diff --git a/src/Doc/Nitpick/document/root.tex b/src/Doc/Nitpick/document/root.tex
--- a/src/Doc/Nitpick/document/root.tex
+++ b/src/Doc/Nitpick/document/root.tex
@@ -1,2758 +1,2750 @@
 \documentclass[a4paper,12pt]{article}
 \usepackage[T1]{fontenc}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{color}
 \usepackage{footmisc}
 \usepackage{graphicx}
 %\usepackage{mathpazo}
 \usepackage{multicol}
 \usepackage{stmaryrd}
 %\usepackage[scaled=.85]{beramono}
 \usepackage{isabelle,iman,pdfsetup}
 
 %\oddsidemargin=4.6mm
 %\evensidemargin=4.6mm
 %\textwidth=150mm
 %\topmargin=4.6mm
 %\headheight=0mm
 %\headsep=0mm
 %\textheight=234mm
 
 \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\unk{{?}}
 \def\unk{{\_}}
 \def\unkef{(\lambda x.\; \unk)}
 \def\undef{(\lambda x.\; \_)}
 %\def\unr{\textit{others}}
 \def\unr{\ldots}
 \def\Abs#1{\hbox{\rm{\guillemetleft}}{\,#1\,}\hbox{\rm{\guillemetright}}}
 \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}
 
 \renewcommand\_{\hbox{\textunderscore\kern-.05ex}}
 
 \begin{document}
 
 %%% TYPESETTING
 %\renewcommand\labelitemi{$\bullet$}
 \renewcommand\labelitemi{\raise.065ex\hbox{\small\textbullet}}
 
 \title{\includegraphics[scale=0.5]{isabelle_logo} \\[4ex]
 Picking Nits \\[\smallskipamount]
 \Large A User's Guide to Nitpick for Isabelle/HOL}
 \author{\hbox{} \\
 Jasmin Blanchette \\
 {\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\
 \hbox{}}
 
 \maketitle
 
 \tableofcontents
 
 \setlength{\parskip}{.7em plus .2em minus .1em}
 \setlength{\parindent}{0pt}
 \setlength{\abovedisplayskip}{\parskip}
 \setlength{\abovedisplayshortskip}{.9\parskip}
 \setlength{\belowdisplayskip}{\parskip}
 \setlength{\belowdisplayshortskip}{.9\parskip}
 
 % General-purpose enum environment with correct spacing
 \newenvironment{enum}%
     {\begin{list}{}{%
         \setlength{\topsep}{.1\parskip}%
         \setlength{\partopsep}{.1\parskip}%
         \setlength{\itemsep}{\parskip}%
         \advance\itemsep by-\parsep}}
     {\end{list}}
 
 \def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin
 \advance\rightskip by\leftmargin}
 \def\post{\vskip0pt plus1ex\endgroup}
 
 \def\prew{\pre\advance\rightskip by-\leftmargin}
 \def\postw{\post}
 
 \section{Introduction}
 \label{introduction}
 
 Nitpick \cite{blanchette-nipkow-2010} 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 provides an automatic mode that can be enabled via the ``Auto Nitpick''
 option under ``Plugins > Plugin Options > Isabelle > General'' in
 Isabelle/jEdit. In this mode, Nitpick is run on every newly entered theorem.
 
 \newbox\boxA
 \setbox\boxA=\hbox{\texttt{nospam}}
 
 \newcommand\authoremail{\texttt{jasmin.blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
 inria\allowbreak .\allowbreak fr}}
 
 To run Nitpick, you must also make sure that the theory \textit{Nitpick} is
 imported---this is rarely a problem in practice since it is part of
 \textit{Main}. The examples presented in this manual can be found
 in Isabelle's \texttt{src/HOL/\allowbreak Nitpick\_\allowbreak Examples/\allowbreak Manual\_Nits.thy} theory.
 The known bugs and limitations at the time of writing are listed in
 \S\ref{known-bugs-and-limitations}. Comments and bug reports concerning
 the tool or the manual should be directed to the author at \authoremail.
 
 \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{Installation}
 \label{installation}
 
 Nitpick is part of Isabelle, so you do not need to install it. It relies on a
 third-party Kodkod front-end called Kodkodi, which in turn requires a Java
 virtual machine. Both are provided as official Isabelle components.
 
 %There are two main ways of installing Kodkodi:
 %
 %\begin{enum}
 %\item[\labelitemi] If you installed an official Isabelle package,
 %it should already include a properly setup version of Kodkodi.
 %
 %\item[\labelitemi] If you use a repository or snapshot version of Isabelle, you
 %an official Isabelle package, you can download the Isabelle-aware Kodkodi package
 %from \url{http://www21.in.tum.de/~blanchet/\#software}. Extract the archive, then add a
 %line to your \texttt{\$ISABELLE\_HOME\_USER\slash etc\slash components}%
 %\footnote{The variable \texttt{\$ISABELLE\_HOME\_USER} is set by Isabelle at
 %startup. Its value can be retrieved by executing \texttt{isabelle}
 %\texttt{getenv} \texttt{ISABELLE\_HOME\_USER} on the command line.}
 %file with the absolute path to Kodkodi. For example, if the
 %\texttt{components} file does not exist yet and you extracted Kodkodi to
 %\texttt{/usr/local/kodkodi-1.5.2}, create it with the single line
 %
 %\prew
 %\texttt{/usr/local/kodkodi-1.5.2}
 %\postw
 %
 %(including an invisible newline character) in it.
 %\end{enum}
 
 To check whether Kodkodi is successfully installed, you can try out the example
 in \S\ref{propositional-logic}.
 
 \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
 as follows:
 
 \prew
 \textbf{theory}~\textit{Scratch} \\
 \textbf{imports}~\textit{Main~Quotient\_Product~RealDef} \\
 \textbf{begin}
 \postw
 
 The results presented here were obtained using the JNI (Java Native Interface)
 version of MiniSat and with multithreading disabled to reduce nondeterminism.
 This was done by adding the line
 
 \prew
 \textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSat\_JNI}, \,\textit{max\_threads}~= 1]
 \postw
 
 after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
 Kodkodi and is precompiled for Linux, Mac~OS~X, and Windows (Cygwin). Other SAT
 solvers can also be used, 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.
 
 \subsection{Propositional Logic}
 \label{propositional-logic}
 
 Let's start with a trivial example from 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 Free variables: \nopagebreak \\
 \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): \\
 \phantom{0}1. $P\,\Longrightarrow\, Q$ \\
 \phantom{0}2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
 \textbf{nitpick}~1 \\[2\smallskipamount]
 {\slshape Nitpick found a counterexample: \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \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 Free variables: \nopagebreak \\
 \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} ``$x \in A\,\Longrightarrow\, (\textrm{THE}~y.\;y \in A) \in A$''
 \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 (10 by default), looking for a finite
 countermodel:
 
 \prew
 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
 \slshape
 Trying 10 scopes: \nopagebreak \\
 \hbox{}\qquad \textit{card}~$'a$~= 1; \\
 \hbox{}\qquad \textit{card}~$'a$~= 2; \\
 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
 \hbox{}\qquad \textit{card}~$'a$~= 10 \\[2\smallskipamount]
 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $A = \{a_2,\, a_3\}$ \\
 \hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
 Total time: 963 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} ``$x \in A\,\Longrightarrow\, (\textrm{THE}~y.\;y \in A) \in A$'' \\
 \textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
 \slshape
 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $A = \{a_2,\, a_3\}$ \\
 \hbox{}\qquad\qquad $x = a_3$ \\
 \hbox{}\qquad Constant: \nopagebreak \\
 \hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;y \in A = a_1$
 \postw
 
 As the result of an optimization, Nitpick directly assigned a value to the
 subterm $\textrm{THE}~y.\;y \in A$, rather than to the \textit{The} constant. We
 can disable this optimization by using the command
 
 \prew
 \textbf{nitpick}~[\textit{dont\_specialize},\, \textit{show\_consts}]
 \postw
 
 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} ``$\exists {!}x.\; x \in A\,\Longrightarrow\, (\textrm{THE}~y.\;y \in A) \in A$''
 \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 `--'
 is entered as \texttt{-} (hyphen).} \\[2\smallskipamount]
 \slshape Nitpick found no counterexample.
 \postw
 
 Let's see if Sledgehammer can find a proof:
 
 \prew
 \textbf{sledgehammer} \\[2\smallskipamount]
 {\slshape Sledgehammer: ``$e$'' on goal \\
 Try this: \textbf{by}~(\textit{metis~theI}) (42 ms)} \\
 \hbox{}\qquad\vdots \\[2\smallskipamount]
 \textbf{by}~(\textit{metis~theI\/})
 \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 Free variable: \nopagebreak \\
 \hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
 \hbox{}\qquad Skolem constants: \nopagebreak \\
 \hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
 \hbox{}\qquad\qquad $y = a_2$
 \postw
 
 (The Isabelle/HOL notation $f(x := y)$ denotes the function that maps $x$ to $y$
 and that otherwise behaves like $f$.)
 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 Skolem constant: \nopagebreak \\
 \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 Free variable: \nopagebreak \\
 \hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
 \hbox{}\qquad Skolem constants: \nopagebreak \\
 \hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
 \hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
 \postw
 
 What happened here is that Nitpick expanded \textit{sym} 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.
 
 \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's approach is to consider finite
 subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined
 value (displayed as `$\unk$'). The type \textit{int} is handled similarly.
 Internally, 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 Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $i = 0$ \\
 \hbox{}\qquad\qquad $j = 1$ \\
 \hbox{}\qquad\qquad $m = 1$ \\
 \hbox{}\qquad\qquad $n = 0$
 \postw
 
 Internally, Nitpick uses either a unary or a binary representation of numbers.
 The unary representation is more efficient but only suitable for numbers very
 close to zero. By default, Nitpick attempts to choose the more appropriate
 encoding by inspecting the formula at hand. This behavior can be overridden by
 passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For
 binary notation, the number of bits to use can be specified using
 the \textit{bits} option. For example:
 
 \prew
 \textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$]
 \postw
 
 With infinite types, we don't always have the luxury of a genuine counterexample
 and must often content ourselves with a potentially spurious one.
 For example:
 
 \prew
 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
 \textbf{nitpick} [\textit{card~nat}~= 50] \\[2\smallskipamount]
 \slshape Warning: The conjecture either trivially holds for the given scopes or lies outside Nitpick's supported
 fragment; only potentially spurious counterexamples may be found \\[2\smallskipamount]
 Nitpick found a potentially spurious counterexample: \\[2\smallskipamount]
 \hbox{}\qquad Free variable: \nopagebreak \\
 \hbox{}\qquad\qquad $P = \textit{False}$
 \postw
 
 The issue 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 fully satisfied by Nitpick,
 the putative lemma can never be falsified.
 
 Some conjectures involving elementary number theory make Nitpick look like a
 giant with feet of clay:
 
 \prew
 \textbf{lemma} ``$P~\textit{Suc\/}$'' \\
 \textbf{nitpick} \\[2\smallskipamount]
 \slshape
 Nitpick found no counterexample
 \postw
 
 On any finite set $N$, \textit{Suc} is a partial function; for example, if $N =
 \{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\,
 \ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when 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 Free variable: \nopagebreak \\
 \hbox{}\qquad\qquad $P = \unkef(\unkef(0 := \unkef(0 := 0)) := \mathit{False})$ \\[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 10 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 Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $\textit{xs} = []$ \\
 \hbox{}\qquad\qquad $\textit{y} = a_1$
 \postw
 
 To see why the counterexample is genuine, we enable \textit{show\_consts}
 and \textit{show\_\allowbreak datatypes}:
 
 \prew
 {\slshape Type:} \\
 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\
 {\slshape Constants:} \\
 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \unkef([] := [a_1, a_1])$ \\
 \hbox{}\qquad $\textit{hd} = \unkef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$
 \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 $[a_1,
 a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not
 representable in the subset of $'a$~\textit{list} considered by Nitpick, which
 is shown under the ``Type'' heading; hence the result is $\unk$. Similarly,
 appending $[a_1, a_1]$ 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 $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe
 that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin
 \mathcal{S}$ 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\_types}] \\[2\smallskipamount]
 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $\textit{xs} = [a_2]$ \\
 \hbox{}\qquad\qquad $\textit{ys} = [a_1]$ \\
 \hbox{}\qquad Types: \\
 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$
 \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, Quotient Types, Records, Rationals, and Reals}
 \label{typedefs-quotient-types-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} ``$\lbrakk A \in X;\> B \in X\rbrakk \,\Longrightarrow\, c \in X$'' \\
 \textbf{nitpick} [\textit{show\_types}] \\[2\smallskipamount]
 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $X = \{\Abs{0},\, \Abs{1}\}$ \\
 \hbox{}\qquad\qquad $c = \Abs{2}$ \\
 \hbox{}\qquad Types: \\
 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$
 \postw
 
 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
 
 Quotient types are handled in much the same way. The following fragment defines
 the integer type \textit{my\_int} by encoding the integer $x$ by a pair of
 natural numbers $(m, n)$ such that $x + n = m$:
 
 \prew
 \textbf{fun} \textit{my\_int\_rel} \textbf{where} \\
 ``$\textit{my\_int\_rel}~(x,\, y)~(u,\, v) = (x + v = u + y)$'' \\[2\smallskipamount]
 %
 \textbf{quotient\_type}~\textit{my\_int} = ``$\textit{nat} \times \textit{nat\/}$''$\;{/}\;$\textit{my\_int\_rel} \\
 \textbf{by}~(\textit{auto simp add\/}:\ \textit{equivp\_def fun\_eq\_iff}) \\[2\smallskipamount]
 %
 \textbf{definition}~\textit{add\_raw}~\textbf{where} \\
 ``$\textit{add\_raw} \,\equiv\, \lambda(x,\, y)~(u,\, v).\; (x + (u\Colon\textit{nat}), y + (v\Colon\textit{nat}))$'' \\[2\smallskipamount]
 %
 \textbf{quotient\_definition} ``$\textit{add\/}\Colon\textit{my\_int} \Rightarrow \textit{my\_int} \Rightarrow \textit{my\_int\/}$'' \textbf{is} \textit{add\_raw} \\[2\smallskipamount]
 %
 \textbf{lemma} ``$\textit{add}~x~y = \textit{add}~x~x$'' \\
 \textbf{nitpick} [\textit{show\_types}] \\[2\smallskipamount]
 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $x = \Abs{(0,\, 0)}$ \\
 \hbox{}\qquad\qquad $y = \Abs{(0,\, 1)}$ \\
 \hbox{}\qquad Types: \\
 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, \unr\}$ \\
 \hbox{}\qquad\qquad $\textit{nat} \times \textit{nat}~[\textsl{boxed\/}] = \{(0,\, 0),\> (1,\, 0),\> \unr\}$ \\
 \hbox{}\qquad\qquad $\textit{my\_int} = \{\Abs{(0,\, 0)},\> \Abs{(0,\, 1)},\> \unr\}$
 \postw
 
 The values $\Abs{(0,\, 0)}$ and $\Abs{(0,\, 1)}$ represent the
 integers $0$ and $-1$, respectively. Other representants would have been
 possible---e.g., $\Abs{(5,\, 5)}$ and $\Abs{(11,\, 12)}$. If we are going to
 use \textit{my\_int} extensively, it pays off to install a term postprocessor
 that converts the pair notation to the standard mathematical notation:
 
 \prew
 $\textbf{ML}~\,\{{*} \\
 \!\begin{aligned}[t]
 %& ({*}~\,\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \\[-2pt]
 %& \phantom{(*}~\,{\rightarrow}\;\textit{term}~\,{*}) \\[-2pt]
 & \textbf{fun}\,~\textit{my\_int\_postproc}~\_~\_~\_~T~(\textit{Const}~\_~\$~(\textit{Const}~\_~\$~\textit{t1}~\$~\textit{t2\/})) = {} \\[-2pt]
 & \phantom{fun}\,~\textit{HOLogic.mk\_number}~T~(\textit{snd}~(\textit{HOLogic.dest\_number~t1}) \\[-2pt]
 & \phantom{fun\,~\textit{HOLogic.mk\_number}~T~(}{-}~\textit{snd}~(\textit{HOLogic.dest\_number~t2\/})) \\[-2pt]
 & \phantom{fun}\!{\mid}\,~\textit{my\_int\_postproc}~\_~\_~\_~\_~t = t \\[-2pt]
 {*}\}\end{aligned}$ \\[2\smallskipamount]
 $\textbf{declaration}~\,\{{*} \\
 \!\begin{aligned}[t]
 & \textit{Nitpick\_Model.register\_term\_postprocessor}~\!\begin{aligned}[t]
   & @\{\textrm{typ}~\textit{my\_int}\} \\[-2pt]
   & \textit{my\_int\_postproc}\end{aligned} \\[-2pt]
 {*}\}\end{aligned}$
 \postw
 
 Records are handled as datatypes with a single constructor:
 
 \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\_types}] \\[2\smallskipamount]
 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
 \hbox{}\qquad Types: \\
 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
 \hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
 & \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
 & \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
 \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\_types}] \\[2\smallskipamount]
 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $x = 1/2$ \\
 \hbox{}\qquad\qquad $y = -1/2$ \\
 \hbox{}\qquad Types: \\
 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
 \hbox{}\qquad\qquad $\textit{int} = \{-3,\, -2,\, -1,\, 0,\, 1,\, 2,\, 3,\, 4,\, \unr\}$ \\
 \hbox{}\qquad\qquad $\textit{real} = \{-3/2,\, -1/2,\, 0,\, 1/2,\, 1,\, 2,\, 3,\, 4,\, \unr\}$
 \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}~= 50, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
 \slshape The inductive predicate ``\textit{even}'' was proved well-founded;
 Nitpick can compute it efficiently \\[2\smallskipamount]
 Trying 1 scope: \\
 \hbox{}\qquad \textit{card nat}~= 50. \\[2\smallskipamount]
 Warning: The conjecture either trivially holds for the given scopes or lies outside Nitpick's supported fragment; only
 potentially spurious counterexamples may be found \\[2\smallskipamount]
 Nitpick found a potentially spurious counterexample for \textit{card nat}~= 50: \\[2\smallskipamount]
 \hbox{}\qquad Empty assignment \\[2\smallskipamount]
 Nitpick could not find a better counterexample
 It checked 1 of 1 scope \\[2\smallskipamount]
 Total time: 1.62 s.
 \postw
 
 No genuine counterexample is possible because Nitpick cannot rule out the
 existence of a natural number $n \ge 50$ 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} ``$\exists n \mathbin{\le} 49.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}] \\[2\smallskipamount]
 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
 \hbox{}\qquad Empty assignment
 \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 Constant: \nopagebreak \\
 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\unkef(\!\begin{aligned}[t]
 & 0 := \unkef(0 := \textit{True},\, 2 := \textit{True}),\, \\[-2pt]
 & 1 := \unkef(0 := \textit{True},\, 2 := \textit{True},\, 4 := \textit{True}),\, \\[-2pt]
 & 2 := \unkef(0 := \textit{True},\, 2 := \textit{True},\, 4 := \textit{True},\, \\[-2pt]
 & \phantom{2 := \unkef(}6 := \textit{True},\, 8 := \textit{True}))\end{aligned}$ \\[2\smallskipamount]
 Total time: 1.87 s.
 \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.
 The predicate $\textit{even}'$ evaluates to either \textit{True} or $\unk$,
 never \textit{False}.
 
 %Some values are marked with superscripted question
 %marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
 %predicate evaluates to $\unk$.
 
 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, 20, 24, and 28
 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 Free variable: \nopagebreak \\
 \hbox{}\qquad\qquad $n = 1$ \\
 \hbox{}\qquad Constants: \nopagebreak \\
 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\unkef(\!\begin{aligned}[t]
 & 0 := \unkef(0 := \mathit{True},\, 2 := \mathit{True}))\end{aligned}$  \\
 \hbox{}\qquad\qquad $\textit{even}' \leq \unkef(\!\begin{aligned}[t]
 & 0 := \mathit{True},\, 1 := \mathit{False},\, 2 := \mathit{True},\, \\[-2pt]
 & 4 := \mathit{True},\, 6 := \mathit{True},\, 8 := \mathit{True})\end{aligned}$
 \postw
 
 Notice the special constraint $\textit{even}' \leq \ldots$ 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} ``$\textit{nats} = (\lambda n.\; n \mathbin\in \{0, 1, 2, 3, 4\})$'' \\
 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
 \slshape Nitpick found a counterexample:
 \\[2\smallskipamount]
 \hbox{}\qquad Constants: \nopagebreak \\
 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \unkef(0 := \unkef,\, 1 := \unkef,\, 2 := \unkef)$ \\
 \hbox{}\qquad\qquad $\textit{nats} \geq \unkef(3 := \textit{True},\, 4 := \textit{False},\, 5 := \textit{True})$
 \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} = 4,\, \textit{show\_consts}] \\[2\smallskipamount]
 \slshape Nitpick found a counterexample:
 \\[2\smallskipamount]
 \hbox{}\qquad Free variable: \nopagebreak \\
 \hbox{}\qquad\qquad $n = 1$ \\
 \hbox{}\qquad Constants: \nopagebreak \\
 \hbox{}\qquad\qquad $\textit{even} = \unkef(0 := True, 1 := False, 2 := True, 3 := False)$ \\
 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = {}$ \\
 \hbox{}\qquad\qquad\quad $\unkef(0 := \textit{False},\, 1 := \textit{True},\, 2 := \textit{False},\, 3 := \textit{False})$ \\
 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \unkef$\\
 \hbox{}\qquad\qquad\quad $(
 \!\begin{aligned}[t]
 & 0 := \unkef(0 := \textit{True},\, 1 := \textit{False},\, 2 := \textit{True},\, 3 := \textit{False}), \\[-2pt]
 & 1 := \unkef(0 := \textit{False},\, 1 := \textit{True},\, 2 := \textit{False},\, 3 := \textit{True}), \\[-2pt]
 & 2 := \unkef(0 := \textit{False},\, 1 := \textit{False},\, 2 := \textit{True},\, 3 := \textit{False}), \\[-2pt]
 & 3 := \unkef(0 := \textit{False},\, 1 := \textit{False},\, 2 := \textit{False},\, 3 := \textit{True}))
 \end{aligned}$ \\
 \hbox{}\qquad\qquad $\textit{odd} \leq \unkef(0 := \textit{False},\, 1 := \textit{True},\, 2 := \textit{False},\, 3 := \textit{True})$
 \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 and 3. 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
 you can disable it by passing the \textit{dont\_star\_linear\_preds} option.
 
 \subsection{Coinductive Datatypes}
 \label{coinductive-datatypes}
 
 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 coinductive lists, or ``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{codatatype} $'a$ \textit{llist} = \textit{LNil}~$\mid$~\textit{LCons}~$'a$~``$'a\;\textit{llist}$'' \\[2\smallskipamount]
 \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 Free variables: \nopagebreak \\
 \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{primcorec}~$\textit{iterates}$~\textbf{where} \\
 ``$\textit{iterates}~f\>a = \textit{LCons}~a~(\textit{iterates}~f\>(f\>a))$'' \\[2\smallskipamount]
 \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 $'a$ passed the monotonicity test; Nitpick might be able to skip
 some scopes \\[2\smallskipamount]
 Trying 10 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} $'a$~= 10, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 10,
 and \textit{bisim\_depth}~= 9 \\[2\smallskipamount]
 Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
 \textit{card}~``\kern1pt$'a~\textit{llist\/}$''~= 2, and \textit{bisim\_\allowbreak
 depth}~= 1:
 \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\
 \hbox{}\qquad\qquad $\textit{b} = a_2$ \\
 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
 Total time: 1.11 s
 \postw
 
 The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
 $\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
 $[a_2]$ as its stem and $[a_1]$ 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 two
 lazy lists
 %
 \begin{gather*}
 \textrm{THE}~\omega.\; \omega = \textit{LCons}~a~(\textit{LCons}~b~\omega) \\
 \textit{LCons}~a~(\textrm{THE}~\omega.\; \omega = \textit{LCons}~b~(\textit{LCons}~a~\omega))
 \end{gather*}
 %
 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 ``quasi genuine'' and Nitpick recommends to try
 again with \textit{bisim\_depth} set to a nonnegative integer.
 
 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\_types}] \\[2\smallskipamount]
 \slshape Nitpick found a quasi genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
 \hbox{}\qquad Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $a = a_1$ \\
 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
 \textit{LCons}~a_1~\omega$ \\
 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
 \hbox{}\qquad Type:\strut \nopagebreak \\
 \hbox{}\qquad\qquad $'a~\textit{llist} =
 \{\!\begin{aligned}[t]
   & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
   & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
 \\[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, prompting
 Nitpick to label the example as only ``quasi genuine.''
 
 A compromise between leaving out the bisimilarity predicate from the Kodkod
 problem and performing the after-the-fact check is to specify a low
 nonnegative \textit{bisim\_depth} value. In general, a value of $K$ means that
 Nitpick will require all lists to be distinguished from each other by their
 prefixes of length $K$. However, 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 10 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}~= 10, \textit{card tm}~= 10, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm\/}$'' = 10 \\[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 Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $\sigma = \unkef(\!\begin{aligned}[t]
 & 0 := \textit{Var}~0,\>
   1 := \textit{Var}~0,\>
   2 := \textit{Var}~0, \\[-2pt]
 & 3 := \textit{Var}~0,\>
   4 := \textit{Var}~0,\>
   5 := \textit{Lam}~(\textit{Lam}~(\textit{Var}~0)))\end{aligned}$ \\
 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
 Total time: 3.08 s
 \postw
 
 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
 $\lambda$-calculus notation, $t$ is
 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is (wrongly) $\lambda x\, y.\> y$.
 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 10 to the type $\textit{nat} \Rightarrow \textit{tm}$
 of the higher-order argument $\sigma$ of \textit{subst}.
 For the formula of interest, knowing 6 values of that type was enough to find
 the counterexample. Without boxing, $6^6 = 46\,656$ values must be
 considered, a hopeless undertaking:
 
 \prew
 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
 {\slshape Nitpick ran out of time after checking 3 of 10 scopes}
 \postw
 
 Boxing can be enabled or disabled globally or on a per-type basis using the
 \textit{box} option. Nitpick usually performs reasonable choices about which
 types should be boxed, but option tweaking sometimes helps.
 
 %A related optimization,
 %``finitization,'' attempts to wrap functions that are constant at all but finitely
 %many points (e.g., finite sets); see the documentation for the \textit{finitize}
 %option in \S\ref{scope-of-search} for details.
 
 \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--10, no fewer than $10^4 = 10\,000$ 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 $1\,000$ scopes to
 exhaust the specification \textit{card}~= 1--10 (10 cardinalies for $'a$
 $\times$ 10 cardinalities for $'b$ $\times$ 10 cardinalities for the datatypes).
 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 $'a$ and $'b$ passed the monotonicity test; Nitpick might be able to skip some scopes.
  \\[2\smallskipamount]
 Trying 10 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} $'a$~= 10, \textit{card} $'b$~= 10,
 \textit{card} \textit{nat}~= 10, \textit{card} ``$('a \times {'}b)$
 \textit{list\/}''~= 10, \\
 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 10, and
 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 10
 \\[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 Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
 \hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
 Total time: 1.63 s.
 \postw
 
 In theory, it should be sufficient to test a single scope:
 
 \prew
 \textbf{nitpick}~[\textit{card}~= 10]
 \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}, and \textit{real}. Thus, given the
 cardinality specification 1--10, a formula involving \textit{nat}, \textit{int},
 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
 consider only 10~scopes instead of $10^4 = 10\,000$. On the other hand,
 \textbf{typedef}s and quotient types are generally nonmonotonic.
 
 \subsection{Inductive Properties}
 \label{inductive-properties}
 
 Inductive properties are a particular pain to prove, because the failure to
 establish an induction step can mean several things:
 %
 \begin{enumerate}
 \item The property is invalid.
 \item The property is valid but is too weak to support the induction step.
 \item The property is valid and strong enough; it's just that we haven't found
 the proof yet.
 \end{enumerate}
 %
 Depending on which scenario applies, we would take the appropriate course of
 action:
 %
 \begin{enumerate}
 \item Repair the statement of the property so that it becomes valid.
 \item Generalize the property and/or prove auxiliary properties.
 \item Work harder on a proof.
 \end{enumerate}
 %
 How can we distinguish between the three scenarios? Nitpick's normal mode of
 operation can often detect scenario 1, and Isabelle's automatic tactics help with
 scenario 3. Using appropriate techniques, it is also often possible to use
 Nitpick to identify scenario 2. Consider the following transition system,
 in which natural numbers represent states:
 
 \prew
 \textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
 ``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
 ``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
 ``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
 \postw
 
 We will try to prove that only even numbers are reachable:
 
 \prew
 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
 \postw
 
 Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
 so let's attempt a proof by induction:
 
 \prew
 \textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
 \textbf{apply}~\textit{auto}
 \postw
 
 This leaves us in the following proof state:
 
 \prew
 {\slshape goal (2 subgoals): \\
 \phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
 \phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
 }
 \postw
 
 If we run Nitpick on the first subgoal, it still won't find any
 counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
 is helpless. However, notice the $n \in \textit{reach}$ assumption, which
 strengthens the induction hypothesis but is not immediately usable in the proof.
 If we remove it and invoke Nitpick, this time we get a counterexample:
 
 \prew
 \textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
 \textbf{nitpick} \\[2\smallskipamount]
 \slshape Nitpick found a counterexample: \\[2\smallskipamount]
 \hbox{}\qquad Skolem constant: \nopagebreak \\
 \hbox{}\qquad\qquad $n = 0$
 \postw
 
 Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
 to strength the lemma:
 
 \prew
 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
 \postw
 
 Unfortunately, the proof by induction still gets stuck, except that Nitpick now
 finds the counterexample $n = 2$. We generalize the lemma further to
 
 \prew
 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
 \postw
 
 and this time \textit{arith} can finish off the subgoals.
 
 \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]
 \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 strings 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 Free variable: \nopagebreak \\
 \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 Free variable: \nopagebreak \\
 \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 Free variable: \nopagebreak \\
 \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{aa\_tree} = \\
 \hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder\/}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}''  \\[2\smallskipamount]
 \textbf{primrec} \textit{data} \textbf{where} \\
 ``$\textit{data}~\Lambda = \unkef$'' $\,\mid$ \\
 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[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 9 of 10 scopes}
 \postw
 
 Furthermore, applying \textit{skew} or \textit{split} on 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 Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $t = N~a_1~1~\Lambda~\Lambda$ \\
 \hbox{}\qquad\qquad $x = a_2$
 \postw
 
 It's hard to see why this is a counterexample. 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 Free variables: \nopagebreak \\
 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
 \hbox{}\qquad\qquad $x = 0$ \\
 \hbox{}\qquad Evaluated term: \\
 \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 nodes 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 8 of 10 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 7 of 10 scopes}
 \postw
 
 We could continue like this and sketch a full-blown theory of AA trees. 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\defl{\{}
 \def\defr{\}}
 
 \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}$\bigr]$\enskip \defl\textit{true}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\enskip \defl\textit{false}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\enskip \defl\textit{smart}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\enskip \defl\textit{#3}\defr} \nopagebreak\\[\parskip]}
 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\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]
 \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}), regression testing (\S\ref{regression-testing}),
 optimizations (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
 
 Nitpick also provides an automatic mode that can be enabled via the
 ``Auto Nitpick'' option under ``Plugins > Plugin Options > Isabelle > General''
 in Isabelle/jEdit. For automatic runs,
 \textit{user\_axioms} (\S\ref{mode-of-operation}),
 \textit{assms} (\S\ref{mode-of-operation}), and \textit{mono}
 (\S\ref{scope-of-search}) are implicitly enabled,
 \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, \textit{max\_threads}
 (\S\ref{optimizations}) is taken to be 1, and \textit{timeout}
 (\S\ref{timeouts}) is superseded by the ``Auto Time Limit'' in
 Isabelle/jEdit. Nitpick's output is also more concise.
 
 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[\labelitemi] \qtybf{string}: A string.
 \item[\labelitemi] \qtybf{string\_list\/}: A space-separated list of strings
 (e.g., ``\textit{ichi ni san}'').
 \item[\labelitemi] \qtybf{bool\/}: \textit{true} or \textit{false}.
 \item[\labelitemi] \qtybf{smart\_bool\/}: \textit{true}, \textit{false}, or \textit{smart}.
 \item[\labelitemi] \qtybf{int\/}: An integer. Negative integers are prefixed with a hyphen.
 \item[\labelitemi] \qtybf{smart\_int\/}: An integer or \textit{smart}.
 \item[\labelitemi] \qtybf{int\_range}: An integer (e.g., 3) or a range
 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' is entered as \texttt{-} (hyphen).
 \item[\labelitemi] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
 \item[\labelitemi] \qtybf{float}: An floating-point number (e.g., 0.5 or 60)
 expressing a number of seconds.
 \item[\labelitemi] \qtybf{const\/}: The name of a HOL constant.
 \item[\labelitemi] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
 \item[\labelitemi] \qtybf{term\_list\/}: A space-separated list of HOL terms (e.g.,
 ``$f~x$''~``$g~y$'').
 \item[\labelitemi] \qtybf{type}: A HOL type.
 \end{enum}
 
 Default values are indicated in curly brackets (\textrm{\{\}}). Boolean options
 have a negated counterpart (e.g., \textit{mono} vs.\
 \textit{non\_mono}). When setting them, ``= \textit{true}'' may be omitted.
 
 \subsection{Mode of Operation}
 \label{mode-of-operation}
 
 \begin{enum}
 \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 ``quasi genuine.'' The \textit{debug}
 (\S\ref{output-format}) option can be used to find out which axioms were
 considered.
 
 \nopagebreak
 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
 (\S\ref{output-format}).}
 
 \optrue{assms}{no\_assms}
 Specifies whether the relevant assumptions in structured proofs should be
 considered. The option is implicitly enabled for automatic runs.
 
 \nopagebreak
 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
 
 \opfalse{spy}{dont\_spy}
 Specifies whether Nitpick should record statistics in
 \texttt{\$ISA\-BELLE\_\allowbreak HOME\_\allowbreak USER/\allowbreak spy\_\allowbreak nitpick}.
 These statistics can be useful to the developer of Nitpick. If you are willing to have your
 interactions recorded in the name of science, please enable this feature and send the statistics
 file every now and then to the author of this manual (\authoremail).
 To change the default value of this option globally, set the environment variable
 \texttt{NITPICK\_SPY} to \texttt{yes}.
 
 \nopagebreak
 {\small See also \textit{debug} (\S\ref{output-format}).}
 
 \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. The files are identified by the extensions
 \texttt{.kki}, \texttt{.cnf}, \texttt{.out}, and
 \texttt{.err}; you may safely remove them after Nitpick has run.
 
 \textbf{Warning:} This option is not thread-safe. Use at your own risks.
 
 \nopagebreak
 {\small See also \textit{debug} (\S\ref{output-format}).}
 \end{enum}
 
 \subsection{Scope of Search}
 \label{scope-of-search}
 
 \begin{enum}
 \oparg{card}{type}{int\_seq}
 Specifies the sequence of cardinalities to use for a given type.
 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{$1$--$n$}.
 
 \nopagebreak
 {\small See also \textit{box} (\S\ref{scope-of-search}) and \textit{mono}
 (\S\ref{scope-of-search}).}
 
 \opdefault{card}{int\_seq}{\upshape 1--10}
 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.
 
 \oparg{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.
 
 \opnodefault{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.
 
 \opsmart{binary\_ints}{unary\_ints}
 Specifies whether natural numbers and integers should be encoded using a unary
 or binary notation. In unary mode, 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:
 %
 $$\hbox{\begin{tabular}{@{}rll@{}}%
 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
 \end{tabular}}$$
 %
 In binary mode, the cardinality specifies the number of distinct values that can
 be constructed. Each of these value is represented by a bit pattern whose length
 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
 Nitpick attempts to choose the more appropriate encoding by inspecting the
 formula at hand, preferring the binary notation for problems involving
 multiplicative operators or large constants.
 
 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
 problems that refer to the types \textit{rat} or \textit{real} or the constants
 \textit{Suc}, \textit{gcd}, or \textit{lcm}.
 
 {\small See also \textit{bits} (\S\ref{scope-of-search}) and
 \textit{show\_types} (\S\ref{output-format}).}
 
 \opdefault{bits}{int\_seq}{\upshape 1--10}
 Specifies the number of bits to use to represent natural numbers and integers in
 binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
 
 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
 
 \opargboolorsmart{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[\labelitemi] \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 ``quasi
 genuine.''
 
 \item[\labelitemi] \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[\labelitemi] \textbf{\textit{smart}:} Try to prove that the inductive
 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
 \textit{size\_change} 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.
 
 \oparg{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}).}
 
 \opdefault{iter}{int\_seq}{\upshape 0{,}1{,}2{,}4{,}8{,}12{,}16{,}20{,}24{,}28}
 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.
 
 \opdefault{bisim\_depth}{int\_seq}{\upshape 9}
 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 ``quasi genuine.'' The iteration counts are automatically bounded by
 the sum of the cardinalities of the coinductive datatypes occurring in the
 formula to falsify.
 
 \opargboolorsmart{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[\labelitemi] \textbf{\textit{true}:} Box the specified type whenever
 practicable.
 \item[\labelitemi] \textbf{\textit{false}:} Never box the type.
 \item[\labelitemi] \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}
 
 \nopagebreak
 {\small See also \textit{finitize} (\S\ref{scope-of-search}), \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.
 
 \opargboolorsmart{finitize}{type}{dont\_finitize}
 Specifies whether Nitpick should attempt to finitize an infinite datatype. The
 option can then take the following values:
 
 \begin{enum}
 \item[\labelitemi] \textbf{\textit{true}:} Finitize the datatype. Since this is
 unsound, counterexamples generated under these conditions are tagged as ``quasi
 genuine.''
 \item[\labelitemi] \textbf{\textit{false}:} Don't attempt to finitize the datatype.
 \item[\labelitemi] \textbf{\textit{smart}:}
 If the datatype's constructors don't appear in the problem, perform a
 monotonicity analysis to detect whether the datatype can be soundly finitized;
 otherwise, don't finitize it.
 \end{enum}
 
 \nopagebreak
 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{mono}
 (\S\ref{scope-of-search}), \textit{verbose} (\S\ref{output-format}), and
 \textit{debug} (\S\ref{output-format}).}
 
 \opsmart{finitize}{dont\_finitize}
 Specifies the default finitization setting to use. This can be overridden on a
 per-type basis using the \textit{finitize}~\qty{type} option described above.
 
 \opargboolorsmart{mono}{type}{non\_mono}
 Specifies whether the given type should be considered monotonic when enumerating
 scopes and finitizing types. 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 can also diminish the chance of
 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}. The
 option is implicitly set to \textit{true} for automatic runs.
 
 \nopagebreak
 {\small See also \textit{card} (\S\ref{scope-of-search}),
 \textit{finitize} (\S\ref{scope-of-search}),
 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
 (\S\ref{output-format}).}
 
 \opsmart{mono}{non\_mono}
 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.
 
 \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{spy} (\S\ref{mode-of-operation}),
 \textit{overlord} (\S\ref{mode-of-operation}), and
 \textit{batch\_size} (\S\ref{optimizations}).}
 
 \opfalse{show\_types}{hide\_types}
 Specifies whether the subsets used to approximate (co)in\-duc\-tive data\-types should
 be displayed as part of counterexamples. Such subsets are sometimes helpful when
 investigating whether a potentially spurious counterexample is genuine, but
 their potential for clutter is real.
 
 \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.
 
 \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.
 
 \opnodefault{show\_all}{bool}
 Abbreviation for \textit{show\_types}, \textit{show\_skolems}, and
 \textit{show\_consts}.
 
 \opdefault{max\_potential}{int}{\upshape 1}
 Specifies the maximum number of potentially spurious 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, such as
 \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that many of
 the counterexamples may be identical.
 
 \nopagebreak
 {\small See also \textit{sat\_solver} (\S\ref{optimizations}).}
 
 \opdefault{max\_genuine}{int}{\upshape 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,
 such as \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that
 many of the counterexamples may be identical.
 
 \nopagebreak
 {\small See also \textit{sat\_solver} (\S\ref{optimizations}).}
 
 \opnodefault{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.
 
 \oparg{atoms}{type}{string\_list}
 Specifies the names to use to refer to the atoms of the given type. By default,
 Nitpick generates names of the form $a_1, \ldots, a_n$, where $a$ is the first
 letter of the type's name.
 
 \opnodefault{atoms}{string\_list}
 Specifies the default names to use to refer to atoms of any type. For example,
 to call the three atoms of type ${'}a$ \textit{ichi}, \textit{ni}, and
 \textit{san} instead of $a_1$, $a_2$, $a_3$, specify the option
 ``\textit{atoms}~${'}a$ = \textit{ichi~ni~san}''. The default names can be
 overridden on a per-type basis using the \textit{atoms}~\qty{type} option
 described above.
 
 \oparg{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 $1,\ldots,1,n_1,\ldots,n_k$.
 
 \opdefault{format}{int\_seq}{\upshape 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}
 
 \subsection{Regression Testing}
 \label{regression-testing}
 
 \begin{enum}
 \opnodefault{expect}{string}
 Specifies the expected outcome, which must be one of the following:
 
 \begin{enum}
 \item[\labelitemi] \textbf{\textit{genuine}:} Nitpick found a genuine counterexample.
 \item[\labelitemi] \textbf{\textit{quasi\_genuine}:} Nitpick found a ``quasi
 genuine'' counterexample (i.e., a counterexample that is genuine unless
 it contradicts a missing axiom or a dangerous option was used inappropriately).
 \item[\labelitemi] \textbf{\textit{potential}:} Nitpick found a potentially
 spurious counterexample.
 \item[\labelitemi] \textbf{\textit{none}:} Nitpick found no counterexample.
 \item[\labelitemi] \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}
 \opdefault{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{MiniSat\_JNI}
 (recommended) or \textit{SAT4J}.
 
 The supported solvers are listed below:
 
 \begin{enum}
 
 \item[\labelitemi] \textbf{\textit{Lingeling\_JNI}:}
 Lingeling is an efficient solver written in C. The JNI (Java Native Interface)
 version of Lingeling is bundled with Kodkodi and is precompiled for Linux and
 Mac~OS~X. It is also available from the Kodkod web site \cite{kodkod-2009}.
 
 \item[\labelitemi] \textbf{\textit{CryptoMiniSat}:} CryptoMiniSat is the winner of
 the 2010 SAT Race. To use CryptoMiniSat, set the environment variable
 \texttt{CRYPTO\-MINISAT\_}\discretionary{}{}{}\texttt{HOME} to the directory that contains the \texttt{crypto\-minisat}
 executable.%
-\footnote{Important note for Cygwin users: The path must be specified using
-native Windows syntax. Make sure to escape backslashes properly.%
-\label{cygwin-paths}}
 The \cpp{} sources and executables for Crypto\-Mini\-Sat are available at
 \url{http://planete.inrialpes.fr/~soos/}\allowbreak\url{CryptoMiniSat2/index.php}.
 Nitpick has been tested with version 2.51.
 
 \item[\labelitemi] \textbf{\textit{CryptoMiniSat\_JNI}:} The JNI (Java Native
 Interface) version of CryptoMiniSat is bundled with Kodkodi and is precompiled
 for Linux and Mac~OS~X. It is also available from the Kodkod web site
 \cite{kodkod-2009}.
 
 \item[\labelitemi] \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.%
-\footref{cygwin-paths}
 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 2.2.
 
 \item[\labelitemi] \textbf{\textit{MiniSat\_JNI}:} The JNI
 version of MiniSat is bundled with Kodkodi and is precompiled for Linux,
 Mac~OS~X, and Windows (Cygwin). It is also available from the Kodkod web site
 \cite{kodkod-2009}. Unlike the standard version of MiniSat, the JNI version can
 be used incrementally.
 
 \item[\labelitemi] \textbf{\textit{Riss3g}:} Riss3g is an efficient solver written in
 \cpp{}. To use Riss3g, set the environment variable \texttt{RISS3G\_HOME} to the
 directory that contains the \texttt{riss3g} executable.%
-\footref{cygwin-paths}
 The \cpp{} sources for Riss3g are available at
 \url{http://tools.computational-logic.org/content/riss3g.php}.
 Nitpick has been tested with the SAT Competition 2013 version.
 
 \item[\labelitemi] \textbf{\textit{zChaff}:} zChaff is an older solver written
 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
 the directory that contains the \texttt{zchaff} executable.%
-\footref{cygwin-paths}
 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[\labelitemi] \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.%
-\footref{cygwin-paths}
 The \cpp{} sources for RSat are available at
 \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been tested with version
 2.01.
 
 \item[\labelitemi] \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.\footref{cygwin-paths}
+executable.
 The BerkMin executables are available at
 \url{http://eigold.tripod.com/BerkMin.html}.
 
 \item[\labelitemi] \textbf{\textit{BerkMin\_Alloy}:} 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.%
-\footref{cygwin-paths}
 
 \item[\labelitemi] \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[\labelitemi] \textbf{\textit{SAT4J\_Light}:} Variant of SAT4J that is
 optimized for small problems. It can also be used incrementally.
 
 \item[\labelitemi] \textbf{\textit{smart}:} If \textit{sat\_solver} is set to
 \textit{smart}, Nitpick selects the first solver among the above that is
 recognized by Isabelle. If \textit{verbose} (\S\ref{output-format}) is enabled,
 Nitpick displays which SAT solver was chosen.
 \end{enum}
 \fussy
 
 \opdefault{batch\_size}{smart\_int}{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 50 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{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}).}
 
 \opnodefault{whack}{term\_list}
 Specifies a list of atomic terms (usually constants, but also free and schematic
 variables) that should be taken as being $\unk$ (unknown). This can be useful to
 reduce the size of the Kodkod problem if you can guess in advance that a
 constant might not be needed to find a countermodel.
 
 {\small See also \textit{debug} (\S\ref{output-format}).}
 
 \opnodefault{need}{term\_list}
 Specifies a list of datatype values (normally ground constructor terms) that
 should be part of the subterm-closed subsets used to approximate datatypes. If
 you know that a value must necessarily belong to the subset of representable
 values that approximates a datatype, specifying it can speed up the search,
 especially for high cardinalities.
 %By default, Nitpick inspects the conjecture to infer needed datatype values.
 
 \opsmart{total\_consts}{partial\_consts}
 Specifies whether constants occurring in the problem other than constructors can
 be assumed to be considered total for the representable values that approximate
 a datatype. This option is highly incomplete; it should be used only for
 problems that do not construct datatype values explicitly. Since this option is
 (in rare cases) unsound, counterexamples generated under these conditions are
 tagged as ``quasi genuine.''
 
 \opdefault{datatype\_sym\_break}{int}{\upshape 5}
 Specifies an upper bound on the number of datatypes for which Nitpick generates
 symmetry breaking predicates. Symmetry breaking can speed up the SAT solver
 considerably, especially for unsatisfiable problems, but too much of it can slow
 it down.
 
 \opdefault{kodkod\_sym\_break}{int}{\upshape 15}
 Specifies an upper bound on the number of relations for which Kodkod generates
 symmetry breaking predicates. Symmetry breaking can speed up the SAT solver
 considerably, especially for unsatisfiable problems, but too much of it can slow
 it down.
 
 \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.
 
 \opdefault{max\_threads}{int}{\upshape 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. The option is implicitly set to 1 for automatic runs.
 
 \nopagebreak
 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and
 \textit{timeout} (\S\ref{timeouts}).}
 \end{enum}
 
 \subsection{Timeouts}
 \label{timeouts}
 
 \begin{enum}
 \opdefault{timeout}{float}{\upshape 30}
 Specifies the maximum number of seconds 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, the ``Auto Time
 Limit'' option under ``Plugins > Plugin Options > Isabelle > General'' is used
 instead.
 
 \nopagebreak
 {\small See also \textit{max\_threads} (\S\ref{optimizations}).}
 
 \opdefault{tac\_timeout}{float}{\upshape 0.5}
 Specifies the maximum number of seconds that should be used by internal
 tactics---\textit{lexicographic\_order} and \textit{size\_change} when checking
 whether a (co)in\-duc\-tive predicate is well-founded or the monotonicity
 inference. Nitpick tries to honor this constraint but offers no guarantees.
 
 \nopagebreak
 {\small See also \textit{wf} (\S\ref{scope-of-search}) and
 \textit{mono} (\S\ref{scope-of-search}).}
 \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{enum}
 \flushitem{\textit{nitpick\_unfold}}
 
 \nopagebreak
 This attribute specifies an equation that Nitpick should use to expand a
 constant. The equation should be logically equivalent to the constant's actual
 definition and should be of the form
 
 \qquad $c~{?}x_1~\ldots~{?}x_n \,=\, t$,
 
 or
 
 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
 
 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
 $t$. Each occurrence of $c$ in the problem is expanded to $\lambda x_1\,\ldots
 x_n.\; t$.
 
 \flushitem{\textit{nitpick\_simp}}
 
 \nopagebreak
 This attribute specifies the equations that constitute the specification of a
 constant. The \textbf{primrec}, \textbf{function}, and
 \textbf{nominal\_\allowbreak primrec} packages automatically attach this
 attribute to their \textit{simps} rules. The equations must be of the form
 
 \qquad $c~t_1~\ldots\ t_n \;\bigl[{=}\; u\bigr]$
 
 or
 
 \qquad $c~t_1~\ldots\ t_n \,\equiv\, u.$
 
 \flushitem{\textit{nitpick\_psimp}}
 
 \nopagebreak
 This attribute specifies the equations that constitute the partial specification
 of a constant. The \textbf{function} package automatically attaches this
 attribute to its \textit{psimps} rules. The conditional equations must be of the
 form
 
 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \;\bigl[{=}\; u\bigr]$
 
 or
 
 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,\equiv\, u$.
 
 \flushitem{\textit{nitpick\_choice\_spec}}
 
 \nopagebreak
 This attribute specifies the (free-form) specification of a constant defined
 using the \textbf{specification} command.
 \end{enum}
 
 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, if the constant was defined using the
 \allowbreak\textbf{specification} command and the
 \textit{nitpick\_choice\_spec} set associated with the constant is not empty, it
 uses these theorems as the specification of the constant.
 
 \item[4.] Otherwise, it looks up the definition of the constant. If the
 \textit{nitpick\_unfold} 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.
 
 \begin{enum}
 \item[1.] If the definition is of the form
 
 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$
 
 or
 
 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{gfp}~(\lambda f.\; t).$
 
 Nitpick assumes that the definition was made using a (co)inductive package
 based on the user-specified introduction rules registered in Isabelle's internal
 \textit{Spec\_Rules} table. The tool uses the introduction rules to ascertain
 whether the definition is well-founded and the definition to generate a
 fixed-point equation or an unrolled equation.
 
 \item[2.] If the definition is compact enough, the constant is \textsl{unfolded}
 wherever it appears; otherwise, it is defined equationally, as with
 the \textit{nitpick\_simp} attribute.
 \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
 
 By default, Nitpick uses the \textit{lfp}-based definition in conjunction with
 the introduction rules. To override this, you can specify an alternative
 definition as follows:
 
 \prew
 \textbf{lemma} $\mathit{odd\_alt\_unfold}$ [\textit{nitpick\_unfold}]:\kern.4em ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
 \postw
 
 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
 = 1$. Alternatively, you can specify an equational specification of the constant:
 
 \prew
 \textbf{lemma} $\mathit{odd\_simp}$ [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
 \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.
 
 Because of its internal three-valued logic, Nitpick tends to lose a
 lot of precision in the presence of partially specified constants. For example,
 
 \prew
 \textbf{lemma} \textit{odd\_simp} [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd~x} = \lnot\, \textit{even}~x$''
 \postw
 
 is superior to
 
 \prew
 \textbf{lemma} \textit{odd\_psimps} [\textit{nitpick\_simp}]: \\
 ``$\textit{even~x} \,\Longrightarrow\, \textit{odd~x} = \textit{False\/}$'' \\
 ``$\lnot\, \textit{even~x} \,\Longrightarrow\, \textit{odd~x} = \textit{True\/}$''
 \postw
 
 Because Nitpick sometimes unfolds definitions but never simplification rules,
 you can ensure that a constant is defined explicitly using the
 \textit{nitpick\_simp}. For example:
 
 \prew
 \textbf{definition}~\textit{optimum} \textbf{where} [\textit{nitpick\_simp}]: \\
 ``$\textit{optimum}~t =
      (\forall u.\; \textit{consistent}~u \mathrel{\land} \textit{alphabet}~t = \textit{alphabet}~u$ \\
 \phantom{``$\textit{optimum}~t = (\forall u.\;$}${\mathrel{\land}}\; \textit{freq}~t = \textit{freq}~u \longrightarrow
          \textit{cost}~t \le \textit{cost}~u)$''
 \postw
 
 In some rare occasions, you might want to provide an inductive or coinductive
 view on top of an existing constant $c$. The easiest way to achieve this is to
 define a new constant $c'$ (co)inductively. Then prove that $c$ equals $c'$
 and let Nitpick know about it:
 
 \prew
 \textbf{lemma} \textit{c\_alt\_unfold} [\textit{nitpick\_unfold}]:\kern.4em ``$c \equiv c'$\kern2pt ''
 \postw
 
 This ensures that Nitpick will substitute $c'$ for $c$ and use the (co)inductive
 definition.
 
 \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 term postprocessors as well as coinductive
 datatypes.
 
 \subsection{Invoking Nitpick}
 \label{invoking-nitpick}
 
 The \textit{Nitpick} structure offers the following functions for invoking your
 favorite counterexample generator:
 
 \prew
 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{mode}
 \rightarrow \textit{int} \rightarrow \textit{int} \rightarrow \textit{int}$ \\
 $\hbox{}\quad{\rightarrow}\; (\textit{term} * \textit{term})~\textit{list}
 \rightarrow \textit{term~list} \rightarrow \textit{term} \rightarrow \textit{string} * \textit{Proof.state}$ \\
 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{mode} \rightarrow \textit{int} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
 \postw
 
 The return value is a new proof state paired with an outcome string
 (``genuine'', ``quasi\_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 where Nitpick is invoked
 on subgoal $i$ of $n$ with no time limit:
 
 \prew
 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout\/}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = {}$ \\
 $\hbox{}\quad\textit{Nitpick.pick\_nits\_in\_subgoal}~\textit{state}~\textit{params}~\textit{Nitpick.Normal}~\textit{i}~\textit{n}$
 \postw
 
 \let\antiq=\textrm
 
 \subsection{Registering Term Postprocessors}
 \label{registering-term-postprocessors}
 
 It is possible to change the output of any term that Nitpick considers a
 datatype by registering a term postprocessor. The interface for registering and
 unregistering postprocessors consists of the following pair of functions defined
 in the \textit{Nitpick\_Model} structure:
 
 \prew
 $\textbf{type}\,~\textit{term\_postprocessor}\,~{=} {}$ \\
 $\hbox{}\quad\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \rightarrow \textit{term}$ \\
 $\textbf{val}\,~\textit{register\_term\_postprocessor} : {}$ \\
 $\hbox{}\quad\textit{typ} \rightarrow \textit{term\_postprocessor} \rightarrow \textit{morphism} \rightarrow \textit{Context.generic}$ \\
 $\hbox{}\quad{\rightarrow}\; \textit{Context.generic}$ \\
 $\textbf{val}\,~\textit{unregister\_term\_postprocessor} : {}$ \\
 $\hbox{}\quad\textit{typ} \rightarrow \textit{morphism} \rightarrow \textit{Context.generic} \rightarrow \textit{Context.generic}$
 \postw
 
 \S\ref{typedefs-quotient-types-records-rationals-and-reals} and
 \texttt{src/HOL/Library/Multiset.thy} illustrate this feature in context.
 
 \subsection{Registering Coinductive Datatypes}
 \label{registering-coinductive-datatypes}
 
 Coinductive datatypes defined using the \textbf{codatatype} command that do not
 involve nested recursion through non-codatatypes are supported by Nitpick.
 If you have defined a custom coinductive datatype, you can tell Nitpick about
 it, so that it can use an efficient Kodkod axiomatization. The interface for
 registering and unregistering coinductive datatypes consists of the following
 pair of functions defined in the \textit{Nitpick\_HOL} structure:
 
 \prew
 $\textbf{val}\,~\textit{register\_codatatype\/} : {}$ \\
 $\hbox{}\quad\textit{morphism} \rightarrow \textit{typ} \rightarrow \textit{string} \rightarrow (\textit{string} \times \textit{typ})\;\textit{list} \rightarrow \textit{Context.generic} {}$ \\
 $\hbox{}\quad{\rightarrow}\; \textit{Context.generic}$ \\
 $\textbf{val}\,~\textit{unregister\_codatatype\/} : {}$ \\
 $\hbox{}\quad\textit{morphism} \rightarrow \textit{typ} \rightarrow \textit{Context.generic} \rightarrow \textit{Context.generic} {}$
 \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
 $\textbf{declaration}~\,\{{*}$ \\
 $\hbox{}\quad\textit{Nitpick\_HOL.register\_codatatype}~@\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}$ \\
 $\hbox{}\qquad\quad @\{\antiq{const\_name}~ \textit{llist\_case}\}$ \\
 $\hbox{}\qquad\quad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])$ \\
 ${*}\}$
 \postw
 
 The \textit{register\_codatatype} function takes a coinductive datatype, its
 case function, and the list of its constructors (in addition to the current
 morphism and generic proof context). 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
 $\textbf{declaration}~\,\{{*}$ \\
 $\hbox{}\quad\textit{Nitpick\_HOL.unregister\_codatatype}~@\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}$ \\
 ${*}\}$
 \postw
 
 Inductive datatypes can be registered as coinductive datatypes, given
 appropriate coinductive constructors. However, doing so precludes
 the use of the inductive constructors---Nitpick will generate an error if they
 are needed.
 
 \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[\labelitemi] 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} ``$\textit{prec}~0 = \textit{undefined\/}$'' \\
 \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 generally considered bad style because they rely on the
 internal representation of functions synthesized by Isabelle, an implementation
 detail.
 
 \item[\labelitemi] Similarly, Nitpick might find spurious counterexamples for
 theorems that rely on the use of the indefinite description operator internally
 by \textbf{specification} and \textbf{quot\_type}.
 
 \item[\labelitemi] Axioms or definitions that restrict the possible values of the
 \textit{undefined} constant or other partially specified built-in Isabelle
 constants (e.g., \textit{Abs\_} and \textit{Rep\_} constants) are in general
 ignored. Again, such nonconservative extensions are generally considered bad
 style.
 
 \item[\labelitemi] Nitpick produces spurious counterexamples when invoked after a
 \textbf{guess} command in a structured proof.
 
 \item[\labelitemi] Datatypes defined using \textbf{datatype} and
 codatatypes defined using \textbf{codatatype} that involve nested (co)recursion
 through non-(co)datatypes are not properly supported and may result in spurious
 counterexamples.
 
 \item[\labelitemi] Types that are registered with several distinct sets of
 constructors, including \textit{enat} if the \textit{Coinductive} entry of
 the \textit{Archive of Formal Proofs} is loaded, can confuse Nitpick.
 
 \item[\labelitemi] The \textit{nitpick\_xxx} attributes and the
 \textit{Nitpick\_xxx.register\_yyy} functions can cause havoc if used
 improperly.
 
 \item[\labelitemi] Although this has never been observed, arbitrary theorem
 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples.
 
 \item[\labelitemi] All constants, types, free variables, and schematic variables
 whose names start with \textit{Nitpick}{.} are reserved for internal use.
 
 \item[\labelitemi] Some users report technical issues with the default SAT
 solver on Windows. Setting the \textit{sat\_solver} option
 (\S\ref{optimizations}) to \textit{MiniSat\_JNI} should solve this.
 \end{enum}
 
 \let\em=\sl
 \bibliography{manual}{}
 \bibliographystyle{abbrv}
 
 \end{document}
diff --git a/src/HOL/Tools/Nitpick/kodkod_sat.ML b/src/HOL/Tools/Nitpick/kodkod_sat.ML
--- a/src/HOL/Tools/Nitpick/kodkod_sat.ML
+++ b/src/HOL/Tools/Nitpick/kodkod_sat.ML
@@ -1,115 +1,112 @@
 (*  Title:      HOL/Tools/Nitpick/kodkod_sat.ML
     Author:     Jasmin Blanchette, TU Muenchen
     Copyright   2009, 2010
 
 Kodkod SAT solver integration.
 *)
 
 signature KODKOD_SAT =
 sig
   val configured_sat_solvers : bool -> string list
   val smart_sat_solver_name : bool -> string
   val sat_solver_spec : string -> string * string list
 end;
 
 structure Kodkod_SAT : KODKOD_SAT =
 struct
 
 open Kodkod
 
 datatype sink = ToStdout | ToFile
 datatype availability = Java | JNI of int list
 datatype mode = Batch | Incremental
 
 datatype sat_solver_info =
   Internal of availability * mode * string list |
   External of string * string * string list |
   ExternalV2 of sink * string * string * string list * string * string * string
 
 (* for compatibility with "SAT_Solver" *)
 val berkmin_exec = getenv "BERKMIN_EXE"
 
 val static_list =
   [("Lingeling_JNI", Internal (JNI [1, 5], Batch, ["Lingeling"])),
    ("CryptoMiniSat", External ("CRYPTOMINISAT_HOME", "cryptominisat", [])),
    ("CryptoMiniSat_JNI", Internal (JNI [1, 5], Batch, ["CryptoMiniSat"])),
    ("MiniSat", ExternalV2 (ToFile, "MINISAT_HOME", "minisat", [], "SAT", "", "UNSAT")),
    ("MiniSat_JNI", Internal (JNI [], Incremental, ["MiniSat"])),
    ("zChaff", ExternalV2 (ToStdout, "ZCHAFF_HOME", "zchaff", [],
                           "Instance Satisfiable", "",
                           "Instance Unsatisfiable")),
    ("RSat", ExternalV2 (ToStdout, "RSAT_HOME", "rsat", ["-s"],
                         "s SATISFIABLE", "v ", "s UNSATISFIABLE")),
    ("Riss3g", External ("RISS3G_HOME", "riss3g", [])),
    ("BerkMin", ExternalV2 (ToStdout, "BERKMIN_HOME",
                            if berkmin_exec = "" then "BerkMin561"
                            else berkmin_exec, [], "Satisfiable          !!",
                            "solution =", "UNSATISFIABLE          !!")),
    ("BerkMin_Alloy", External ("BERKMINALLOY_HOME", "berkmin", [])),
    ("SAT4J", Internal (Java, Incremental, ["DefaultSAT4J"])),
    ("SAT4J_Light", Internal (Java, Incremental, ["LightSAT4J"]))]
 
 fun dynamic_entry_for_external name dev home exec args markers =
   let
-    fun make_args dir () =
+    fun make_args () =
       let
         val serial_str = serial_string ()
         val base = name ^ serial_str
         val out_file = base ^ ".out"
-        val dir_sep =
-          if String.isSubstring "\\" dir andalso not (String.isSubstring "/" dir)
-          then "\\" else "/"
+        val exe = Path.variable home + Path.platform_exe (Path.basic exec)
       in
-       [if null markers then "External" else "ExternalV2",
-        dir ^ dir_sep ^ exec, base ^ ".cnf"] @
-        (if null markers then []
-         else [if dev = ToFile then out_file else ""]) @ markers @
+       [if null markers then "External" else "ExternalV2"] @
+       [File.platform_path exe, base ^ ".cnf"] @
+       (if null markers then [] else [if dev = ToFile then out_file else ""]) @ markers @
        (if dev = ToFile then [out_file] else []) @ args
       end
-  in case getenv home of "" => NONE | dir => SOME (name, make_args dir) end
+  in if getenv home = "" then NONE else SOME (name, make_args) end
 
 fun dynamic_entry_for_info incremental (name, Internal (Java, mode, ss)) =
     if incremental andalso mode = Batch then NONE else SOME (name, K ss)
   | dynamic_entry_for_info incremental (name, Internal (JNI from_version, mode, ss)) =
     if (incremental andalso mode = Batch) orelse
        is_less (dict_ord int_ord (kodkodi_version (), from_version))
     then NONE
     else
       let
         val lib_paths = getenv "KODKODI_JAVA_LIBRARY_PATH" |> space_explode ":"
       in
         if exists (fn path => File.exists (Path.explode (path ^ "/"))) lib_paths
         then SOME (name, K ss)
         else NONE
       end
   | dynamic_entry_for_info false (name, External (home, exec, args)) =
     dynamic_entry_for_external name ToStdout home exec args []
   | dynamic_entry_for_info false
         (name, ExternalV2 (dev, home, exec, args, m1, m2, m3)) =
     dynamic_entry_for_external name dev home exec args [m1, m2, m3]
   | dynamic_entry_for_info true _ = NONE
 
 fun dynamic_list incremental =
   map_filter (dynamic_entry_for_info incremental) static_list
 
 val configured_sat_solvers = map fst o dynamic_list
 val smart_sat_solver_name = fst o hd o dynamic_list
 
 fun sat_solver_spec name =
   let
     val dyns = dynamic_list false
     fun enum_solvers solvers =
       commas (distinct (op =) (map (quote o fst) solvers))
   in
     (name, the (AList.lookup (op =) dyns name) ())
     handle Option.Option =>
            error (if AList.defined (op =) static_list name then
                     "The SAT solver " ^ quote name ^ " is not configured. The \
                     \following solvers are configured:\n" ^
                     enum_solvers dyns
                   else
                     "Unknown SAT solver " ^ quote name ^ "\nThe following \
                     \solvers are supported:\n" ^ enum_solvers static_list)
   end
 
 end;