qtybf{string\_list\/}: A space-separated list of strings
1883 (e.g., ``\textit{ichi ni san}'').
1884 \item[$\bullet$] \qtybf{bool\/}: \textit{true} or \textit{false}.
1885 \item[$\bullet$] \qtybf{bool\_or\_smart\/}: \textit{true}, \textit{false}, or \textit{smart}.
1886 \item[$\bullet$] \qtybf{int\/}: An integer. Negative integers are prefixed with a hyphen.
1887 \item[$\bullet$] \qtybf{int\_or\_smart\/}: An integer or \textit{smart}.
1888 \item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
1889 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
1890 \item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
1891 \item[$\bullet$] \qtybf{time}: An integer (e.g., 60) or floating-point number
1892 (e.g., 0.5) expressing a number of seconds, or the keyword \textit{none}
1894 \item[$\bullet$] \qtybf{const\/}: The name of a HOL constant.
1895 \item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
1896 \item[$\bullet$] \qtybf{term\_list\/}: A space-separated list of HOL terms (e.g.,
1897 ``$f~x$''~``$g~y$'').
1898 \item[$\bullet$] \qtybf{type}: A HOL type.
1901 Default values are indicated in square brackets. Boolean options have a negated
1902 counterpart (e.g., \textit{blocking} vs.\ \textit{non\_blocking}). When setting
1903 Boolean options, ``= \textit{true}'' may be omitted.
1905 \subsection{Mode of Operation}
1906 \label{mode-of-operation}
1909 \optrue{blocking}{non\_blocking}
1910 Specifies whether the \textbf{nitpick} command should operate synchronously.
1911 The asynchronous (non-blocking) mode lets the user start proving the putative
1912 theorem while Nitpick looks for a counterexample, but it can also be more
1913 confusing. For technical reasons, automatic runs currently always block.
1915 \optrue{falsify}{satisfy}
1916 Specifies whether Nitpick should look for falsifying examples (countermodels) or
1917 satisfying examples (models). This manual assumes throughout that
1918 \textit{falsify} is enabled.
1920 \opsmart{user\_axioms}{no\_user\_axioms}
1921 Specifies whether the user-defined axioms (specified using
1922 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
1923 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
1924 the constants that occur in the formula to falsify. The option is implicitly set
1925 to \textit{true} for automatic runs.
1927 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might
1928 nonetheless ignore some polymorphic axioms. Counterexamples generated under
1929 these conditions are tagged as ``quasi genuine.'' The \textit{debug}
1930 (\S\ref{output-format}) option can be used to find out which axioms were
1934 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
1935 (\S\ref{output-format}).}
1937 \optrue{assms}{no\_assms}
1938 Specifies whether the relevant assumptions in structured proofs should be
1939 considered. The option is implicitly enabled for automatic runs.
1942 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
1944 \opfalse{overlord}{no\_overlord}
1945 Specifies whether Nitpick should put its temporary files in
1946 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
1947 debugging Nitpick but also unsafe if several instances of the tool are run
1948 simultaneously. The files are identified by the extensions
1949 \texttt{.kki}, \texttt{.cnf}, \texttt{.out}, and
1950 \texttt{.err}; you may safely remove them after Nitpick has run.
1953 {\small See also \textit{debug} (\S\ref{output-format}).}
1956 \subsection{Scope of Search}
1957 \label{scope-of-search}
1960 \oparg{card}{type}{int\_seq}
1961 Specifies the sequence of cardinalities to use for a given type.
1962 For free types, and often also for \textbf{typedecl}'d types, it usually makes
1963 sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
1966 {\small See also \textit{box} (\S\ref{scope-of-search}) and \textit{mono}
1967 (\S\ref{scope-of-search}).}
1969 \opdefault{card}{int\_seq}{$\mathbf{1}$--$\mathbf{10}$}
1970 Specifies the default sequence of cardinalities to use. This can be overridden
1971 on a per-type basis using the \textit{card}~\qty{type} option described above.
1973 \oparg{max}{const}{int\_seq}
1974 Specifies the sequence of maximum multiplicities to use for a given
1975 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
1976 number of distinct values that it can construct. Nonsensical values (e.g.,
1977 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for
1978 datatypes equipped with several constructors.
1980 \opnodefault{max}{int\_seq}
1981 Specifies the default sequence of maximum multiplicities to use for
1982 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
1983 basis using the \textit{max}~\qty{const} option described above.
1985 \opsmart{binary\_ints}{unary\_ints}
1986 Specifies whether natural numbers and integers should be encoded using a unary
1987 or binary notation. In unary mode, the cardinality fully specifies the subset
1988 used to approximate the type. For example:
1990 $$\hbox{\begin{tabular}{@{}rll@{}}%
1991 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
1992 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
1993 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
1998 $$\hbox{\begin{tabular}{@{}rll@{}}%
1999 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
2000 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
2003 In binary mode, the cardinality specifies the number of distinct values that can
2004 be constructed. Each of these value is represented by a bit pattern whose length
2005 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
2006 Nitpick attempts to choose the more appropriate encoding by inspecting the
2007 formula at hand, preferring the binary notation for problems involving
2008 multiplicative operators or large constants.
2010 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
2011 problems that refer to the types \textit{rat} or \textit{real} or the constants
2012 \textit{Suc}, \textit{gcd}, or \textit{lcm}.
2014 {\small See also \textit{bits} (\S\ref{scope-of-search}) and
2015 \textit{show\_datatypes} (\S\ref{output-format}).}
2017 \opdefault{bits}{int\_seq}{$\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{8},\mathbf{10},\mathbf{12},\mathbf{14},\mathbf{16}$}
2018 Specifies the number of bits to use to represent natural numbers and integers in
2019 binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
2021 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
2023 \opargboolorsmart{wf}{const}{non\_wf}
2024 Specifies whether the specified (co)in\-duc\-tively defined predicate is
2025 well-founded. The option can take the following values:
2028 \item[$\bullet$] \textbf{\textit{true}:} Tentatively treat the (co)in\-duc\-tive
2029 predicate as if it were well-founded. Since this is generally not sound when the
2030 predicate is not well-founded, the counterexamples are tagged as ``quasi
2033 \item[$\bullet$] \textbf{\textit{false}:} Treat the (co)in\-duc\-tive predicate
2034 as if it were not well-founded. The predicate is then unrolled as prescribed by
2035 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
2038 \item[$\bullet$] \textbf{\textit{smart}:} Try to prove that the inductive
2039 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
2040 \textit{size\_change} tactics. If this succeeds (or the predicate occurs with an
2041 appropriate polarity in the formula to falsify), use an efficient fixed-point
2042 equation as specification of the predicate; otherwise, unroll the predicates
2043 according to the \textit{iter}~\qty{const} and \textit{iter} options.
2047 {\small See also \textit{iter} (\S\ref{scope-of-search}),
2048 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
2049 (\S\ref{timeouts}).}
2051 \opsmart{wf}{non\_wf}
2052 Specifies the default wellfoundedness setting to use. This can be overridden on
2053 a per-predicate basis using the \textit{wf}~\qty{const} option above.
2055 \oparg{iter}{const}{int\_seq}
2056 Specifies the sequence of iteration counts to use when unrolling a given
2057 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
2058 predicates that occur negatively and coinductive predicates that occur
2059 positively in the formula to falsify and that cannot be proved to be
2060 well-founded, but this behavior is influenced by the \textit{wf} option. The
2061 iteration counts are automatically bounded by the cardinality of the predicate's
2064 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
2065 \textit{star\_linear\_preds} (\S\ref{optimizations}).}
2067 \opdefault{iter}{int\_seq}{$\mathbf{0{,}1{,}2{,}4{,}8{,}12{,}16{,}20{,}24{,}28}$}
2068 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
2069 predicates. This can be overridden on a per-predicate basis using the
2070 \textit{iter} \qty{const} option above.
2072 \opdefault{bisim\_depth}{int\_seq}{$\mathbf{9}$}
2073 Specifies the sequence of iteration counts to use when unrolling the
2074 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
2075 of $-1$ means that no predicate is generated, in which case Nitpick performs an
2076 after-the-fact check to see if the known coinductive datatype values are
2077 bidissimilar. If two values are found to be bisimilar, the counterexample is
2078 tagged as ``quasi genuine.'' The iteration counts are automatically bounded by
2079 the sum of the cardinalities of the coinductive datatypes occurring in the
2082 \opargboolorsmart{box}{type}{dont\_box}
2083 Specifies whether Nitpick should attempt to wrap (``box'') a given function or
2084 product type in an isomorphic datatype internally. Boxing is an effective mean
2085 to reduce the search space and speed up Nitpick, because the isomorphic datatype
2086 is approximated by a subset of the possible function or pair values.
2087 Like other drastic optimizations, it can also prevent the discovery of
2088 counterexamples. The option can take the following values:
2091 \item[$\bullet$] \textbf{\textit{true}:} Box the specified type whenever
2093 \item[$\bullet$] \textbf{\textit{false}:} Never box the type.
2094 \item[$\bullet$] \textbf{\textit{smart}:} Box the type only in contexts where it
2095 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
2096 higher-order functions are good candidates for boxing.
2100 {\small See also \textit{finitize} (\S\ref{scope-of-search}), \textit{verbose}
2101 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}).}
2103 \opsmart{box}{dont\_box}
2104 Specifies the default boxing setting to use. This can be overridden on a
2105 per-type basis using the \textit{box}~\qty{type} option described above.
2107 \opargboolorsmart{finitize}{type}{dont\_finitize}
2108 Specifies whether Nitpick should attempt to finitize a given type, which can be
2109 a function type or an infinite ``shallow datatype'' (an infinite datatype whose
2110 constructors don't appear in the problem).
2112 For function types, Nitpick performs a monotonicity analysis to detect functions
2113 that are constant at all but finitely many points (e.g., finite sets) and treats
2114 such occurrences specially, thereby increasing the precision. The option can
2115 then take the following values:
2118 \item[$\bullet$] \textbf{\textit{false}:} Don't attempt to finitize the type.
2119 \item[$\bullet$] \textbf{\textit{true}} or \textbf{\textit{smart}:} Finitize the
2120 type wherever possible.
2123 The semantics of the option is somewhat different for infinite shallow
2127 \item[$\bullet$] \textbf{\textit{true}:} Finitize the datatype. Since this is
2128 unsound, counterexamples generated under these conditions are tagged as ``quasi
2130 \item[$\bullet$] \textbf{\textit{false}:} Don't attempt to finitize the datatype.
2131 \item[$\bullet$] \textbf{\textit{smart}:} Perform a monotonicity analysis to
2132 detect whether the datatype can be safely finitized before finitizing it.
2135 Like other drastic optimizations, finitization can sometimes prevent the
2136 discovery of counterexamples.
2139 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{mono}
2140 (\S\ref{scope-of-search}), \textit{verbose} (\S\ref{output-format}), and
2141 \textit{debug} (\S\ref{output-format}).}
2143 \opsmart{finitize}{dont\_finitize}
2144 Specifies the default finitization setting to use. This can be overridden on a
2145 per-type basis using the \textit{finitize}~\qty{type} option described above.
2147 \opargboolorsmart{mono}{type}{non\_mono}
2148 Specifies whether the given type should be considered monotonic when enumerating
2149 scopes and finitizing types. If the option is set to \textit{smart}, Nitpick
2150 performs a monotonicity check on the type. Setting this option to \textit{true}
2151 can reduce the number of scopes tried, but it can also diminish the chance of
2152 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}. The
2153 option is implicitly set to \textit{true} for automatic runs.
2156 {\small See also \textit{card} (\S\ref{scope-of-search}),
2157 \textit{finitize} (\S\ref{scope-of-search}),
2158 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
2159 (\S\ref{output-format}).}
2161 \opsmart{mono}{non\_mono}
2162 Specifies the default monotonicity setting to use. This can be overridden on a
2163 per-type basis using the \textit{mono}~\qty{type} option described above.
2165 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
2166 Specifies whether type variables with the same sort constraints should be
2167 merged. Setting this option to \textit{true} can reduce the number of scopes
2168 tried and the size of the generated Kodkod formulas, but it also diminishes the
2169 theoretical chance of finding a counterexample.
2171 {\small See also \textit{mono} (\S\ref{scope-of-search}).}
2173 \opargbool{std}{type}{non\_std}
2174 Specifies whether the given (recursive) datatype should be given standard
2175 models. Nonstandard models are unsound but can help debug structural induction
2176 proofs, as explained in \S\ref{inductive-properties}.
2178 \optrue{std}{non\_std}
2179 Specifies the default standardness to use. This can be overridden on a per-type
2180 basis using the \textit{std}~\qty{type} option described above.
2183 \subsection{Output Format}
2184 \label{output-format}
2187 \opfalse{verbose}{quiet}
2188 Specifies whether the \textbf{nitpick} command should explain what it does. This
2189 option is useful to determine which scopes are tried or which SAT solver is
2190 used. This option is implicitly disabled for automatic runs.
2192 \opfalse{debug}{no\_debug}
2193 Specifies whether Nitpick should display additional debugging information beyond
2194 what \textit{verbose} already displays. Enabling \textit{debug} also enables
2195 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
2196 option is implicitly disabled for automatic runs.
2199 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
2200 \textit{batch\_size} (\S\ref{optimizations}).}
2202 \opfalse{show\_datatypes}{hide\_datatypes}
2203 Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
2204 be displayed as part of counterexamples. Such subsets are sometimes helpful when
2205 investigating whether a potential counterexample is genuine or spurious, but
2206 their potential for clutter is real.
2208 \opfalse{show\_consts}{hide\_consts}
2209 Specifies whether the values of constants occurring in the formula (including
2210 its axioms) should be displayed along with any counterexample. These values are
2211 sometimes helpful when investigating why a counterexample is
2212 genuine, but they can clutter the output.
2214 \opnodefault{show\_all}{bool}
2215 Abbreviation for \textit{show\_datatypes} and \textit{show\_consts}.
2217 \opdefault{max\_potential}{int}{$\mathbf{1}$}
2218 Specifies the maximum number of potential counterexamples to display. Setting
2219 this option to 0 speeds up the search for a genuine counterexample. This option
2220 is implicitly set to 0 for automatic runs. If you set this option to a value
2221 greater than 1, you will need an incremental SAT solver, such as
2222 \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that many of
2223 the counterexamples may be identical.
2226 {\small See also \textit{check\_potential} (\S\ref{authentication}) and
2227 \textit{sat\_solver} (\S\ref{optimizations}).}
2229 \opdefault{max\_genuine}{int}{$\mathbf{1}$}
2230 Specifies the maximum number of genuine counterexamples to display. If you set
2231 this option to a value greater than 1, you will need an incremental SAT solver,
2232 such as \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that
2233 many of the counterexamples may be identical.
2236 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and
2237 \textit{sat\_solver} (\S\ref{optimizations}).}
2239 \opnodefault{eval}{term\_list}
2240 Specifies the list of terms whose values should be displayed along with
2241 counterexamples. This option suffers from an ``observer effect'': Nitpick might
2242 find different counterexamples for different values of this option.
2244 \oparg{atoms}{type}{string\_list}
2245 Specifies the names to use to refer to the atoms of the given type. By default,
2246 Nitpick generates names of the form $a_1, \ldots, a_n$, where $a$ is the first
2247 letter of the type's name.
2249 \opnodefault{atoms}{string\_list}
2250 Specifies the default names to use to refer to atoms of any type. For example,
2251 to call the three atoms of type ${'}a$ \textit{ichi}, \textit{ni}, and
2252 \textit{san} instead of $a_1$, $a_2$, $a_3$, specify the option
2253 ``\textit{atoms}~${'}a$ = \textit{ichi~ni~san}''. The default names can be
2254 overridden on a per-type basis using the \textit{atoms}~\qty{type} option
2257 \oparg{format}{term}{int\_seq}
2258 Specifies how to uncurry the value displayed for a variable or constant.
2259 Uncurrying sometimes increases the readability of the output for high-arity
2260 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
2261 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
2262 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
2263 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
2264 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
2265 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
2266 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
2267 arguments that are not accounted for are left alone, as if the specification had
2268 been $1,\ldots,1,n_1,\ldots,n_k$.
2270 \opdefault{format}{int\_seq}{$\mathbf{1}$}
2271 Specifies the default format to use. Irrespective of the default format, the
2272 extra arguments to a Skolem constant corresponding to the outer bound variables
2273 are kept separated from the remaining arguments, the \textbf{for} arguments of
2274 an inductive definitions are kept separated from the remaining arguments, and
2275 the iteration counter of an unrolled inductive definition is shown alone. The
2276 default format can be overridden on a per-variable or per-constant basis using
2277 the \textit{format}~\qty{term} option described above.
2280 \subsection{Authentication}
2281 \label{authentication}
2284 \opfalse{check\_potential}{trust\_potential}
2285 Specifies whether potential counterexamples should be given to Isabelle's
2286 \textit{auto} tactic to assess their validity. If a potential counterexample is
2287 shown to be genuine, Nitpick displays a message to this effect and terminates.
2290 {\small See also \textit{max\_potential} (\S\ref{output-format}).}
2292 \opfalse{check\_genuine}{trust\_genuine}
2293 Specifies whether genuine and quasi genuine counterexamples should be given to
2294 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
2295 counterexample is shown to be spurious, the user is kindly asked to send a bug
2296 report to the author at
2297 \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
2300 {\small See also \textit{max\_genuine} (\S\ref{output-format}).}
2302 \opnodefault{expect}{string}
2303 Specifies the expected outcome, which must be one of the following:
2306 \item[$\bullet$] \textbf{\textit{genuine}:} Nitpick found a genuine counterexample.
2307 \item[$\bullet$] \textbf{\textit{quasi\_genuine}:} Nitpick found a ``quasi
2308 genuine'' counterexample (i.e., a counterexample that is genuine unless
2309 it contradicts a missing axiom or a dangerous option was used inappropriately).
2310 \item[$\bullet$] \textbf{\textit{potential}:} Nitpick found a potential counterexample.
2311 \item[$\bullet$] \textbf{\textit{none}:} Nitpick found no counterexample.
2312 \item[$\bullet$] \textbf{\textit{unknown}:} Nitpick encountered some problem (e.g.,
2313 Kodkod ran out of memory).
2316 Nitpick emits an error if the actual outcome differs from the expected outcome.
2317 This option is useful for regression testing.
2320 \subsection{Optimizations}
2321 \label{optimizations}
2323 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
2328 \opdefault{sat\_solver}{string}{smart}
2329 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
2330 to be faster than their Java counterparts, but they can be more difficult to
2331 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
2332 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
2333 you will need an incremental SAT solver, such as \textit{MiniSat\_JNI}
2334 (recommended) or \textit{SAT4J}.
2336 The supported solvers are listed below:
2340 \item[$\bullet$] \textbf{\textit{MiniSat}:} MiniSat is an efficient solver
2341 written in \cpp{}. To use MiniSat, set the environment variable
2342 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
2344 \footnote{Important note for Cygwin users: The path must be specified using
2345 native Windows syntax. Make sure to escape backslashes properly.%
2346 \label{cygwin-paths}}
2347 The \cpp{} sources and executables for MiniSat are available at
2348 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
2349 and 2.0 beta (2007-07-21).
2351 \item[$\bullet$] \textbf{\textit{MiniSat\_JNI}:} The JNI (Java Native Interface)
2352 version of MiniSat is bundled with Kodkodi and is precompiled for the major
2353 platforms. It is also available from \texttt{native\-solver.\allowbreak tgz},
2354 which you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
2355 version of MiniSat, the JNI version can be used incrementally.
2357 \item[$\bullet$] \textbf{\textit{CryptoMiniSat}:} CryptoMiniSat is the winner of
2358 the 2010 SAT Race. To use CryptoMiniSat, set the environment variable
2359 \texttt{CRYPTO\-MINISAT\_}\discretionary{}{}{}\texttt{HOME} to the directory that contains the \texttt{crypto\-minisat}
2361 \footref{cygwin-paths}
2362 The \cpp{} sources and executables for Crypto\-Mini\-Sat are available at
2363 \url{http://planete.inrialpes.fr/~soos/}\allowbreak\url{CryptoMiniSat2/index.php}.
2364 Nitpick has been tested with version 2.51.
2366 \item[$\bullet$] \textbf{\textit{PicoSAT}:} PicoSAT is an efficient solver
2367 written in C. You can install a standard version of
2368 PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
2369 that contains the \texttt{picosat} executable.%
2370 \footref{cygwin-paths}
2371 The C sources for PicoSAT are
2372 available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
2373 Nitpick has been tested with version 913.
2375 \item[$\bullet$] \textbf{\textit{zChaff}:} zChaff is an efficient solver written
2376 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
2377 the directory that contains the \texttt{zchaff} executable.%
2378 \footref{cygwin-paths}
2379 The \cpp{} sources and executables for zChaff are available at
2380 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
2381 versions 2004-05-13, 2004-11-15, and 2007-03-12.
2383 \item[$\bullet$] \textbf{\textit{zChaff\_JNI}:} The JNI version of zChaff is
2384 bundled with Kodkodi and is precompiled for the major
2385 platforms. It is also available from \texttt{native\-solver.\allowbreak tgz},
2386 which you will find on Kodkod's web site \cite{kodkod-2009}.
2388 \item[$\bullet$] \textbf{\textit{RSat}:} RSat is an efficient solver written in
2389 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
2390 directory that contains the \texttt{rsat} executable.%
2391 \footref{cygwin-paths}
2392 The \cpp{} sources for RSat are available at
2393 \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been tested with version
2396 \item[$\bullet$] \textbf{\textit{BerkMin}:} BerkMin561 is an efficient solver
2397 written in C. To use BerkMin, set the environment variable
2398 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
2399 executable.\footref{cygwin-paths}
2400 The BerkMin executables are available at
2401 \url{http://eigold.tripod.com/BerkMin.html}.
2403 \item[$\bullet$] \textbf{\textit{BerkMin\_Alloy}:} Variant of BerkMin that is
2404 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
2405 version of BerkMin, set the environment variable
2406 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
2408 \footref{cygwin-paths}
2410 \item[$\bullet$] \textbf{\textit{Jerusat}:} Jerusat 1.3 is an efficient solver
2411 written in C. To use Jerusat, set the environment variable
2412 \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
2414 \footref{cygwin-paths}
2415 The C sources for Jerusat are available at
2416 \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
2418 \item[$\bullet$] \textbf{\textit{SAT4J}:} SAT4J is a reasonably efficient solver
2419 written in Java that can be used incrementally. It is bundled with Kodkodi and
2420 requires no further installation or configuration steps. Do not attempt to
2421 install the official SAT4J packages, because their API is incompatible with
2424 \item[$\bullet$] \textbf{\textit{SAT4J\_Light}:} Variant of SAT4J that is
2425 optimized for small problems. It can also be used incrementally.
2427 \item[$\bullet$] \textbf{\textit{smart}:} If \textit{sat\_solver} is set to
2428 \textit{smart}, Nitpick selects the first solver among the above that is
2429 recognized by Isabelle. If \textit{verbose} (\S\ref{output-format}) is enabled,
2430 Nitpick displays which SAT solver was chosen.
2434 \opdefault{batch\_size}{int\_or\_smart}{smart}
2435 Specifies the maximum number of Kodkod problems that should be lumped together
2436 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
2437 together ensures that Kodkodi is launched less often, but it makes the verbose
2438 output less readable and is sometimes detrimental to performance. If
2439 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
2440 \textit{debug} (\S\ref{output-format}) is set and 50 otherwise.
2442 \optrue{destroy\_constrs}{dont\_destroy\_constrs}
2443 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
2444 be rewritten to use (automatically generated) discriminators and destructors.
2445 This optimization can drastically reduce the size of the Boolean formulas given
2449 {\small See also \textit{debug} (\S\ref{output-format}).}
2451 \optrue{specialize}{dont\_specialize}
2452 Specifies whether functions invoked with static arguments should be specialized.
2453 This optimization can drastically reduce the search space, especially for
2454 higher-order functions.
2457 {\small See also \textit{debug} (\S\ref{output-format}) and
2458 \textit{show\_consts} (\S\ref{output-format}).}
2460 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
2461 Specifies whether Nitpick should use Kodkod's transitive closure operator to
2462 encode non-well-founded ``linear inductive predicates,'' i.e., inductive
2463 predicates for which each the predicate occurs in at most one assumption of each
2464 introduction rule. Using the reflexive transitive closure is in principle
2465 equivalent to setting \textit{iter} to the cardinality of the predicate's
2466 domain, but it is usually more efficient.
2468 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
2469 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
2471 {\small See also \textit{debug} (\S\ref{output-format}).}
2473 \opnodefault{whack}{term\_list}
2474 Specifies a list of atomic terms (usually constants, but also free and schematic
2475 variables) that should be taken as being $\unk$ (unknown). This can be useful to
2476 reduce the size of the Kodkod problem if you can guess in advance that a
2477 constant might not be needed to find a countermodel.
2479 {\small See also \textit{debug} (\S\ref{output-format}).}
2481 \optrue{peephole\_optim}{no\_peephole\_optim}
2482 Specifies whether Nitpick should simplify the generated Kodkod formulas using a
2483 peephole optimizer. These optimizations can make a significant difference.
2484 Unless you are tracking down a bug in Nitpick or distrust the peephole
2485 optimizer, you should leave this option enabled.
2487 \opdefault{datatype\_sym\_break}{int}{5}
2488 Specifies an upper bound on the number of datatypes for which Nitpick generates
2489 symmetry breaking predicates. Symmetry breaking can speed up the SAT solver
2490 considerably, especially for unsatisfiable problems, but too much of it can slow
2493 \opdefault{kodkod\_sym\_break}{int}{15}
2494 Specifies an upper bound on the number of relations for which Kodkod generates
2495 symmetry breaking predicates. Symmetry breaking can speed up the SAT solver
2496 considerably, especially for unsatisfiable problems, but too much of it can slow
2499 \opdefault{max\_threads}{int}{0}
2500 Specifies the maximum number of threads to use in Kodkod. If this option is set
2501 to 0, Kodkod will compute an appropriate value based on the number of processor
2502 cores available. The option is implicitly set to 1 for automatic runs.
2505 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
2506 \textit{timeout} (\S\ref{timeouts}).}
2509 \subsection{Timeouts}
2513 \opdefault{timeout}{time}{$\mathbf{30}$}
2514 Specifies the maximum number of seconds that the \textbf{nitpick} command should
2515 spend looking for a counterexample. Nitpick tries to honor this constraint as
2516 well as it can but offers no guarantees. For automatic runs,
2517 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
2518 a time slot whose length is specified by the ``Auto Counterexample Time
2519 Limit'' option in Proof General.
2522 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
2524 \opdefault{tac\_timeout}{time}{$\mathbf{0.5}$}
2525 Specifies the maximum number of seconds that the \textit{auto} tactic should use
2526 when checking a counterexample, and similarly that \textit{lexicographic\_order}
2527 and \textit{size\_change} should use when checking whether a (co)in\-duc\-tive
2528 predicate is well-founded. Nitpick tries to honor this constraint as well as it
2529 can but offers no guarantees.
2532 {\small See also \textit{wf} (\S\ref{scope-of-search}),
2533 \textit{check\_potential} (\S\ref{authentication}),
2534 and \textit{check\_genuine} (\S\ref{authentication}).}
2537 \section{Attribute Reference}
2538 \label{attribute-reference}
2540 Nitpick needs to consider the definitions of all constants occurring in a
2541 formula in order to falsify it. For constants introduced using the
2542 \textbf{definition} command, the definition is simply the associated
2543 \textit{\_def} axiom. In contrast, instead of using the internal representation
2544 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
2545 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural
2546 equational specification entered by the user.
2548 Behind the scenes, Isabelle's built-in packages and theories rely on the
2549 following attributes to affect Nitpick's behavior:
2552 \flushitem{\textit{nitpick\_def}}
2555 This attribute specifies an alternative definition of a constant. The
2556 alternative definition should be logically equivalent to the constant's actual
2557 axiomatic definition and should be of the form
2559 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
2561 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
2564 \flushitem{\textit{nitpick\_simp}}
2567 This attribute specifies the equations that constitute the specification of a
2568 constant. The \textbf{primrec}, \textbf{function}, and
2569 \textbf{nominal\_\allowbreak primrec} packages automatically attach this
2570 attribute to their \textit{simps} rules. The equations must be of the form
2572 \qquad $c~t_1~\ldots\ t_n \;\bigl[{=}\; u\bigr]$
2576 \qquad $c~t_1~\ldots\ t_n \,\equiv\, u.$
2578 \flushitem{\textit{nitpick\_psimp}}
2581 This attribute specifies the equations that constitute the partial specification
2582 of a constant. The \textbf{function} package automatically attaches this
2583 attribute to its \textit{psimps} rules. The conditional equations must be of the
2586 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \;\bigl[{=}\; u\bigr]$
2590 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,\equiv\, u$.
2592 \flushitem{\textit{nitpick\_choice\_spec}}
2595 This attribute specifies the (free-form) specification of a constant defined
2596 using the \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command.
2599 When faced with a constant, Nitpick proceeds as follows:
2602 \item[1.] If the \textit{nitpick\_simp} set associated with the constant
2603 is not empty, Nitpick uses these rules as the specification of the constant.
2605 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
2606 the constant is not empty, it uses these rules as the specification of the
2609 \item[3.] Otherwise, if the constant was defined using the
2610 \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command and the
2611 \textit{nitpick\_choice\_spec} set associated with the constant is not empty, it
2612 uses these theorems as the specification of the constant.
2614 \item[4.] Otherwise, it looks up the definition of the constant. If the
2615 \textit{nitpick\_def} set associated with the constant is not empty, it uses the
2616 latest rule added to the set as the definition of the constant; otherwise it
2617 uses the actual definition axiom.
2620 \item[1.] If the definition is of the form
2622 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$
2626 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{gfp}~(\lambda f.\; t).$
2628 Nitpick assumes that the definition was made using a (co)inductive package
2629 based on the user-specified introduction rules registered in Isabelle's internal
2630 \textit{Spec\_Rules} table. The tool uses the introduction rules to ascertain
2631 whether the definition is well-founded and the definition to generate a
2632 fixed-point equation or an unrolled equation.
2634 \item[2.] If the definition is compact enough, the constant is \textsl{unfolded}
2635 wherever it appears; otherwise, it is defined equationally, as with
2636 the \textit{nitpick\_simp} attribute.
2640 As an illustration, consider the inductive definition
2643 \textbf{inductive}~\textit{odd}~\textbf{where} \\
2644 ``\textit{odd}~1'' $\,\mid$ \\
2645 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
2648 By default, Nitpick uses the \textit{lfp}-based definition in conjunction with
2649 the introduction rules. To override this, you can specify an alternative
2650 definition as follows:
2653 \textbf{lemma} $\mathit{odd\_alt\_def}$ [\textit{nitpick\_def}]:\kern.4em ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
2656 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
2657 = 1$. Alternatively, you can specify an equational specification of the constant:
2660 \textbf{lemma} $\mathit{odd\_simp}$ [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
2663 Such tweaks should be done with great care, because Nitpick will assume that the
2664 constant is completely defined by its equational specification. For example, if
2665 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
2666 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
2667 (\S\ref{output-format}) option is extremely useful to understand what is going
2668 on when experimenting with \textit{nitpick\_} attributes.
2670 Because of its internal three-valued logic, Nitpick tends to lose a
2671 lot of precision in the presence of partially specified constants. For example,
2674 \textbf{lemma} \textit{odd\_simp} [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd~x} = \lnot\, \textit{even}~x$''
2680 \textbf{lemma} \textit{odd\_psimps} [\textit{nitpick\_simp}]: \\
2681 ``$\textit{even~x} \,\Longrightarrow\, \textit{odd~x} = \textit{False\/}$'' \\
2682 ``$\lnot\, \textit{even~x} \,\Longrightarrow\, \textit{odd~x} = \textit{True\/}$''
2685 Because Nitpick sometimes unfolds definitions but never simplification rules,
2686 you can ensure that a constant is defined explicitly using the
2687 \textit{nitpick\_simp}. For example:
2690 \textbf{definition}~\textit{optimum} \textbf{where} [\textit{nitpick\_simp}]: \\
2691 ``$\textit{optimum}~t =
2692 (\forall u.\; \textit{consistent}~u \mathrel{\land} \textit{alphabet}~t = \textit{alphabet}~u$ \\
2693 \phantom{``$\textit{optimum}~t = (\forall u.\;$}${\mathrel{\land}}\; \textit{freq}~t = \textit{freq}~u \longrightarrow
2694 \textit{cost}~t \le \textit{cost}~u)$''
2697 In some rare occasions, you might want to provide an inductive or coinductive
2698 view on top of an existing constant $c$. The easiest way to achieve this is to
2699 define a new constant $c'$ (co)inductively. Then prove that $c$ equals $c'$
2700 and let Nitpick know about it:
2703 \textbf{lemma} \textit{c\_alt\_def} [\textit{nitpick\_def}]:\kern.4em ``$c \equiv c'$\kern2pt ''
2706 This ensures that Nitpick will substitute $c'$ for $c$ and use the (co)inductive
2709 \section{Standard ML Interface}
2710 \label{standard-ml-interface}
2712 Nitpick provides a rich Standard ML interface used mainly for internal purposes
2713 and debugging. Among the most interesting functions exported by Nitpick are
2714 those that let you invoke the tool programmatically and those that let you
2715 register and unregister custom coinductive datatypes as well as term
2718 \subsection{Invocation of Nitpick}
2719 \label{invocation-of-nitpick}
2721 The \textit{Nitpick} structure offers the following functions for invoking your
2722 favorite counterexample generator:
2725 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
2726 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
2727 \hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
2728 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
2729 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
2732 The return value is a new proof state paired with an outcome string
2733 (``genuine'', ``quasi\_genuine'', ``potential'', ``none'', or ``unknown''). The
2734 \textit{params} type is a large record that lets you set Nitpick's options. The
2735 current default options can be retrieved by calling the following function
2736 defined in the \textit{Nitpick\_Isar} structure:
2739 $\textbf{val}\,~\textit{default\_params} :\,
2740 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
2743 The second argument lets you override option values before they are parsed and
2744 put into a \textit{params} record. Here is an example:
2747 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout\/}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
2748 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
2749 & \textit{state}~\textit{params}~\textit{false} \\[-2pt]
2750 & \textit{subgoal}\end{aligned}$
2755 \subsection{Registration of Coinductive Datatypes}
2756 \label{registration-of-coinductive-datatypes}
2758 If you have defined a custom coinductive datatype, you can tell Nitpick about
2759 it, so that it can use an efficient Kodkod axiomatization similar to the one it
2760 uses for lazy lists. The interface for registering and unregistering coinductive
2761 datatypes consists of the following pair of functions defined in the
2762 \textit{Nitpick\_HOL} structure:
2765 $\textbf{val}\,~\textit{register\_codatatype\/} : {}$ \\
2766 $\hbox{}\quad\textit{morphism} \rightarrow \textit{typ} \rightarrow \textit{string} \rightarrow (\textit{string} \times \textit{typ})\;\textit{list} \rightarrow \textit{Context.generic} {}$ \\
2767 $\hbox{}\quad{\rightarrow}\; \textit{Context.generic}$ \\
2768 $\textbf{val}\,~\textit{unregister\_codatatype\/} : {}$ \\
2769 $\hbox{}\quad\textit{morphism} \rightarrow \textit{typ} \rightarrow \textit{Context.generic} \rightarrow \textit{Context.generic} {}$
2772 The type $'a~\textit{llist}$ of lazy lists is already registered; had it
2773 not been, you could have told Nitpick about it by adding the following line
2774 to your theory file:
2777 $\textbf{declaration}~\,\{{*}$ \\
2778 $\hbox{}\quad\textit{Nitpick\_HOL.register\_codatatype}~@\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}$ \\
2779 $\hbox{}\qquad\quad @\{\antiq{const\_name}~ \textit{llist\_case}\}$ \\
2780 $\hbox{}\qquad\quad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])$ \\
2784 The \textit{register\_codatatype} function takes a coinductive datatype, its
2785 case function, and the list of its constructors (in addition to the current
2786 morphism and generic proof context). The case function must take its arguments
2787 in the order that the constructors are listed. If no case function with the
2788 correct signature is available, simply pass the empty string.
2790 On the other hand, if your goal is to cripple Nitpick, add the following line to
2791 your theory file and try to check a few conjectures about lazy lists:
2794 $\textbf{declaration}~\,\{{*}$ \\
2795 $\hbox{}\quad\textit{Nitpick\_HOL.unregister\_codatatype}~@\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}$ \\
2799 Inductive datatypes can be registered as coinductive datatypes, given
2800 appropriate coinductive constructors. However, doing so precludes
2801 the use of the inductive constructors---Nitpick will generate an error if they
2804 \subsection{Registration of Term Postprocessors}
2805 \label{registration-of-term-postprocessors}
2807 It is possible to change the output of any term that Nitpick considers a
2808 datatype by registering a term postprocessor. The interface for registering and
2809 unregistering postprocessors consists of the following pair of functions defined
2810 in the \textit{Nitpick\_Model} structure:
2813 $\textbf{type}\,~\textit{term\_postprocessor}\,~{=} {}$ \\
2814 $\hbox{}\quad\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \rightarrow \textit{term}$ \\
2815 $\textbf{val}\,~\textit{register\_term\_postprocessor} : {}$ \\
2816 $\hbox{}\quad\textit{typ} \rightarrow \textit{term\_postprocessor} \rightarrow \textit{morphism} \rightarrow \textit{Context.generic}$ \\
2817 $\hbox{}\quad{\rightarrow}\; \textit{Context.generic}$ \\
2818 $\textbf{val}\,~\textit{unregister\_term\_postprocessor} : {}$ \\
2819 $\hbox{}\quad\textit{typ} \rightarrow \textit{morphism} \rightarrow \textit{Context.generic} \rightarrow \textit{Context.generic}$
2822 \S\ref{typedefs-quotient-types-records-rationals-and-reals} and
2823 \texttt{src/HOL/Library/Multiset.thy} illustrate this feature in context.
2825 \section{Known Bugs and Limitations}
2826 \label{known-bugs-and-limitations}
2828 Here are the known bugs and limitations in Nitpick at the time of writing:
2831 \item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
2832 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
2833 Nitpick to generate spurious counterexamples for theorems that refer to values
2834 for which the function is not defined. For example:
2837 \textbf{primrec} \textit{prec} \textbf{where} \\
2838 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
2839 \textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
2840 \textbf{nitpick} \\[2\smallskipamount]
2841 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2:
2843 \\[2\smallskipamount]
2844 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
2845 \textbf{by}~(\textit{auto simp}:~\textit{prec\_def})
2848 Such theorems are generally considered bad style because they rely on the
2849 internal representation of functions synthesized by Isabelle, an implementation
2852 \item[$\bullet$] Similarly, Nitpick might find spurious counterexamples for
2853 theorems that rely on the use of the indefinite description operator internally
2854 by \textbf{specification} and \textbf{quot\_type}.
2856 \item[$\bullet$] Axioms or definitions that restrict the possible values of the
2857 \textit{undefined} constant or other partially specified built-in Isabelle
2858 constants (e.g., \textit{Abs\_} and \textit{Rep\_} constants) are in general
2859 ignored. Again, such nonconservative extensions are generally considered bad
2862 \item[$\bullet$] Nitpick maintains a global cache of wellfoundedness conditions,
2863 which can become invalid if you change the definition of an inductive predicate
2864 that is registered in the cache. To clear the cache,
2865 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
2868 \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
2869 \textbf{guess} command in a structured proof.
2871 \item[$\bullet$] The \textit{nitpick\_xxx} attributes and the
2872 \textit{Nitpick\_xxx.register\_yyy} functions can cause havoc if used
2875 \item[$\bullet$] Although this has never been observed, arbitrary theorem
2876 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
2878 \item[$\bullet$] All constants, types, free variables, and schematic variables
2879 whose names start with \textit{Nitpick}{.} are reserved for internal use.
2883 \bibliography{../manual}{}
2884 \bibliographystyle{abbrv}