doc-src/Nitpick/nitpick.tex
 author blanchet Tue, 27 Oct 2009 16:52:06 +0100 changeset 33556 cba22e2999d5 parent 33232 f93390060bbe child 33557 107f3df799f6 permissions -rw-r--r--
renamed Nitpick option "coalesce_type_vars" to "merge_type_vars" (shorter) and cleaned up old hacks that are no longer necessary

\documentclass[a4paper,12pt]{article}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[french,english]{babel}
\usepackage{color}
\usepackage{graphicx}
%\usepackage{mathpazo}
\usepackage{multicol}
\usepackage{stmaryrd}
%\usepackage[scaled=.85]{beramono}
\usepackage{../iman,../pdfsetup}

%\oddsidemargin=4.6mm
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\def\Colon{\mathord{:\mkern-1.5mu:}}
%\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}}
%\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}}
\def\lparr{\mathopen{(\mkern-4mu\mid}}
\def\rparr{\mathclose{\mid\mkern-4mu)}}

\def\undef{\textit{undefined}}
\def\unk{{?}}
%\def\unr{\textit{others}}
\def\unr{\ldots}
\def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
\def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}

\hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
counter-example counter-examples data-type data-types co-data-type
co-data-types in-duc-tive co-in-duc-tive}

\urlstyle{tt}

\begin{document}

\title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
Picking Nits \\[\smallskipamount]
\Large A User's Guide to Nitpick for Isabelle/HOL 2010}
\author{\hbox{} \\
Jasmin Christian Blanchette \\
{\normalsize Fakult\"at f\"ur Informatik, Technische Universit\"at M\"unchen} \\
\hbox{}}

\maketitle

\tableofcontents

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\def\post{\vskip0pt plus1ex\endgroup}

\def\postw{\post}

\section{Introduction}
\label{introduction}

Nitpick \cite{blanchette-nipkow-2009} is a counterexample generator for
Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
first-order relational model finder developed by the Software Design Group at
MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
borrows many ideas and code fragments, but it benefits from Kodkod's
optimizations and a new encoding scheme. The name Nitpick is shamelessly
appropriated from a now retired Alloy precursor.

Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
theorem and wait a few seconds. Nonetheless, there are situations where knowing
how it works under the hood and how it reacts to various options helps
increase the test coverage. This manual also explains how to install the tool on
your workstation. Should the motivation fail you, think of the many hours of
hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.

Another common use of Nitpick is to find out whether the axioms of a locale are
satisfiable, while the locale is being developed. To check this, it suffices to
write

\prew
\textbf{lemma}~$\textit{False}$'' \\
\textbf{nitpick}~[\textit{show\_all}]
\postw

after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
must find a model for the axioms. If it finds no model, we have an indication
that the axioms might be unsatisfiable.

Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
machine called \texttt{java}. The examples presented in this manual can be found
in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.

\newbox\boxA
\setbox\boxA=\hbox{\texttt{nospam}}

The known bugs and limitations at the time of writing are listed in
\S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
or this manual should be directed to
\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
in.\allowbreak tum.\allowbreak de}.

\vskip2.5\smallskipamount

\textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
suggesting several textual improvements.
% and Perry James for reporting a typo.

\section{First Steps}
\label{first-steps}

This section introduces Nitpick by presenting small examples. If possible, you
should try out the examples on your workstation. Your theory file should start
the standard way:

\prew
\textbf{theory}~\textit{Scratch} \\
\textbf{imports}~\textit{Main} \\
\textbf{begin}
\postw

The results presented here were obtained using the JNI version of MiniSat and
the line

\prew
\postw

after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
be installed, as explained in \S\ref{optimizations}. If you have already
configured SAT solvers in Isabelle (e.g., for Refute), these will also be
available to Nitpick.

Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
Nitpick also provides an automatic mode that can be enabled by specifying

\prew
\textbf{nitpick\_params} [\textit{auto}]
\postw

at the beginning of the theory file. In this mode, Nitpick is run for up to 5
seconds (by default) on every newly entered theorem, much like Auto Quickcheck.

\subsection{Propositional Logic}
\label{propositional-logic}

\prew
\textbf{lemma}~$P \longleftrightarrow Q$'' \\
\textbf{nitpick}
\postw

You should get the following output:

\prew
\slshape
Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $P = \textit{True}$ \\
\hbox{}\qquad\qquad $Q = \textit{False}$
\postw

Nitpick can also be invoked on individual subgoals, as in the example below:

\prew
\textbf{apply}~\textit{auto} \\[2\smallskipamount]
{\slshape goal (2 subgoals): \\
\ 1. $P\,\Longrightarrow\, Q$ \\
\ 2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
\textbf{nitpick}~1 \\[2\smallskipamount]
{\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $P = \textit{True}$ \\
\hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
\textbf{nitpick}~2 \\[2\smallskipamount]
{\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $P = \textit{False}$ \\
\hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
\textbf{oops}
\postw

\subsection{Type Variables}
\label{type-variables}

If you are left unimpressed by the previous example, don't worry. The next
one is more mind- and computer-boggling:

\prew
\textbf{lemma} $P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
\postw
\pagebreak[2] %% TYPESETTING

The putative lemma involves the definite description operator, {THE}, presented
in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
lemma is merely asserting the indefinite description operator axiom with {THE}
substituted for {SOME}.

The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
containing type variables, Nitpick enumerates the possible domains for each type
variable, up to a given cardinality (8 by default), looking for a finite
countermodel:

\prew
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
\slshape
Trying 8 scopes: \nopagebreak \\
\hbox{}\qquad \textit{card}~$'a$~= 1; \\
\hbox{}\qquad \textit{card}~$'a$~= 2; \\
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
\hbox{}\qquad \textit{card}~$'a$~= 8. \\[2\smallskipamount]
Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
\hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
Total time: 580 ms.
\postw

Nitpick found a counterexample in which $'a$ has cardinality 3. (For
cardinalities 1 and 2, the formula holds.) In the counterexample, the three
values of type $'a$ are written $a_1$, $a_2$, and $a_3$.

The message Trying $n$ scopes: {\ldots}''\ is shown only if the option
\textit{verbose} is enabled. You can specify \textit{verbose} each time you
invoke \textbf{nitpick}, or you can set it globally using the command

\prew
\textbf{nitpick\_params} [\textit{verbose}]
\postw

This command also displays the current default values for all of the options
supported by Nitpick. The options are listed in \S\ref{option-reference}.

\subsection{Constants}
\label{constants}

By just looking at Nitpick's output, it might not be clear why the
counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
this time telling it to show the values of the constants that occur in the
formula:

\prew
\textbf{lemma}~$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
\textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
\slshape
Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
\hbox{}\qquad\qquad $x = a_3$ \\
\hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
\postw

We can see more clearly now. Since the predicate $P$ isn't true for a unique
value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
$a_1$. Since $P~a_1$ is false, the entire formula is falsified.

As an optimization, Nitpick's preprocessor introduced the special constant
\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
$\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
satisfying $P~y$. We disable this optimization by passing the
\textit{full\_descrs} option:

\prew
\textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
\slshape
Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
\hbox{}\qquad\qquad $x = a_3$ \\
\hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
\postw

As the result of another optimization, Nitpick directly assigned a value to the
subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
disable this second optimization by using the command

\prew
\textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
\textit{show\_consts}]
\postw

we finally get \textit{The}:

\prew
\slshape Constant: \nopagebreak \\
\hbox{}\qquad \mathit{The} = \undef{} (\!\begin{aligned}[t]% & \{\} := a_3,\> \{a_3\} := a_3,\> \{a_2\} := a_2, \\[-2pt] %% TYPESETTING & \{a_2, a_3\} := a_1,\> \{a_1\} := a_1,\> \{a_1, a_3\} := a_3, \\[-2pt] & \{a_1, a_2\} := a_3,\> \{a_1, a_2, a_3\} := a_3)\end{aligned}
\postw

Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
just like before.\footnote{The \undef{} symbol's presence is explained as
follows: In higher-order logic, any function can be built from the undefined
function using repeated applications of the function update operator $f(x := y)$, just like any list can be built from the empty list using $x \mathbin{\#} xs$.}

Our misadventures with THE suggest adding $\exists!x{.}$' (there exists a
unique $x$ such that'') at the front of our putative lemma's assumption:

\prew
\textbf{lemma}~$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
\postw

The fix appears to work:

\prew
\textbf{nitpick} \\[2\smallskipamount]
\slshape Nitpick found no counterexample.
\postw

We can further increase our confidence in the formula by exhausting all
cardinalities up to 50:

\prew
\textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol --'
can be entered as \texttt{-} (hyphen) or
\texttt{\char\\\char\<midarrow\char\>}.} \\[2\smallskipamount]
\slshape Nitpick found no counterexample.
\postw

Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:

\prew
\textbf{sledgehammer} \\[2\smallskipamount]
{\slshape Sledgehammer: external prover $e$'' for subgoal 1: \\
$\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
\textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
{\slshape No subgoals!}% \\[2\smallskipamount]
%\textbf{done}
\postw

This must be our lucky day.

\subsection{Skolemization}
\label{skolemization}

Are all invertible functions onto? Let's find out:

\prew
\textbf{lemma} $\exists g.\; \forall x.~g~(f~x) = x \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
\hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
\hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
\hbox{}\qquad\qquad $y = a_2$
\postw

Although $f$ is the only free variable occurring in the formula, Nitpick also
displays values for the bound variables $g$ and $y$. These values are available
to Nitpick because it performs skolemization as a preprocessing step.

In the previous example, skolemization only affected the outermost quantifiers.
This is not always the case, as illustrated below:

\prew
\textbf{lemma} $\exists x.\; \forall f.\; f~x = x$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
\hbox{}\qquad\qquad \lambda x.\; f = \undef{}(\!\begin{aligned}[t] & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt] & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}
\postw

The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
$x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.

The source of the Skolem constants is sometimes more obscure:

\prew
\textbf{lemma} $\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
\hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
\hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
\hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
\postw

What happened here is that Nitpick expanded the \textit{sym} constant to its
definition:

\prew
$\mathit{sym}~r \,\equiv\, \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
\postw

As their names suggest, the Skolem constants $\mathit{sym}.x$ and
$\mathit{sym}.y$ are simply the bound variables $x$ and $y$
from \textit{sym}'s definition.

Although skolemization is a useful optimization, you can disable it by invoking
Nitpick with \textit{dont\_skolemize}. See \S\ref{optimizations} for details.

\subsection{Natural Numbers and Integers}
\label{natural-numbers-and-integers}

Because of the axiom of infinity, the type \textit{nat} does not admit any
finite models. To deal with this, Nitpick considers prefixes $\{0,\, 1,\, \ldots,\, K - 1\}$ of \textit{nat} (where $K = \textit{card}~\textit{nat}$) and
maps all other numbers to the undefined value ($\unk$). The type \textit{int} is
handled in a similar way: If $K = \textit{card}~\textit{int}$, the subset of
\textit{int} known to Nitpick is $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}$. Undefined values lead to a three-valued logic.

Here is an example involving \textit{int}:

\prew
\textbf{lemma} $\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $i = 0$ \\
\hbox{}\qquad\qquad $j = 1$ \\
\hbox{}\qquad\qquad $m = 1$ \\
\hbox{}\qquad\qquad $n = 0$
\postw

With infinite types, we don't always have the luxury of a genuine counterexample
and must often content ourselves with a potential one. The tedious task of
finding out whether the potential counterexample is in fact genuine can be
outsourced to \textit{auto} by passing the option \textit{check\_potential}. For
example:

\prew
\textbf{lemma} $\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
\textbf{nitpick} [\textit{card~nat}~= 100,\, \textit{check\_potential}] \\[2\smallskipamount]
\slshape Nitpick found a potential counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
Confirmation by \textit{auto}'': The above counterexample is genuine.
\postw

You might wonder why the counterexample is first reported as potential. The root
of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
\textit{False}; but otherwise, it does not know anything about values of $n \ge \textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
\textit{True}. Since the assumption can never be satisfied, the putative lemma
can never be falsified.

Incidentally, if you distrust the so-called genuine counterexamples, you can
enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
aware that \textit{auto} will often fail to prove that the counterexample is
genuine or spurious.

Some conjectures involving elementary number theory make Nitpick look like a
giant with feet of clay:

\prew
\textbf{lemma} $P~\textit{Suc}$'' \\
\textbf{nitpick} [\textit{card} = 1--6] \\[2\smallskipamount]
\slshape
Nitpick found no counterexample.
\postw

For any cardinality $k$, \textit{Suc} is the partial function $\{0 \mapsto 1,\, 1 \mapsto 2,\, \ldots,\, k - 1 \mapsto \unk\}$, which evaluates to $\unk$ when
it is passed as argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$.
The next example is similar:

\prew
\textbf{lemma} $P~(\textit{op}~{+}\Colon \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
\textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
{\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
\hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
\textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
{\slshape Nitpick found no counterexample.}
\postw

The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
$\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0, 1\}$.

Because numbers are infinite and are approximated using a three-valued logic,
there is usually no need to systematically enumerate domain sizes. If Nitpick
cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
example above is an exception to this principle.) Nitpick nonetheless enumerates
all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
cardinalities are fast to handle and give rise to simpler counterexamples. This
is explained in more detail in \S\ref{scope-monotonicity}.

\subsection{Inductive Datatypes}
\label{inductive-datatypes}

Like natural numbers and integers, inductive datatypes with recursive
constructors admit no finite models and must be approximated by a subterm-closed
subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
Nitpick looks for all counterexamples that can be built using at most 10
different lists.

Let's see with an example involving \textit{hd} (which returns the first element
of a list) and $@$ (which concatenates two lists):

\prew
\textbf{lemma} $\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
\hbox{}\qquad\qquad $\textit{xs} = []$ \\
\hbox{}\qquad\qquad $\textit{y} = a_3$
\postw

To see why the counterexample is genuine, we enable \textit{show\_consts}
and \textit{show\_\allowbreak datatypes}:

\prew
{\slshape Datatype:} \\
\hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_3, a_3],\, [a_3],\, \unr\}$ \\
{\slshape Constants:} \\
\hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_3, a_3],\> [a_3, a_3] := \unk,\> [a_3] := \unk)$ \\
\hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_3, a_3] := a_3,\> [a_3] := a_3)$
\postw

Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
including $a_2$.

The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
append operator whose second argument is fixed to be $[y, y]$. Appending $[a_3, a_3]$ to $[a_3]$ would normally give $[a_3, a_3, a_3]$, but this value is not
representable in the subset of $'a$~\textit{list} considered by Nitpick, which
is shown under the Datatype'' heading; hence the result is $\unk$. Similarly,
appending $[a_3, a_3]$ to itself gives $\unk$.

Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
considers the following subsets:

\kern-.5\smallskipamount %% TYPESETTING

\prew
\begin{multicols}{3}
$\{[],\, [a_1],\, [a_2]\}$; \\
$\{[],\, [a_1],\, [a_3]\}$; \\
$\{[],\, [a_2],\, [a_3]\}$; \\
$\{[],\, [a_1],\, [a_1, a_1]\}$; \\
$\{[],\, [a_1],\, [a_2, a_1]\}$; \\
$\{[],\, [a_1],\, [a_3, a_1]\}$; \\
$\{[],\, [a_2],\, [a_1, a_2]\}$; \\
$\{[],\, [a_2],\, [a_2, a_2]\}$; \\
$\{[],\, [a_2],\, [a_3, a_2]\}$; \\
$\{[],\, [a_3],\, [a_1, a_3]\}$; \\
$\{[],\, [a_3],\, [a_2, a_3]\}$; \\
$\{[],\, [a_3],\, [a_3, a_3]\}$.
\end{multicols}
\postw

\kern-2\smallskipamount %% TYPESETTING

All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
are listed and only those. As an example of a non-subterm-closed subset,
consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_3]\}$, and observe
that $[a_1, a_3]$ (i.e., $a_1 \mathbin{\#} [a_3]$) has $[a_3] \notin \mathcal{S}$ as a subterm.

Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:

\prew
\textbf{lemma} $\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
\\
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
\hbox{}\qquad\qquad $\textit{xs} = [a_2]$ \\
\hbox{}\qquad\qquad $\textit{ys} = [a_3]$ \\
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
\hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_3],\, [a_2],\, \unr\}$
\postw

Because datatypes are approximated using a three-valued logic, there is usually
no need to systematically enumerate cardinalities: If Nitpick cannot find a
genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
unlikely that one could be found for smaller cardinalities.

\subsection{Typedefs, Records, Rationals, and Reals}
\label{typedefs-records-rationals-and-reals}

Nitpick generally treats types declared using \textbf{typedef} as datatypes
whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
For example:

\prew
\textbf{typedef}~\textit{three} = $\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
\textbf{by}~\textit{blast} \\[2\smallskipamount]
\textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} \kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
\textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} $B \,\equiv\, \textit{Abs\_three}~1$'' \\
\textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} $C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
\textbf{lemma} $\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $P = \{\Abs{1},\, \Abs{0}\}$ \\
\hbox{}\qquad\qquad $x = \Abs{2}$ \\
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
\hbox{}\qquad\qquad $\textit{three} = \{\Abs{2},\, \Abs{1},\, \Abs{0},\, \unr\}$
\postw

%% MARK
In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.

%% MARK
Records, which are implemented as \textbf{typedef}s behind the scenes, are
handled in much the same way:

\prew
\textbf{record} \textit{point} = \\
\hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
\hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
\textbf{lemma} $\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
\hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
\hbox{}\qquad\qquad $\textit{point} = \{\lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\> \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr,\, \unr\}$\kern-1pt %% QUIET
\postw

Finally, Nitpick provides rudimentary support for rationals and reals using a
similar approach:

\prew
\textbf{lemma} $4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $x = 1/2$ \\
\hbox{}\qquad\qquad $y = -1/2$ \\
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
\hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
\postw

\subsection{Inductive and Coinductive Predicates}
\label{inductive-and-coinductive-predicates}

Inductively defined predicates (and sets) are particularly problematic for
counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
the problem is that they are defined using a least fixed point construction.

Nitpick's philosophy is that not all inductive predicates are equal. Consider
the \textit{even} predicate below:

\prew
\textbf{inductive}~\textit{even}~\textbf{where} \\
\textit{even}~0'' $\,\mid$ \\
\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
\postw

This predicate enjoys the desirable property of being well-founded, which means
that the introduction rules don't give rise to infinite chains of the form

\prew
$\cdots\,\Longrightarrow\, \textit{even}~k'' \,\Longrightarrow\, \textit{even}~k' \,\Longrightarrow\, \textit{even}~k.$
\postw

For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
$k/2 + 1$:

\prew
$\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots \,\Longrightarrow\, \textit{even}~(k - 2) \,\Longrightarrow\, \textit{even}~k.$
\postw

Wellfoundedness is desirable because it enables Nitpick to use a very efficient
fixed point computation.%
\footnote{If an inductive predicate is
well-founded, then it has exactly one fixed point, which is simultaneously the
least and the greatest fixed point. In these circumstances, the computation of
the least fixed point amounts to the computation of an arbitrary fixed point,
which can be performed using a straightforward recursive equation.}
Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
just as Isabelle's \textbf{function} package usually discharges termination
proof obligations automatically.

Let's try an example:

\prew
\textbf{lemma} $\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
\textbf{nitpick}~[\textit{card nat}~= 100,\, \textit{verbose}] \\[2\smallskipamount]
\slshape The inductive predicate \textit{even}'' was proved well-founded.
Nitpick can compute it efficiently. \\[2\smallskipamount]
Trying 1 scope: \\
Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
Nitpick could not find a better counterexample. \\[2\smallskipamount]
Total time: 2274 ms.
\postw

No genuine counterexample is possible because Nitpick cannot rule out the
existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
$\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
existential quantifier:

\prew
\textbf{lemma} $\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
\textbf{nitpick}~[\textit{card nat}~= 100] \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\postw

So far we were blessed by the wellfoundedness of \textit{even}. What happens if
we use the following definition instead?

\prew
\textbf{inductive} $\textit{even}'$ \textbf{where} \\
$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
$\textit{even}'~2$'' $\,\mid$ \\
$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
\postw

This definition is not well-founded: From $\textit{even}'~0$ and
$\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
predicates $\textit{even}$ and $\textit{even}'$ are equivalent.

Let's check a property involving $\textit{even}'$. To make up for the
foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
\textit{nat}'s cardinality to a mere 10:

\prew
\textbf{lemma}~$\exists n \in \{0, 2, 4, 6, 8\}.\; \lnot\;\textit{even}'~n$'' \\
\textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
\slshape
The inductive predicate $\textit{even}'\!$'' could not be proved well-founded.
Nitpick might need to unroll it. \\[2\smallskipamount]
Trying 6 scopes: \\
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = \undef(\!\begin{aligned}[t] & 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt] & 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt] & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned} \\[2\smallskipamount]
Total time: 1140 ms.
\postw

Nitpick's output is very instructive. First, it tells us that the predicate is
unrolled, meaning that it is computed iteratively from the empty set. Then it
lists six scopes specifying different bounds on the numbers of iterations:\ 0,
1, 2, 4, 8, and~9.

The output also shows how each iteration contributes to $\textit{even}'$. The
notation $\lambda i.\; \textit{even}'$ indicates that the value of the
predicate depends on an iteration counter. Iteration 0 provides the basis
elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
iterations would not contribute any new elements.

Some values are marked with superscripted question
marks~(\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
\textit{True} or $\unk$, never \textit{False}.

When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
iterations. However, these numbers are bounded by the cardinality of the
predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
ever needed to compute the value of a \textit{nat} predicate. You can specify
the number of iterations using the \textit{iter} option, as explained in
\S\ref{scope-of-search}.

In the next formula, $\textit{even}'$ occurs both positively and negatively:

\prew
\textbf{lemma} $\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
\textbf{nitpick} [\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $n = 1$ \\
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = \undef(\!\begin{aligned}[t] & 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}  \\
\hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
\postw

Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\, 8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
fixed point (not necessarily the least one). It is used to falsify
$\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
$\textit{even}'~(n - 2)$.

Coinductive predicates are handled dually. For example:

\prew
\textbf{coinductive} \textit{nats} \textbf{where} \\
$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
\textbf{lemma} $\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
\slshape Nitpick found a counterexample:
\\[2\smallskipamount]
\hbox{}\qquad\qquad \lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t] & 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt] & \unr\})\end{aligned} \\
\hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
\postw

As a special case, Nitpick uses Kodkod's transitive closure operator to encode
negative occurrences of non-well-founded linear inductive predicates,'' i.e.,
inductive predicates for which each the predicate occurs in at most one
assumption of each introduction rule. For example:

\prew
\textbf{inductive} \textit{odd} \textbf{where} \\
$\textit{odd}~1$'' $\,\mid$ \\
$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
\textbf{lemma}~$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
\slshape Nitpick found a counterexample:
\\[2\smallskipamount]
\hbox{}\qquad\qquad $n = 1$ \\
\hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
\hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
\hbox{}\qquad\qquad \textit{odd}_{\textsl{step}} = \! \!\begin{aligned}[t] & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt] & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3), (3, 5), \\[-2pt] & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt] & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned} \\
\hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
\postw

\noindent
In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
$\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
elements from known ones. The set $\textit{odd}$ consists of all the values
reachable through the reflexive transitive closure of
$\textit{odd}_{\textrm{step}}$ starting with any element from
$\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
transitive closure to encode linear predicates is normally either more thorough
or more efficient than unrolling (depending on the value of \textit{iter}), but
for those cases where it isn't you can disable it by passing the
\textit{dont\_star\_linear\_preds} option.

\subsection{Coinductive Datatypes}
\label{coinductive-datatypes}

While Isabelle regrettably lacks a high-level mechanism for defining coinductive
datatypes, the \textit{Coinductive\_List} theory provides a coinductive lazy
list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
these lazy lists seamlessly and provides a hook, described in
\S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
datatypes.

(Co)intuitively, a coinductive datatype is similar to an inductive datatype but
allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a, \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0, 1, 2, 3, \ldots]$ can be defined as lazy lists using the
$\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
$\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist} \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.

Although it is otherwise no friend of infinity, Nitpick can find counterexamples
involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
finite lists:

\prew
\textbf{lemma} $\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
\hbox{}\qquad\qquad $\textit{a} = a_1$ \\
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
\postw

The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
infinite list $[a_1, a_1, a_1, \ldots]$.

The next example is more interesting:

\prew
\textbf{lemma}~$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
\slshape The type \kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
some scopes. \\[2\smallskipamount]
Trying 8 scopes: \\
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} \kern1pt$'a~\textit{list}$''~= 1,
and \textit{bisim\_depth}~= 0. \\
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} \kern1pt$'a~\textit{list}$''~= 8,
and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
\textit{card}~\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
depth}~= 1:
\\[2\smallskipamount]
\hbox{}\qquad\qquad $\textit{a} = a_2$ \\
\hbox{}\qquad\qquad $\textit{b} = a_1$ \\
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
\hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_1~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega)$ \\[2\smallskipamount]
Total time: 726 ms.
\postw

The lazy list $\textit{xs}$ is simply $[a_2, a_2, a_2, \ldots]$, whereas
$\textit{ys}$ is $[a_1, a_2, a_2, a_2, \ldots]$, i.e., a lasso-shaped list with
$[a_1]$ as its stem and $[a_2]$ as its cycle. In general, the list segment
within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
the segment leading to the binder is the stem.

A salient property of coinductive datatypes is that two objects are considered
equal if and only if they lead to the same observations. For example, the lazy
lists $\textrm{THE}~\omega.\; \omega = \textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
$\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega = \textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
concept of equality for coinductive datatypes is called bisimulation and is
defined coinductively.

Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
the Kodkod problem to ensure that distinct objects lead to different
observations. This precaution is somewhat expensive and often unnecessary, so it
can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
bisimilarity check is then performed \textsl{after} the counterexample has been
found to ensure correctness. If this after-the-fact check fails, the
counterexample is tagged as likely genuine'' and Nitpick recommends to try
again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
check for the previous example saves approximately 150~milli\-seconds; the speed
gains can be more significant for larger scopes.

The next formula illustrates the need for bisimilarity (either as a Kodkod
predicate or as an after-the-fact check) to prevent spurious counterexamples:

\prew
\textbf{lemma} $\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
\textbf{nitpick} [\textit{bisim\_depth} = $-1$,\, \textit{show\_datatypes}] \\[2\smallskipamount]
\slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
\hbox{}\qquad\qquad $a = a_2$ \\
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
\hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
\hbox{}\qquad\qquad 'a~\textit{llist} = \{\!\begin{aligned}[t] & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega, \\[-2pt] & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega,\> \unr\}\end{aligned}
\\[2\smallskipamount]
Try again with \textit{bisim\_depth}'' set to a nonnegative value to confirm
that the counterexample is genuine. \\[2\smallskipamount]
{\upshape\textbf{nitpick}} \\[2\smallskipamount]
\slshape Nitpick found no counterexample.
\postw

In the first \textbf{nitpick} invocation, the after-the-fact check discovered
that the two known elements of type $'a~\textit{llist}$ are bisimilar.

A compromise between leaving out the bisimilarity predicate from the Kodkod
problem and performing the after-the-fact check is to specify a lower
nonnegative \textit{bisim\_depth} value than the default one provided by
Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
be distinguished from each other by their prefixes of length $K$. Be aware that
setting $K$ to a too low value can overconstrain Nitpick, preventing it from
finding any counterexamples.

\subsection{Boxing}
\label{boxing}

Nitpick normally maps function and product types directly to the corresponding
Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
off to treat these types in the same way as plain datatypes, by approximating
them by a subset of a given cardinality. This technique is called boxing'' and
is particularly useful for functions passed as arguments to other functions, for
high-arity functions, and for large tuples. Under the hood, boxing involves
wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
isomorphic datatypes, as can be seen by enabling the \textit{debug} option.

To illustrate boxing, we consider a formalization of $\lambda$-terms represented
using de Bruijn's notation:

\prew
\textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
\postw

The $\textit{lift}~t~k$ function increments all variables with indices greater
than or equal to $k$ by one:

\prew
\textbf{primrec} \textit{lift} \textbf{where} \\
$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
\postw

The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
term $t$ has a loose variable with index $k$ or more:

\prew
\textbf{primrec}~\textit{loose} \textbf{where} \\
$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
\postw

Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
on $t$:

\prew
\textbf{primrec}~\textit{subst} \textbf{where} \\
$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
\phantom{}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
\postw

A substitution is a function that maps variable indices to terms. Observe that
$\sigma$ is a function passed as argument and that Nitpick can't optimize it
away, because the recursive call for the \textit{Lam} case involves an altered
version. Also notice the \textit{lift} call, which increments the variable
indices when moving under a \textit{Lam}.

A reasonable property to expect of substitution is that it should leave closed
terms unchanged. Alas, even this simple property does not hold:

\pre
\textbf{lemma}~$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
\slshape
Trying 8 scopes: \nopagebreak \\
\hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} $\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
\hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} $\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
\hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} $\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
and \textit{card}~$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
\hbox{}\qquad\qquad \sigma = \undef(\!\begin{aligned}[t] & 0 := \textit{Var}~0,\> 1 := \textit{Var}~0,\> 2 := \textit{Var}~0, \\[-2pt] & 3 := \textit{Var}~0,\> 4 := \textit{Var}~0,\> 5 := \textit{Var}~0)\end{aligned} \\
\hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
Total time: $4679$ ms.
\postw

Using \textit{eval}, we find out that $\textit{subst}~\sigma~t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
$\lambda$-term notation, $t$~is
$\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
replaced with $\textit{lift}~(\sigma~m)~0$.

An interesting aspect of Nitpick's verbose output is that it assigned inceasing
cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
For the formula of interest, knowing 6 values of that type was enough to find
the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
considered, a hopeless undertaking:

\prew
\textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
{\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
\postw

{\looseness=-1
Boxing can be enabled or disabled globally or on a per-type basis using the
\textit{box} option. Moreover, setting the cardinality of a function or
product type implicitly enables boxing for that type. Nitpick usually performs
reasonable choices about which types should be boxed, but option tweaking
sometimes helps.

}

\subsection{Scope Monotonicity}
\label{scope-monotonicity}

The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
and \textit{max}) controls which scopes are actually tested. In general, to
exhaust all models below a certain cardinality bound, the number of scopes that
Nitpick must consider increases exponentially with the number of type variables
(and \textbf{typedecl}'d types) occurring in the formula. Given the default
cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
considered for a formula involving $'a$, $'b$, $'c$, and $'d$.

Fortunately, many formulas exhibit a property called \textsl{scope
monotonicity}, meaning that if the formula is falsifiable for a given scope,
it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.

Consider the formula

\prew
\textbf{lemma}~$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
\postw

where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
$'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
that any counterexample found with a small scope would still be a counterexample
in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
by the larger scope. Nitpick comes to the same conclusion after a careful
inspection of the formula and the relevant definitions:

\prew
\textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
\slshape
The types \kern1pt$'a$'' and \kern1pt$'b$'' passed the monotonicity test.
Nitpick might be able to skip some scopes.
\\[2\smallskipamount]
Trying 8 scopes: \\
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
\textit{card} \textit{nat}~= 1, \textit{card} $('a \times {'}b)$
\textit{list}''~= 1, \\
\hbox{}\qquad\quad \textit{card} \kern1pt$'a$ \textit{list}''~= 1, and
\textit{card} \kern1pt$'b$ \textit{list}''~= 1. \\
\hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
\textit{card} \textit{nat}~= 2, \textit{card} $('a \times {'}b)$
\textit{list}''~= 2, \\
\hbox{}\qquad\quad \textit{card} \kern1pt$'a$ \textit{list}''~= 2, and
\textit{card} \kern1pt$'b$ \textit{list}''~= 2. \\
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
\textit{card} \textit{nat}~= 8, \textit{card} $('a \times {'}b)$
\textit{list}''~= 8, \\
\hbox{}\qquad\quad \textit{card} \kern1pt$'a$ \textit{list}''~= 8, and
\textit{card} \kern1pt$'b$ \textit{list}''~= 8.
\\[2\smallskipamount]
Nitpick found a counterexample for
\textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
\textit{card} \textit{nat}~= 5, \textit{card} $('a \times {'}b)$
\textit{list}''~= 5, \textit{card} \kern1pt$'a$ \textit{list}''~= 5, and
\textit{card} \kern1pt$'b$ \textit{list}''~= 5:
\\[2\smallskipamount]
\hbox{}\qquad\qquad $\textit{xs} = [a_4, a_5]$ \\
\hbox{}\qquad\qquad $\textit{ys} = [b_3, b_3]$ \\[2\smallskipamount]
Total time: 1636 ms.
\postw

In theory, it should be sufficient to test a single scope:

\prew
\textbf{nitpick}~[\textit{card}~= 8]
\postw

However, this is often less efficient in practice and may lead to overly complex
counterexamples.

If the monotonicity check fails but we believe that the formula is monotonic (or
we don't mind missing some counterexamples), we can pass the
\textit{mono} option. To convince yourself that this option is risky,
simply consider this example from \S\ref{skolemization}:

\prew
\textbf{lemma} $\exists g.\; \forall x\Colon 'b.~g~(f~x) = x \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
\textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
{\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
\textbf{nitpick} \\[2\smallskipamount]
\slshape
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
\hbox{}\qquad $\vdots$
\postw

(It turns out the formula holds if and only if $\textit{card}~'a \le \textit{card}~'b$.) Although this is rarely advisable, the automatic
monotonicity checks can be disabled by passing \textit{non\_mono}
(\S\ref{optimizations}).

As insinuated in \S\ref{natural-numbers-and-integers} and
\S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
are normally monotonic and treated as such. The same is true for record types,
\textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
\textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
consider only 8~scopes instead of $32\,768$.

\section{Case Studies}
\label{case-studies}

As a didactic device, the previous section focused mostly on toy formulas whose
validity can easily be assessed just by looking at the formula. We will now
review two somewhat more realistic case studies that are within Nitpick's
reach:\ a context-free grammar modeled by mutually inductive sets and a
functional implementation of AA trees. The results presented in this
section were produced with the following settings:

\prew
\textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
\postw

\subsection{A Context-Free Grammar}
\label{a-context-free-grammar}

Our first case study is taken from section 7.4 in the Isabelle tutorial
\cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
Ullman, produces all strings with an equal number of $a$'s and $b$'s:

\prew
\begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
$S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
$A$ & $::=$ & $aS \mid bAA$ \\
$B$ & $::=$ & $bS \mid aBB$
\end{tabular}
\postw

The intuition behind the grammar is that $A$ generates all string with one more
$a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.

The alphabet consists exclusively of $a$'s and $b$'s:

\prew
\textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
\postw

Strings over the alphabet are represented by \textit{alphabet list}s.
Nonterminals in the grammar become sets of strings. The production rules
presented above can be expressed as a mutually inductive definition:

\prew
\textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
\textit{R1}:\kern.4em $[] \in S$'' $\,\mid$ \\
\textit{R2}:\kern.4em $w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
\textit{R3}:\kern.4em $w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
\textit{R4}:\kern.4em $w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
\textit{R5}:\kern.4em $w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
\textit{R6}:\kern.4em $\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
\postw

The conversion of the grammar into the inductive definition was done manually by
Joe Blow, an underpaid undergraduate student. As a result, some errors might
have sneaked in.

Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
d'\^etre}. A good approach is to state desirable properties of the specification
(here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
as $b$'s) and check them with Nitpick. If the properties are correctly stated,
counterexamples will point to bugs in the specification. For our grammar
example, we will proceed in two steps, separating the soundness and the
completeness of the set $S$. First, soundness:

\prew
\textbf{theorem}~\textit{S\_sound}: \\
$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $w = [b]$
\postw

It would seem that $[b] \in S$. How could this be? An inspection of the
introduction rules reveals that the only rule with a right-hand side of the form
$b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
\textit{R5}:

\prew
$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
\postw

On closer inspection, we can see that this rule is wrong. To match the
production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
again:

\prew
\textbf{nitpick} \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $w = [a, a, b]$
\postw

Some detective work is necessary to find out what went wrong here. To get $[a, a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
from \textit{R6}:

\prew
$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
\postw

Now, this formula must be wrong: The same assumption occurs twice, and the
variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
the assumptions should have been a $w$.

With the correction made, we don't get any counterexample from Nitpick. Let's
move on and check completeness:

\prew
\textbf{theorem}~\textit{S\_complete}: \\
$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x\mathbin{\leftarrow} w.\; x = b] \longrightarrow w \in S$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $w = [b, b, a, a]$
\postw

Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
$a$'s and $b$'s. But since our inductive definition passed the soundness check,
the introduction rules we have are probably correct. Perhaps we simply lack an
introduction rule. Comparing the grammar with the inductive definition, our
suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
without which the grammar cannot generate two or more $b$'s in a row. So we add
the rule

\prew
$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
\postw

With this last change, we don't get any counterexamples from Nitpick for either
soundness or completeness. We can even generalize our result to cover $A$ and
$B$ as well:

\prew
\textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape Nitpick found no counterexample.
\postw

\subsection{AA Trees}
\label{aa-trees}

AA trees are a kind of balanced trees discovered by Arne Andersson that provide
similar performance to red-black trees, but with a simpler implementation
\cite{andersson-1993}. They can be used to store sets of elements equipped with
a total order $<$. We start by defining the datatype and some basic extractor
functions:

\prew
\textbf{datatype} $'a$~\textit{tree} = $\Lambda$ $\mid$ $N$ \kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} \kern1pt$'a$ \textit{tree}'' \kern1pt$'a$ \textit{tree}''  \\[2\smallskipamount]
\textbf{primrec} \textit{data} \textbf{where} \\
$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
\textbf{primrec} \textit{dataset} \textbf{where} \\
$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
\textbf{primrec} \textit{level} \textbf{where} \\
$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
\textbf{primrec} \textit{left} \textbf{where} \\
$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
\textbf{primrec} \textit{right} \textbf{where} \\
$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
$\textit{right}~(N~\_~\_~\_~u) = u$''
\postw

The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
as follows \cite{wikipedia-2009-aa-trees}:

\kern.2\parskip %% TYPESETTING

\pre
Each node has a level field, and the following invariants must remain true for
the tree to be valid:

\raggedright

\kern-.4\parskip %% TYPESETTING

\begin{enum}
\item[]
\begin{enum}
\item[1.] The level of a leaf node is one.
\item[2.] The level of a left child is strictly less than that of its parent.
\item[3.] The level of a right child is less than or equal to that of its parent.
\item[4.] The level of a right grandchild is strictly less than that of its grandparent.
\item[5.] Every node of level greater than one must have two children.
\end{enum}
\end{enum}
\post

\kern.4\parskip %% TYPESETTING

The \textit{wf} predicate formalizes this description:

\prew
\textbf{primrec} \textit{wf} \textbf{where} \\
$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
$\textit{wf}~(N~\_~k~t~u) =$ \\
\phantom{}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
\phantom{$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
\phantom{$($}$\textrm{else}$ \\
\hbox{}\phantom{$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k \mathrel{\land} \textit{level}~u \le k$ \\
\hbox{}\phantom{$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
\postw

Rebalancing the tree upon insertion and removal of elements is performed by two
auxiliary functions called \textit{skew} and \textit{split}, defined below:

\prew
\textbf{primrec} \textit{skew} \textbf{where} \\
$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
$\textit{skew}~(N~x~k~t~u) = {}$ \\
\phantom{}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k = \textit{level}~t~\textrm{then}$ \\
\phantom{(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~ (\textit{right}~t)~u)$ \\
\phantom{(}$\textrm{else}$ \\
\phantom{(\quad}$N~x~k~t~u)$''
\postw

\prew
\textbf{primrec} \textit{split} \textbf{where} \\
$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
$\textit{split}~(N~x~k~t~u) = {}$ \\
\phantom{}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k = \textit{level}~(\textit{right}~u)~\textrm{then}$ \\
\phantom{(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~ (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
\phantom{(}$\textrm{else}$ \\
\phantom{(\quad}$N~x~k~t~u)$''
\postw

Performing a \textit{skew} or a \textit{split} should have no impact on the set
of elements stored in the tree:

\prew
\textbf{theorem}~\textit{dataset\_skew\_split}:\\
$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
\textbf{nitpick} \\[2\smallskipamount]
{\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
\postw

Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
should not alter the tree:

\prew
\textbf{theorem}~\textit{wf\_skew\_split}:\\
$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
\textbf{nitpick} \\[2\smallskipamount]
{\slshape Nitpick found no counterexample.}
\postw

Insertion is implemented recursively. It preserves the sort order:

\prew
\textbf{primrec}~\textit{insort} \textbf{where} \\
$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
$\textit{insort}~(N~y~k~t~u)~x =$ \\
\phantom{}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
\phantom{$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
\postw

Notice that we deliberately commented out the application of \textit{skew} and
\textit{split}. Let's see if this causes any problems:

\prew
\textbf{theorem}~\textit{wf\_insort}:\kern.4em $\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
\textbf{nitpick} \\[2\smallskipamount]
\slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
\hbox{}\qquad\qquad $t = N~a_3~1~\Lambda~\Lambda$ \\
\hbox{}\qquad\qquad $x = a_4$ \\[2\smallskipamount]
Hint: Maybe you forgot a type constraint?
\postw

It's hard to see why this is a counterexample. The hint is of no help here. To
improve readability, we will restrict the theorem to \textit{nat}, so that we
don't need to look up the value of the $\textit{op}~{<}$ constant to find out
which element is smaller than the other. In addition, we will tell Nitpick to
display the value of $\textit{insort}~t~x$ using the \textit{eval} option. This
gives

\prew
\textbf{theorem} \textit{wf\_insort\_nat}:\kern.4em $\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
\textbf{nitpick} [\textit{eval} = $\textit{insort}~t~x$''] \\[2\smallskipamount]
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
\hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
\hbox{}\qquad\qquad $x = 0$ \\
\hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
\postw

Nitpick's output reveals that the element $0$ was added as a left child of $1$,
where both have a level of 1. This violates the second AA tree invariant, which
states that a left child's level must be less than its parent's. This shouldn't
come as a surprise, considering that we commented out the tree rebalancing code.
Reintroducing the code seems to solve the problem:

\prew
\textbf{theorem}~\textit{wf\_insort}:\kern.4em $\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
\textbf{nitpick} \\[2\smallskipamount]
{\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
\postw

Insertion should transform the set of elements represented by the tree in the
obvious way:

\prew
\textbf{theorem} \textit{dataset\_insort}:\kern.4em
$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
\textbf{nitpick} \\[2\smallskipamount]
{\slshape Nitpick ran out of time after checking 5 of 8 scopes.}
\postw

We could continue like this and sketch a complete theory of AA trees without
performing a single proof. Once the definitions and main theorems are in place
and have been thoroughly tested using Nitpick, we could start working on the
proofs. Developing theories this way usually saves time, because faulty theorems
and definitions are discovered much earlier in the process.

\section{Option Reference}
\label{option-reference}

\def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
\def\qty#1{$\left<\textit{#1}\right>$}
\def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
\def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\ops#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
\def\opu#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
\def\opusmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}

Nitpick's behavior can be influenced by various options, which can be specified
in brackets after the \textbf{nitpick} command. Default values can be set
using \textbf{nitpick\_\allowbreak params}. For example:

\prew
\textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
\postw

The options are categorized as follows:\ mode of operation
(\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
format (\S\ref{output-format}), automatic counterexample checks
(\S\ref{authentication}), optimizations
(\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).

The number of options can be overwhelming at first glance. Do not let that worry
you: Nitpick's defaults have been chosen so that it almost always does the right
thing, and the most important options have been covered in context in
\S\ref{first-steps}.

The descriptions below refer to the following syntactic quantities:

\begin{enum}
\item[$\bullet$] \qtybf{string}: A string.
\item[$\bullet$] \qtybf{bool}: \textit{true} or \textit{false}.
\item[$\bullet$] \qtybf{bool\_or\_smart}: \textit{true}, \textit{false}, or \textit{smart}.
\item[$\bullet$] \qtybf{int}: An integer. Negative integers are prefixed with a hyphen.
\item[$\bullet$] \qtybf{int\_or\_smart}: An integer or \textit{smart}.
\item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
of nonnegative integers (e.g., $1$--$4$). The range symbol --' can be entered as \texttt{-} (hyphen) or \texttt{\char\\\char\<midarrow\char\>}.

\item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
\item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
(milliseconds), or the keyword \textit{none} ($\infty$ years).
\item[$\bullet$] \qtybf{const}: The name of a HOL constant.
\item[$\bullet$] \qtybf{term}: A HOL term (e.g., $f~x$'').
\item[$\bullet$] \qtybf{term\_list}: A space-separated list of HOL terms (e.g.,
$f~x$''~$g~y$'').
\item[$\bullet$] \qtybf{type}: A HOL type.
\end{enum}

Default values are indicated in square brackets. Boolean options have a negated
counterpart (e.g., \textit{auto} vs.\ \textit{no\_auto}). When setting Boolean
options, = \textit{true}'' may be omitted.

\subsection{Mode of Operation}
\label{mode-of-operation}

\begin{enum}
\opfalse{auto}{no\_auto}
Specifies whether Nitpick should be run automatically on newly entered theorems.
For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation}) and
\textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
\textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
(\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
\textit{auto\_timeout} (\S\ref{timeouts}) is used as the time limit instead of
\textit{timeout} (\S\ref{timeouts}). The output is also more concise.

\nopagebreak

\optrue{blocking}{non\_blocking}
Specifies whether the \textbf{nitpick} command should operate synchronously.
The asynchronous (non-blocking) mode lets the user start proving the putative
theorem while Nitpick looks for a counterexample, but it can also be more
confusing. For technical reasons, automatic runs currently always block.

\nopagebreak

\optrue{falsify}{satisfy}
Specifies whether Nitpick should look for falsifying examples (countermodels) or
satisfying examples (models). This manual assumes throughout that
\textit{falsify} is enabled.

\opsmart{user\_axioms}{no\_user\_axioms}
Specifies whether the user-defined axioms (specified using
\textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
the constants that occur in the formula to falsify. The option is implicitly set
to \textit{true} for automatic runs.

\textbf{Warning:} If the option is set to \textit{true}, Nitpick might
nonetheless ignore some polymorphic axioms. Counterexamples generated under
these conditions are tagged as likely genuine.'' The \textit{debug}
(\S\ref{output-format}) option can be used to find out which axioms were
considered.

\nopagebreak
(\S\ref{mode-of-operation}), and \textit{debug} (\S\ref{output-format}).}

\optrue{assms}{no\_assms}
Specifies whether the relevant assumptions in structured proof should be
considered. The option is implicitly enabled for automatic runs.

\nopagebreak
and \textit{user\_axioms} (\S\ref{mode-of-operation}).}

\opfalse{overlord}{no\_overlord}
Specifies whether Nitpick should put its temporary files in
\texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for debugging Nitpick but also unsafe if several instances of the tool are run simultaneously. \nopagebreak {\small See also \textit{debug} (\S\ref{output-format}).} \end{enum} \subsection{Scope of Search} \label{scope-of-search} \begin{enum} \opu{card}{type}{int\_seq} Specifies the sequence of cardinalities to use for a given type. For \textit{nat} and \textit{int}, the cardinality fully specifies the subset used to approximate the type. For example: % $$\hbox{\begin{tabular}{@{}rll@{}}% \textit{card nat} = 4 & induces & \{0,\, 1,\, 2,\, 3\} \\ \textit{card int} = 4 & induces & \{-1,\, 0,\, +1,\, +2\} \\ \textit{card int} = 5 & induces & \{-2,\, -1,\, 0,\, +1,\, +2\}.% \end{tabular}}$$ % In general: % $$\hbox{\begin{tabular}{@{}rll@{}}% \textit{card nat} = K & induces & \{0,\, \ldots,\, K - 1\} \\ \textit{card int} = K & induces & \{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.% \end{tabular}}$$ % For free types, and often also for \textbf{typedecl}'d types, it usually makes sense to specify cardinalities as a range of the form \textit{$1$--$n$}. Although function and product types are normally mapped directly to the corresponding Kodkod concepts, setting the cardinality of such types is also allowed and implicitly enables boxing'' for them, as explained in the description of the \textit{box}~\qty{type} and \textit{box} (\S\ref{scope-of-search}) options. \nopagebreak {\small See also \textit{mono} (\S\ref{scope-of-search}).} \opt{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$} Specifies the default sequence of cardinalities to use. This can be overridden on a per-type basis using the \textit{card}~\qty{type} option described above. \opu{max}{const}{int\_seq} Specifies the sequence of maximum multiplicities to use for a given (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the number of distinct values that it can construct. Nonsensical values (e.g., \textit{max}~[]~$=$~2) are silently repaired. This option is only available for datatypes equipped with several constructors. \ops{max}{int\_seq} Specifies the default sequence of maximum multiplicities to use for (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor basis using the \textit{max}~\qty{const} option described above. \opusmart{wf}{const}{non\_wf} Specifies whether the specified (co)in\-duc\-tively defined predicate is well-founded. The option can take the following values: \begin{enum} \item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive predicate as if it were well-founded. Since this is generally not sound when the predicate is not well-founded, the counterexamples are tagged as likely genuine.'' \item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate as if it were not well-founded. The predicate is then unrolled as prescribed by the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter} options. \item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive predicate is well-founded using Isabelle's \textit{lexicographic\_order} and \textit{sizechange} tactics. If this succeeds (or the predicate occurs with an appropriate polarity in the formula to falsify), use an efficient fixed point equation as specification of the predicate; otherwise, unroll the predicates according to the \textit{iter}~\qty{const} and \textit{iter} options. \end{enum} \nopagebreak {\small See also \textit{iter} (\S\ref{scope-of-search}), \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout} (\S\ref{timeouts}).} \opsmart{wf}{non\_wf} Specifies the default wellfoundedness setting to use. This can be overridden on a per-predicate basis using the \textit{wf}~\qty{const} option above. \opu{iter}{const}{int\_seq} Specifies the sequence of iteration counts to use when unrolling a given (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive predicates that occur negatively and coinductive predicates that occur positively in the formula to falsify and that cannot be proved to be well-founded, but this behavior is influenced by the \textit{wf} option. The iteration counts are automatically bounded by the cardinality of the predicate's domain. {\small See also \textit{wf} (\S\ref{scope-of-search}) and \textit{star\_linear\_preds} (\S\ref{optimizations}).} \opt{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$} Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive predicates. This can be overridden on a per-predicate basis using the \textit{iter} \qty{const} option above. \opt{bisim\_depth}{int\_seq}{$\mathbf{7}$} Specifies the sequence of iteration counts to use when unrolling the bisimilarity predicate generated by Nitpick for coinductive datatypes. A value of$-1$means that no predicate is generated, in which case Nitpick performs an after-the-fact check to see if the known coinductive datatype values are bidissimilar. If two values are found to be bisimilar, the counterexample is tagged as likely genuine.'' The iteration counts are automatically bounded by the sum of the cardinalities of the coinductive datatypes occurring in the formula to falsify. \opusmart{box}{type}{dont\_box} Specifies whether Nitpick should attempt to wrap (box'') a given function or product type in an isomorphic datatype internally. Boxing is an effective mean to reduce the search space and speed up Nitpick, because the isomorphic datatype is approximated by a subset of the possible function or pair values; like other drastic optimizations, it can also prevent the discovery of counterexamples. The option can take the following values: \begin{enum} \item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever practicable. \item[$\bullet$] \textbf{\textit{false}}: Never box the type. \item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it is likely to help. For example,$n$-tuples where$n > 2$and arguments to higher-order functions are good candidates for boxing. \end{enum} Setting the \textit{card}~\qty{type} option for a function or product type implicitly enables boxing for that type. \nopagebreak {\small See also \textit{verbose} (\S\ref{output-format}) and \textit{debug} (\S\ref{output-format}).} \opsmart{box}{dont\_box} Specifies the default boxing setting to use. This can be overridden on a per-type basis using the \textit{box}~\qty{type} option described above. \opusmart{mono}{type}{non\_mono} Specifies whether the specified type should be considered monotonic when enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a monotonicity check on the type. Setting this option to \textit{true} can reduce the number of scopes tried, but it also diminishes the theoretical chance of finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}. \nopagebreak {\small See also \textit{card} (\S\ref{scope-of-search}), \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose} (\S\ref{output-format}).} \opsmart{mono}{non\_box} Specifies the default monotonicity setting to use. This can be overridden on a per-type basis using the \textit{mono}~\qty{type} option described above. \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars} Specifies whether type variables with the same sort constraints should be merged. Setting this option to \textit{true} can reduce the number of scopes tried and the size of the generated Kodkod formulas, but it also diminishes the theoretical chance of finding a counterexample. {\small See also \textit{mono} (\S\ref{scope-of-search}).} \end{enum} \subsection{Output Format} \label{output-format} \begin{enum} \opfalse{verbose}{quiet} Specifies whether the \textbf{nitpick} command should explain what it does. This option is useful to determine which scopes are tried or which SAT solver is used. This option is implicitly disabled for automatic runs. \nopagebreak {\small See also \textit{auto} (\S\ref{mode-of-operation}).} \opfalse{debug}{no\_debug} Specifies whether Nitpick should display additional debugging information beyond what \textit{verbose} already displays. Enabling \textit{debug} also enables \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug} option is implicitly disabled for automatic runs. \nopagebreak {\small See also \textit{auto} (\S\ref{mode-of-operation}), \textit{overlord} (\S\ref{mode-of-operation}), and \textit{batch\_size} (\S\ref{optimizations}).} \optrue{show\_skolems}{hide\_skolem} Specifies whether the values of Skolem constants should be displayed as part of counterexamples. Skolem constants correspond to bound variables in the original formula and usually help us to understand why the counterexample falsifies the formula. \nopagebreak {\small See also \textit{skolemize} (\S\ref{optimizations}).} \opfalse{show\_datatypes}{hide\_datatypes} Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should be displayed as part of counterexamples. Such subsets are sometimes helpful when investigating whether a potential counterexample is genuine or spurious, but their potential for clutter is real. \opfalse{show\_consts}{hide\_consts} Specifies whether the values of constants occurring in the formula (including its axioms) should be displayed along with any counterexample. These values are sometimes helpful when investigating why a counterexample is genuine, but they can clutter the output. \opfalse{show\_all}{dont\_show\_all} Enabling this option effectively enables \textit{show\_skolems}, \textit{show\_datatypes}, and \textit{show\_consts}. \opt{max\_potential}{int}{$\mathbf{1}$} Specifies the maximum number of potential counterexamples to display. Setting this option to 0 speeds up the search for a genuine counterexample. This option is implicitly set to 0 for automatic runs. If you set this option to a value greater than 1, you will need an incremental SAT solver: For efficiency, it is recommended to install the JNI version of MiniSat and set \textit{sat\_solver} = \textit{MiniSatJNI}. Also be aware that many of the counterexamples may look identical, unless the \textit{show\_all} (\S\ref{output-format}) option is enabled. \nopagebreak {\small See also \textit{auto} (\S\ref{mode-of-operation}), \textit{check\_potential} (\S\ref{authentication}), and \textit{sat\_solver} (\S\ref{optimizations}).} \opt{max\_genuine}{int}{$\mathbf{1}$} Specifies the maximum number of genuine counterexamples to display. If you set this option to a value greater than 1, you will need an incremental SAT solver: For efficiency, it is recommended to install the JNI version of MiniSat and set \textit{sat\_solver} = \textit{MiniSatJNI}. Also be aware that many of the counterexamples may look identical, unless the \textit{show\_all} (\S\ref{output-format}) option is enabled. \nopagebreak {\small See also \textit{check\_genuine} (\S\ref{authentication}) and \textit{sat\_solver} (\S\ref{optimizations}).} \ops{eval}{term\_list} Specifies the list of terms whose values should be displayed along with counterexamples. This option suffers from an observer effect'': Nitpick might find different counterexamples for different values of this option. \opu{format}{term}{int\_seq} Specifies how to uncurry the value displayed for a variable or constant. Uncurrying sometimes increases the readability of the output for high-arity functions. For example, given the variable$y \mathbin{\Colon} {'a}\Rightarrow
{'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
{'g}$, setting \textit{format}~$y$= 3 tells Nitpick to group the last three arguments, as if the type had been${'a}\Rightarrow {'b}\Rightarrow
{'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list of values$n_1,\ldots,n_k$tells Nitpick to show the last$n_k$arguments as an$n_k$-tuple, the previous$n_{k-1}$arguments as an$n_{k-1}$-tuple, and so on; arguments that are not accounted for are left alone, as if the specification had been$1,\ldots,1,n_1,\ldots,n_k$. \nopagebreak {\small See also \textit{uncurry} (\S\ref{optimizations}).} \opt{format}{int\_seq}{$\mathbf{1}$} Specifies the default format to use. Irrespective of the default format, the extra arguments to a Skolem constant corresponding to the outer bound variables are kept separated from the remaining arguments, the \textbf{for} arguments of an inductive definitions are kept separated from the remaining arguments, and the iteration counter of an unrolled inductive definition is shown alone. The default format can be overridden on a per-variable or per-constant basis using the \textit{format}~\qty{term} option described above. \end{enum} %% MARK: Authentication \subsection{Authentication} \label{authentication} \begin{enum} \opfalse{check\_potential}{trust\_potential} Specifies whether potential counterexamples should be given to Isabelle's \textit{auto} tactic to assess their validity. If a potential counterexample is shown to be genuine, Nitpick displays a message to this effect and terminates. \nopagebreak {\small See also \textit{max\_potential} (\S\ref{output-format}) and \textit{auto\_timeout} (\S\ref{timeouts}).} \opfalse{check\_genuine}{trust\_genuine} Specifies whether genuine and likely genuine counterexamples should be given to Isabelle's \textit{auto} tactic to assess their validity. If a genuine'' counterexample is shown to be spurious, the user is kindly asked to send a bug report to the author at \texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}. \nopagebreak {\small See also \textit{max\_genuine} (\S\ref{output-format}) and \textit{auto\_timeout} (\S\ref{timeouts}).} \ops{expect}{string} Specifies the expected outcome, which must be one of the following: \begin{enum} \item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample. \item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a likely genuine'' counterexample (i.e., a counterexample that is genuine unless it contradicts a missing axiom or a dangerous option was used inappropriately). \item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample. \item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample. \item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g., Kodkod ran out of memory). \end{enum} Nitpick emits an error if the actual outcome differs from the expected outcome. This option is useful for regression testing. \end{enum} \subsection{Optimizations} \label{optimizations} \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}} \sloppy \begin{enum} \opt{sat\_solver}{string}{smart} Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend to be faster than their Java counterparts, but they can be more difficult to install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1, you will need an incremental SAT solver, such as \textit{MiniSatJNI} (recommended) or \textit{SAT4J}. The supported solvers are listed below: \begin{enum} \item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver written in \cpp{}. To use MiniSat, set the environment variable \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat} executable. The \cpp{} sources and executables for MiniSat are available at \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14 and 2.0 beta (2007-07-21). \item[$\bullet$] \textbf{\textit{MiniSatJNI}}: The JNI (Java Native Interface) version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard version of MiniSat, the JNI version can be used incrementally. \item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver written in C. It is bundled with Kodkodi and requires no further installation or configuration steps. Alternatively, you can install a standard version of PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory that contains the \texttt{picosat} executable. The C sources for PicoSAT are available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi. Nitpick has been tested with version 913. \item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to the directory that contains the \texttt{zchaff} executable. The \cpp{} sources and executables for zChaff are available at \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with versions 2004-05-13, 2004-11-15, and 2007-03-12. \item[$\bullet$] \textbf{\textit{zChaffJNI}}: The JNI version of zChaff is bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on Kodkod's web site \cite{kodkod-2009}. \item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the directory that contains the \texttt{rsat} executable. The \cpp{} sources for RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been tested with version 2.01. \item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver written in C. To use BerkMin, set the environment variable \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561} executable. The BerkMin executables are available at \url{http://eigold.tripod.com/BerkMin.html}. \item[$\bullet$] \textbf{\textit{BerkMinAlloy}}: Variant of BerkMin that is included with Alloy 4 and calls itself sat56'' in its banner text. To use this version of BerkMin, set the environment variable \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin} executable. \item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver written in C. To use Jerusat, set the environment variable \texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3} executable. The C sources for Jerusat are available at \url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}. \item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver written in Java that can be used incrementally. It is bundled with Kodkodi and requires no further installation or configuration steps. Do not attempt to install the official SAT4J packages, because their API is incompatible with Kodkod. \item[$\bullet$] \textbf{\textit{SAT4JLight}}: Variant of SAT4J that is optimized for small problems. It can also be used incrementally. \item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an experimental solver written in \cpp. To use HaifaSat, set the environment variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the \texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at \url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}. \item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to \textit{smart}, Nitpick selects the first solver among MiniSatJNI, MiniSat, PicoSAT, zChaffJNI, zChaff, RSat, BerkMin, BerkMinAlloy, and Jerusat that is recognized by Isabelle. If none is found, it falls back on SAT4J, which should always be available. If \textit{verbose} is enabled, Nitpick displays which SAT solver was chosen. \end{enum} \fussy \opt{batch\_size}{int\_or\_smart}{smart} Specifies the maximum number of Kodkod problems that should be lumped together when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems together ensures that Kodkodi is launched less often, but it makes the verbose output less readable and is sometimes detrimental to performance. If \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if \textit{debug} (\S\ref{output-format}) is set and 64 otherwise. \optrue{destroy\_constrs}{dont\_destroy\_constrs} Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should be rewritten to use (automatically generated) discriminators and destructors. This optimization can drastically reduce the size of the Boolean formulas given to the SAT solver. \nopagebreak {\small See also \textit{debug} (\S\ref{output-format}).} \optrue{specialize}{dont\_specialize} Specifies whether functions invoked with static arguments should be specialized. This optimization can drastically reduce the search space, especially for higher-order functions. \nopagebreak {\small See also \textit{debug} (\S\ref{output-format}) and \textit{show\_consts} (\S\ref{output-format}).} \optrue{skolemize}{dont\_skolemize} Specifies whether the formula should be skolemized. For performance reasons, (positive)$\forall$-quanti\-fiers that occur in the scope of a higher-order (positive)$\exists$-quanti\-fier are left unchanged. \nopagebreak {\small See also \textit{debug} (\S\ref{output-format}) and \textit{show\_skolems} (\S\ref{output-format}).} \optrue{star\_linear\_preds}{dont\_star\_linear\_preds} Specifies whether Nitpick should use Kodkod's transitive closure operator to encode non-well-founded linear inductive predicates,'' i.e., inductive predicates for which each the predicate occurs in at most one assumption of each introduction rule. Using the reflexive transitive closure is in principle equivalent to setting \textit{iter} to the cardinality of the predicate's domain, but it is usually more efficient. {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug} (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).} \optrue{uncurry}{dont\_uncurry} Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no tangible effect on efficiency, but it creates opportunities for the boxing optimization. \nopagebreak {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug} (\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).} \optrue{fast\_descrs}{full\_descrs} Specifies whether Nitpick should optimize the definite and indefinite description operators (THE and SOME). The optimized versions usually help Nitpick generate more counterexamples or at least find them faster, but only the unoptimized versions are complete when all types occurring in the formula are finite. {\small See also \textit{debug} (\S\ref{output-format}).} \optrue{peephole\_optim}{no\_peephole\_optim} Specifies whether Nitpick should simplify the generated Kodkod formulas using a peephole optimizer. These optimizations can make a significant difference. Unless you are tracking down a bug in Nitpick or distrust the peephole optimizer, you should leave this option enabled. \opt{sym\_break}{int}{20} Specifies an upper bound on the number of relations for which Kodkod generates symmetry breaking predicates. According to the Kodkod documentation \cite{kodkod-2009-options}, in general, the higher this value, the more symmetries will be broken, and the faster the formula will be solved. But, setting the value too high may have the opposite effect and slow down the solving.'' \opt{sharing\_depth}{int}{3} Specifies the depth to which Kodkod should check circuits for equivalence during the translation to SAT. The default of 3 is the same as in Alloy. The minimum allowed depth is 1. Increasing the sharing may result in a smaller SAT problem, but can also slow down Kodkod. \opfalse{flatten\_props}{dont\_flatten\_props} Specifies whether Kodkod should try to eliminate intermediate Boolean variables. Although this might sound like a good idea, in practice it can drastically slow down Kodkod. \opt{max\_threads}{int}{0} Specifies the maximum number of threads to use in Kodkod. If this option is set to 0, Kodkod will compute an appropriate value based on the number of processor cores available. \nopagebreak {\small See also \textit{batch\_size} (\S\ref{optimizations}) and \textit{timeout} (\S\ref{timeouts}).} \end{enum} \subsection{Timeouts} \label{timeouts} \begin{enum} \opt{timeout}{time}{$\mathbf{30}$s} Specifies the maximum amount of time that the \textbf{nitpick} command should spend looking for a counterexample. Nitpick tries to honor this constraint as well as it can but offers no guarantees. For automatic runs, \textit{auto\_timeout} is used instead. \nopagebreak {\small See also \textit{auto} (\S\ref{mode-of-operation}) and \textit{max\_threads} (\S\ref{optimizations}).} \opt{auto\_timeout}{time}{$\mathbf{5}$s} Specifies the maximum amount of time that Nitpick should use to find a counterexample when running automatically. Nitpick tries to honor this constraint as well as it can but offers no guarantees. \nopagebreak {\small See also \textit{auto} (\S\ref{mode-of-operation}).} \opt{tac\_timeout}{time}{$\mathbf{500}$\,ms} Specifies the maximum amount of time that the \textit{auto} tactic should use when checking a counterexample, and similarly that \textit{lexicographic\_order} and \textit{sizechange} should use when checking whether a (co)in\-duc\-tive predicate is well-founded. Nitpick tries to honor this constraint as well as it can but offers no guarantees. \nopagebreak {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{check\_potential} (\S\ref{authentication}), and \textit{check\_genuine} (\S\ref{authentication}).} \end{enum} \section{Attribute Reference} \label{attribute-reference} Nitpick needs to consider the definitions of all constants occurring in a formula in order to falsify it. For constants introduced using the \textbf{definition} command, the definition is simply the associated \textit{\_def} axiom. In contrast, instead of using the internal representation of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and \textbf{nominal\_primrec} packages, Nitpick relies on the more natural equational specification entered by the user. Behind the scenes, Isabelle's built-in packages and theories rely on the following attributes to affect Nitpick's behavior: \begin{itemize} \flushitem{\textit{nitpick\_def}} \nopagebreak This attribute specifies an alternative definition of a constant. The alternative definition should be logically equivalent to the constant's actual axiomatic definition and should be of the form \qquad$c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$, where${?}x_1, \ldots, {?}x_n$are distinct variables and$c$does not occur in$t$. \flushitem{\textit{nitpick\_simp}} \nopagebreak This attribute specifies the equations that constitute the specification of a constant. For functions defined using the \textbf{primrec}, \textbf{function}, and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the \textit{simps} rules. The equations must be of the form \qquad$c~t_1~\ldots\ t_n \,=\, u.$\flushitem{\textit{nitpick\_psimp}} \nopagebreak This attribute specifies the equations that constitute the partial specification of a constant. For functions defined using the \textbf{function} package, this corresponds to the \textit{psimps} rules. The conditional equations must be of the form \qquad$\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$. \flushitem{\textit{nitpick\_intro}} \nopagebreak This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate. For predicates defined using the \textbf{inductive} or \textbf{coinductive} command, this corresponds to the \textit{intros} rules. The introduction rules must be of the form \qquad$\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
\ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\
\ldots\ u_n$, where the$P_i$'s are side conditions that do not involve$c$and$M$is an optional monotonic operator. The order of the assumptions is irrelevant. \end{itemize} When faced with a constant, Nitpick proceeds as follows: \begin{enum} \item[1.] If the \textit{nitpick\_simp} set associated with the constant is not empty, Nitpick uses these rules as the specification of the constant. \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with the constant is not empty, it uses these rules as the specification of the constant. \item[3.] Otherwise, it looks up the definition of the constant: \begin{enum} \item[1.] If the \textit{nitpick\_def} set associated with the constant is not empty, it uses the latest rule added to the set as the definition of the constant; otherwise it uses the actual definition axiom. \item[2.] If the definition is of the form \qquad$c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$, then Nitpick assumes that the definition was made using an inductive package and based on the introduction rules marked with \textit{nitpick\_\allowbreak ind\_\allowbreak intros} tries to determine whether the definition is well-founded. \end{enum} \end{enum} As an illustration, consider the inductive definition \prew \textbf{inductive}~\textit{odd}~\textbf{where} \\ \textit{odd}~1''$\,\mid$\\ \textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$'' \postw Isabelle automatically attaches the \textit{nitpick\_intro} attribute to the above rules. Nitpick then uses the \textit{lfp}-based definition in conjunction with these rules. To override this, we can specify an alternative definition as follows: \prew \textbf{lemma}$\mathit{odd\_def}'$[\textit{nitpick\_def}]: $\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$'' \postw Nitpick then expands all occurrences of$\mathit{odd}~n$to$n~\textrm{mod}~2
= 1$. Alternatively, we can specify an equational specification of the constant: \prew \textbf{lemma}$\mathit{odd\_simp}'$[\textit{nitpick\_simp}]: $\textit{odd}~n = (n~\textrm{mod}~2 = 1)$'' \postw Such tweaks should be done with great care, because Nitpick will assume that the constant is completely defined by its equational specification. For example, if 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$\textit{odd}~n$arbitrarily for even values of$n$. The \textit{debug} (\S\ref{output-format}) option is extremely useful to understand what is going on when experimenting with \textit{nitpick\_} attributes. \section{Standard ML Interface} \label{standard-ml-interface} Nitpick provides a rich Standard ML interface used mainly for internal purposes and debugging. Among the most interesting functions exported by Nitpick are those that let you invoke the tool programmatically and those that let you register and unregister custom coinductive datatypes. \subsection{Invocation of Nitpick} \label{invocation-of-nitpick} The \textit{Nitpick} structure offers the following functions for invoking your favorite counterexample generator: \prew$\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
\hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
\hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$\\$\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
\hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$\postw The return value is a new proof state paired with an outcome string (genuine'', likely\_genuine'', potential'', none'', or unknown''). The \textit{params} type is a large record that lets you set Nitpick's options. The current default options can be retrieved by calling the following function defined in the \textit{Nitpick\_Isar} structure: \prew$\textbf{val}\,~\textit{default\_params} :\,
\textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$\postw The second argument lets you override option values before they are parsed and put into a \textit{params} record. Here is an example: \prew$\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{}\textrm{timeout}\textrm{''},\, \textrm{}\textrm{none}\textrm{''})]$\\$\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
& \textit{state}~\textit{params}~\textit{false} \\[-2pt]
& \textit{subgoal}\end{aligned}$\postw \subsection{Registration of Coinductive Datatypes} \label{registration-of-coinductive-datatypes} \let\antiq=\textrm If you have defined a custom coinductive datatype, you can tell Nitpick about it, so that it can use an efficient Kodkod axiomatization similar to the one it uses for lazy lists. The interface for registering and unregistering coinductive datatypes consists of the following pair of functions defined in the \textit{Nitpick} structure: \prew$\textbf{val}\,~\textit{register\_codatatype} :\,
\textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$\\$\textbf{val}\,~\textit{unregister\_codatatype} :\,
\textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$\postw The type$'a~\textit{llist}$of lazy lists is already registered; had it not been, you could have told Nitpick about it by adding the following line to your theory file: \prew$\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
& \textit{Nitpick.register\_codatatype} \\[-2pt]
& \qquad @\{\antiq{typ}~\kern1pt'a~\textit{llist}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
& \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$\postw The \textit{register\_codatatype} function takes a coinductive type, its case function, and the list of its constructors. The case function must take its arguments in the order that the constructors are listed. If no case function with the correct signature is available, simply pass the empty string. On the other hand, if your goal is to cripple Nitpick, add the following line to your theory file and try to check a few conjectures about lazy lists: \prew$\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~
\kern1pt'a~\textit{list}\textrm{''}\}\ \,{*}\}$\postw \section{Known Bugs and Limitations} \label{known-bugs-and-limitations} Here are the known bugs and limitations in Nitpick at the time of writing: \begin{enum} \item[$\bullet$] Underspecified functions defined using the \textbf{primrec}, \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead Nitpick to generate spurious counterexamples for theorems that refer to values for which the function is not defined. For example: \prew \textbf{primrec} \textit{prec} \textbf{where} \\ $\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount] \textbf{lemma} $\textit{prec}~0 = \undef$'' \\ \textbf{nitpick} \\[2\smallskipamount] \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2: \nopagebreak \\[2\smallskipamount] \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount] \textbf{by}~(\textit{auto simp}: \textit{prec\_def}) \postw Such theorems are considered bad style because they rely on the internal representation of functions synthesized by Isabelle, which is an implementation detail. \item[$\bullet$] Nitpick maintains a global cache of wellformedness conditions, which can become invalid if you change the definition of an inductive predicate that is registered in the cache. To clear the cache, run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g., 501$\,\textit{ms}$). \item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a \textbf{guess} command in a structured proof. \item[$\bullet$] The \textit{nitpick\_} attributes and the \textit{Nitpick.register\_} functions can cause havoc if used improperly. \item[$\bullet$] Local definitions are not supported and result in an error. \item[$\bullet\$] All constants and types whose names start with
\textit{Nitpick}{.} are reserved for internal use.
\end{enum}

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