--- a/doc-src/Nitpick/Makefile Tue Aug 28 13:04:15 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,33 +0,0 @@
-
-## targets
-
-default: dvi
-
-
-## dependencies
-
-include ../Makefile.in
-
-NAME = nitpick
-FILES = nitpick.tex ../iman.sty ../manual.bib
-
-dvi: $(NAME).dvi
-
-$(NAME).dvi: $(FILES) isabelle_nitpick.eps
- $(LATEX) $(NAME)
- $(BIBTEX) $(NAME)
- $(LATEX) $(NAME)
- $(LATEX) $(NAME)
- $(SEDINDEX) $(NAME)
- $(LATEX) $(NAME)
-
-pdf: $(NAME).pdf
-
-$(NAME).pdf: $(FILES) isabelle_nitpick.pdf
- $(PDFLATEX) $(NAME)
- $(BIBTEX) $(NAME)
- $(PDFLATEX) $(NAME)
- $(PDFLATEX) $(NAME)
- $(SEDINDEX) $(NAME)
- $(FIXBOOKMARKS) $(NAME).out
- $(PDFLATEX) $(NAME)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Nitpick/document/build Tue Aug 28 13:09:01 2012 +0200
@@ -0,0 +1,24 @@
+#!/bin/bash
+
+set -e
+
+FORMAT="$1"
+VARIANT="$2"
+
+"$ISABELLE_TOOL" logo -o isabelle_nitpick.pdf "Nitpick"
+"$ISABELLE_TOOL" logo -o isabelle_nitpick.eps "Nitpick"
+
+cp "$ISABELLE_HOME/doc-src/iman.sty" .
+cp "$ISABELLE_HOME/doc-src/manual.bib" .
+
+"$ISABELLE_TOOL" latex -o sty
+cp "$ISABELLE_HOME/doc-src/pdfsetup.sty" .
+
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_TOOL" latex -o bbl
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_HOME/doc-src/sedindex" root
+[ -f root.out ] && "$ISABELLE_HOME/doc-src/fixbookmarks" root.out
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_TOOL" latex -o "$FORMAT"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Nitpick/document/root.tex Tue Aug 28 13:09:01 2012 +0200
@@ -0,0 +1,2906 @@
+\documentclass[a4paper,12pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[english,french]{babel}
+\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\unkef{(\lambda x.\; \unk)}
+\def\undef{(\lambda x.\; \_)}
+%\def\unr{\textit{others}}
+\def\unr{\ldots}
+\def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
+\def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
+
+\hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
+counter-example counter-examples data-type data-types co-data-type
+co-data-types in-duc-tive co-in-duc-tive}
+
+\urlstyle{tt}
+
+\begin{document}
+
+%%% TYPESETTING
+%\renewcommand\labelitemi{$\bullet$}
+\renewcommand\labelitemi{\raise.065ex\hbox{\small\textbullet}}
+
+\selectlanguage{english}
+
+\title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
+Picking Nits \\[\smallskipamount]
+\Large A User's Guide to Nitpick for Isabelle/HOL}
+\author{\hbox{} \\
+Jasmin Christian 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.
+
+You can also invoke Nitpick from the ``Commands'' submenu of the
+``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c C-a
+C-n. This is equivalent to entering the \textbf{nitpick} command with no
+arguments in the theory text.
+
+Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
+Nitpick also provides an automatic mode that can be enabled via the ``Auto
+Nitpick'' option from the ``Isabelle'' menu in Proof General. In this mode,
+Nitpick is run on every newly entered theorem. The time limit for Auto Nitpick
+and other automatic tools can be set using the ``Auto Tools Time Limit'' option.
+
+\newbox\boxA
+\setbox\boxA=\hbox{\texttt{nospam}}
+
+\newcommand\authoremail{\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
+in.\allowbreak tum.\allowbreak de}}
+
+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\_Examples/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 either
+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}
+
+Sledgehammer is part of Isabelle, so you don't need to install it. However, it
+relies on a third-party Kodkod front-end called Kodkodi as well as a Java
+virtual machine called \texttt{java} (version 1.5 or above).
+
+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-1.5.1}, create it with the single line
+
+\prew
+\texttt{/usr/local/kodkodi-1.5.1}
+\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 installed, as explained in \S\ref{optimizations}. If you
+have already configured SAT solvers in Isabelle (e.g., for Refute), these will
+also be available to Nitpick.
+
+\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 `--'
+can be entered as \texttt{-} (hyphen) or
+\texttt{\char`\\\char`\<emdash\char`\>}.} \\[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. The tedious
+task of finding out whether the potentially spurious counterexample is in fact
+genuine can be delegated to \textit{auto} by passing \textit{check\_potential}.
+For example:
+
+\prew
+\textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
+\textbf{nitpick} [\textit{card~nat}~= 50, \textit{check\_potential}] \\[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 $P = \textit{False}$ \\[2\smallskipamount]
+Confirmation by ``\textit{auto}'': The above counterexample is genuine.
+\postw
+
+You might wonder why the counterexample is first reported as potentially
+spurious. The root of the problem is that the bound variable in $\forall n.\;
+\textit{Suc}~n \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds
+an $n$ such that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
+\textit{False}; but otherwise, it does not know anything about values of $n \ge
+\textit{card~nat}$ and must therefore evaluate the assumption to~$\unk$, not
+\textit{True}. Since the assumption can never be satisfied, the putative lemma
+can never be falsified.
+
+Incidentally, if you distrust the so-called genuine counterexamples, you can
+enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
+aware that \textit{auto} will usually fail to prove that the counterexample is
+genuine or spurious.
+
+Some conjectures involving elementary number theory make Nitpick look like a
+giant with feet of clay:
+
+\prew
+\textbf{lemma} ``$P~\textit{Suc\/}$'' \\
+\textbf{nitpick} \\[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 Datatype:} \\
+\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 ``Datatype'' 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\_datatypes}] \\[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 Datatypes: \\
+\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\_datatypes}] \\[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 Datatypes: \\
+\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\_datatypes}] \\[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 Datatypes: \\
+\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\_datatypes}] \\[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 Datatypes: \\
+\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\_datatypes}] \\[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 Datatypes: \\
+\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 $\textit{even} = (λx. ?)(0 := True, 1 := False, 2 := True, 3 := False)$ \\
+\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}
+
+While Isabelle regrettably lacks a high-level mechanism for defining coinductive
+datatypes, the \textit{Coinductive\_List} theory from Andreas Lochbihler's
+\textit{Coinductive} AFP entry \cite{lochbihler-2010} provides a coinductive
+``lazy list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick
+supports these lazy lists seamlessly and provides a hook, described in
+\S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
+datatypes.
+
+(Co)intuitively, a coinductive datatype is similar to an inductive datatype but
+allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
+\ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
+1, 2, 3, \ldots]$ can be defined as lazy lists using the
+$\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
+$\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
+\mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
+
+Although it is otherwise no friend of infinity, Nitpick can find counterexamples
+involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
+finite lists:
+
+\prew
+\textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs\/}$'' \\
+\textbf{nitpick} \\[2\smallskipamount]
+\slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
+\hbox{}\qquad 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{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\_datatypes}] \\[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 Codatatype:\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 ``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 lower
+nonnegative \textit{bisim\_depth} value than the default one provided by
+Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
+be distinguished from each other by their prefixes of length $K$. Be aware that
+setting $K$ to a too low value can overconstrain Nitpick, preventing it from
+finding any counterexamples.
+
+\subsection{Boxing}
+\label{boxing}
+
+Nitpick normally maps function and product types directly to the corresponding
+Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
+cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
+\Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
+off to treat these types in the same way as plain datatypes, by approximating
+them by a subset of a given cardinality. This technique is called ``boxing'' and
+is particularly useful for functions passed as arguments to other functions, for
+high-arity functions, and for large tuples. Under the hood, boxing involves
+wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
+isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
+
+To illustrate boxing, we consider a formalization of $\lambda$-terms represented
+using de Bruijn's notation:
+
+\prew
+\textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
+\postw
+
+The $\textit{lift}~t~k$ function increments all variables with indices greater
+than or equal to $k$ by one:
+
+\prew
+\textbf{primrec} \textit{lift} \textbf{where} \\
+``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
+``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
+``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
+\postw
+
+The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
+term $t$ has a loose variable with index $k$ or more:
+
+\prew
+\textbf{primrec}~\textit{loose} \textbf{where} \\
+``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
+``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
+``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
+\postw
+
+Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
+on $t$:
+
+\prew
+\textbf{primrec}~\textit{subst} \textbf{where} \\
+``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
+``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
+\phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
+``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
+\postw
+
+A substitution is a function that maps variable indices to terms. Observe that
+$\sigma$ is a function passed as argument and that Nitpick can't optimize it
+away, because the recursive call for the \textit{Lam} case involves an altered
+version. Also notice the \textit{lift} call, which increments the variable
+indices when moving under a \textit{Lam}.
+
+A reasonable property to expect of substitution is that it should leave closed
+terms unchanged. Alas, even this simple property does not hold:
+
+\pre
+\textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
+\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
+\slshape
+Trying 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.
+
+A similar technique can be employed for structural induction. The
+following mini formalization of full binary trees will serve as illustration:
+
+\prew
+\textbf{datatype} $\kern1pt'a$~\textit{bin\_tree} = $\textit{Leaf}~{\kern1pt'a}$ $\mid$ $\textit{Branch}$ ``\kern1pt$'a$ \textit{bin\_tree}'' ``\kern1pt$'a$ \textit{bin\_tree}'' \\[2\smallskipamount]
+\textbf{primrec}~\textit{labels}~\textbf{where} \\
+``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
+``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
+\textbf{primrec}~\textit{swap}~\textbf{where} \\
+``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
+\phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
+``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
+\postw
+
+The \textit{labels} function returns the set of labels occurring on leaves of a
+tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
+labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
+obtained by swapping $a$ and $b$:
+
+\prew
+\textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
+\postw
+
+Nitpick can't find any counterexample, so we proceed with induction
+(this time favoring a more structured style):
+
+\prew
+\textbf{proof}~(\textit{induct}~$t$) \\
+\hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
+\textbf{next} \\
+\hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
+\postw
+
+Nitpick can't find any counterexample at this point either, but it makes the
+following suggestion:
+
+\prew
+\slshape
+Hint: To check that the induction hypothesis is general enough, try this command:
+\textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
+\postw
+
+If we follow the hint, we get a ``nonstandard'' counterexample for the step:
+
+\prew
+\slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
+\hbox{}\qquad Free variables: \nopagebreak \\
+\hbox{}\qquad\qquad $a = a_1$ \\
+\hbox{}\qquad\qquad $b = a_2$ \\
+\hbox{}\qquad\qquad $t = \xi_1$ \\
+\hbox{}\qquad\qquad $u = \xi_2$ \\
+\hbox{}\qquad Datatype: \nopagebreak \\
+\hbox{}\qquad\qquad $'a~\textit{bin\_tree} =
+\{\!\begin{aligned}[t]
+& \xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \\[-2pt]
+& \textit{Branch}~\xi_1~\xi_2,\> \unr\}\end{aligned}$ \\
+\hbox{}\qquad {\slshape Constants:} \nopagebreak \\
+\hbox{}\qquad\qquad $\textit{labels} = \unkef
+ (\!\begin{aligned}[t]%
+ & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
+ & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
+\hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \unkef
+ (\!\begin{aligned}[t]%
+ & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
+ & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount]
+The existence of a nonstandard model suggests that the induction hypothesis is not general enough or may even
+be wrong. See the Nitpick manual's ``Inductive Properties'' section for details (``\textit{isabelle doc nitpick}'').
+\postw
+
+Reading the Nitpick manual is a most excellent idea.
+But what's going on? The \textit{non\_std} option told the tool to look for
+nonstandard models of binary trees, which means that new ``nonstandard'' trees
+$\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees
+generated by the \textit{Leaf} and \textit{Branch} constructors.%
+\footnote{Notice the similarity between allowing nonstandard trees here and
+allowing unreachable states in the preceding example (by removing the ``$n \in
+\textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
+set of objects over which the induction is performed while doing the step
+in order to test the induction hypothesis's strength.}
+Unlike standard trees, these new trees contain cycles. We will see later that
+every property of acyclic trees that can be proved without using induction also
+holds for cyclic trees. Hence,
+%
+\begin{quote}
+\textsl{If the induction
+hypothesis is strong enough, the induction step will hold even for nonstandard
+objects, and Nitpick won't find any nonstandard counterexample.}
+\end{quote}
+%
+But here the tool find some nonstandard trees $t = \xi_1$
+and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in
+\textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$.
+Because neither tree contains both $a$ and $b$, the induction hypothesis tells
+us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
+and as a result we know nothing about the labels of the tree
+$\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
+$\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
+labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
+\textit{labels}$ $(\textit{swap}~u~a~b)$.
+
+The solution is to ensure that we always know what the labels of the subtrees
+are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
+$t$ in the statement of the lemma:
+
+\prew
+\textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
+\phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
+\phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
+\phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
+\phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
+\postw
+
+This time, Nitpick won't find any nonstandard counterexample, and we can perform
+the induction step using \textit{auto}.
+
+\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}), automatic counterexample checks
+(\S\ref{authentication}), optimizations
+(\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
+
+You can instruct Nitpick to run automatically on newly entered theorems by
+enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
+General. 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{blocking}
+(\S\ref{mode-of-operation}), \textit{verbose} (\S\ref{output-format}), and
+\textit{debug} (\S\ref{output-format}) are disabled, \textit{max\_threads}
+(\S\ref{optimizations}) is taken to be 1, \textit{max\_potential}
+(\S\ref{output-format}) is taken to be 0, and \textit{timeout}
+(\S\ref{timeouts}) is superseded by the ``Auto Tools Time Limit'' in
+Proof General's ``Isabelle'' menu. 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 `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<emdash\char`\>}.
+\item[\labelitemi] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
+\item[\labelitemi] \qtybf{float\_or\_none}: An integer (e.g., 60) or floating-point number
+(e.g., 0.5) expressing a number of seconds, or the keyword \textit{none}
+($\infty$ 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{blocking} vs.\
+\textit{non\_blocking}). When setting them, ``= \textit{true}'' may be omitted.
+
+\subsection{Mode of Operation}
+\label{mode-of-operation}
+
+\begin{enum}
+\optrue{blocking}{non\_blocking}
+Specifies whether the \textbf{nitpick} command should operate synchronously.
+The asynchronous (non-blocking) mode lets the user start proving the putative
+theorem while Nitpick looks for a counterexample, but it can also be more
+confusing. For technical reasons, automatic runs currently always block.
+
+\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{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.
+
+\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\_datatypes} (\S\ref{output-format}).}
+
+\opdefault{bits}{int\_seq}{\upshape 1,2,3,4,6,8,10,12,14,16}
+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}).}
+
+\opargbool{std}{type}{non\_std}
+Specifies whether the given (recursive) datatype should be given standard
+models. Nonstandard models are unsound but can help debug structural induction
+proofs, as explained in \S\ref{inductive-properties}.
+
+\optrue{std}{non\_std}
+Specifies the default standardness to use. This can be overridden on a per-type
+basis using the \textit{std}~\qty{type} option described above.
+\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{overlord} (\S\ref{mode-of-operation}) and
+\textit{batch\_size} (\S\ref{optimizations}).}
+
+\opfalse{show\_datatypes}{hide\_datatypes}
+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\_datatypes}, \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{check\_potential} (\S\ref{authentication}) and
+\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{check\_genuine} (\S\ref{authentication}) and
+\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{Authentication}
+\label{authentication}
+
+\begin{enum}
+\opfalse{check\_potential}{trust\_potential}
+Specifies whether potentially spurious counterexamples should be given to
+Isabelle's \textit{auto} tactic to assess their validity. If a potentially
+spurious counterexample is shown to be genuine, Nitpick displays a message to
+this effect and terminates.
+
+\nopagebreak
+{\small See also \textit{max\_potential} (\S\ref{output-format}).}
+
+\opfalse{check\_genuine}{trust\_genuine}
+Specifies whether genuine and quasi genuine counterexamples should be given to
+Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
+counterexample is shown to be spurious, the user is kindly asked to send a bug
+report to the author at \authoremail.
+
+\nopagebreak
+{\small See also \textit{max\_genuine} (\S\ref{output-format}).}
+
+\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{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{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{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{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}
+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\_or\_none}{\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,
+\textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
+a time slot whose length is specified by the ``Auto Counterexample Time
+Limit'' option in Proof General.
+
+\nopagebreak
+{\small See also \textit{max\_threads} (\S\ref{optimizations}).}
+
+\opdefault{tac\_timeout}{float\_or\_none}{\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, \textit{auto} tactic when
+checking a counterexample, 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}),
+\textit{mono} (\S\ref{scope-of-search}),
+\textit{check\_potential} (\S\ref{authentication}),
+and \textit{check\_genuine} (\S\ref{authentication}).}
+\end{enum}
+
+\section{Attribute Reference}
+\label{attribute-reference}
+
+Nitpick needs to consider the definitions of all constants occurring in a
+formula in order to falsify it. For constants introduced using the
+\textbf{definition} command, the definition is simply the associated
+\textit{\_def} axiom. In contrast, instead of using the internal representation
+of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
+\textbf{nominal\_primrec} packages, Nitpick relies on the more natural
+equational specification entered by the user.
+
+Behind the scenes, Isabelle's built-in packages and theories rely on the
+following attributes to affect Nitpick's behavior:
+
+\begin{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 \hbox{(\textbf{ax\_})}\allowbreak\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
+\hbox{(\textbf{ax\_})}\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 coinductive datatypes as well as term
+postprocessors.
+
+\subsection{Invocation of Nitpick}
+\label{invocation-of-nitpick}
+
+The \textit{Nitpick} structure offers the following functions for invoking your
+favorite counterexample generator:
+
+\prew
+$\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
+\hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{mode}
+\rightarrow \textit{int} \rightarrow \textit{int} \rightarrow \textit{int}$ \\
+$\hbox{}\quad{\rightarrow}\; (\textit{term} * \textit{term})~\textit{list}
+\rightarrow \textit{term~list} \rightarrow \textit{term} \rightarrow \textit{string} * \textit{Proof.state}$ \\
+$\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
+\hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{mode} \rightarrow \textit{int} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
+\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{Registration of Coinductive Datatypes}
+\label{registration-of-coinductive-datatypes}
+
+If you have defined a custom coinductive datatype, you can tell Nitpick about
+it, so that it can use an efficient Kodkod axiomatization similar to the one it
+uses for lazy lists. The interface for registering and unregistering coinductive
+datatypes consists of the following pair of functions defined in the
+\textit{Nitpick\_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.
+
+\subsection{Registration of Term Postprocessors}
+\label{registration-of-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.
+
+\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 maintains a global cache of wellfoundedness conditions,
+which can become invalid if you change the definition of an inductive predicate
+that is registered in the cache. To clear the cache,
+run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
+$0.51$).
+
+\item[\labelitemi] Nitpick produces spurious counterexamples when invoked after a
+\textbf{guess} command in a structured proof.
+
+\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.
+\end{enum}
+
+\let\em=\sl
+\bibliography{manual}{}
+\bibliographystyle{abbrv}
+
+\end{document}
--- a/doc-src/Nitpick/nitpick.tex Tue Aug 28 13:04:15 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2906 +0,0 @@
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-\usepackage{amssymb}
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-%\usepackage{mathpazo}
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-%\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\unkef{(\lambda x.\; \unk)}
-\def\undef{(\lambda x.\; \_)}
-%\def\unr{\textit{others}}
-\def\unr{\ldots}
-\def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
-\def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
-
-\hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
-counter-example counter-examples data-type data-types co-data-type
-co-data-types in-duc-tive co-in-duc-tive}
-
-\urlstyle{tt}
-
-\begin{document}
-
-%%% TYPESETTING
-%\renewcommand\labelitemi{$\bullet$}
-\renewcommand\labelitemi{\raise.065ex\hbox{\small\textbullet}}
-
-\selectlanguage{english}
-
-\title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
-Picking Nits \\[\smallskipamount]
-\Large A User's Guide to Nitpick for Isabelle/HOL}
-\author{\hbox{} \\
-Jasmin Christian 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.
-
-You can also invoke Nitpick from the ``Commands'' submenu of the
-``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c C-a
-C-n. This is equivalent to entering the \textbf{nitpick} command with no
-arguments in the theory text.
-
-Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
-Nitpick also provides an automatic mode that can be enabled via the ``Auto
-Nitpick'' option from the ``Isabelle'' menu in Proof General. In this mode,
-Nitpick is run on every newly entered theorem. The time limit for Auto Nitpick
-and other automatic tools can be set using the ``Auto Tools Time Limit'' option.
-
-\newbox\boxA
-\setbox\boxA=\hbox{\texttt{nospam}}
-
-\newcommand\authoremail{\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
-in.\allowbreak tum.\allowbreak de}}
-
-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\_Examples/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 either
-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}
-
-Sledgehammer is part of Isabelle, so you don't need to install it. However, it
-relies on a third-party Kodkod front-end called Kodkodi as well as a Java
-virtual machine called \texttt{java} (version 1.5 or above).
-
-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-1.5.1}, create it with the single line
-
-\prew
-\texttt{/usr/local/kodkodi-1.5.1}
-\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 installed, as explained in \S\ref{optimizations}. If you
-have already configured SAT solvers in Isabelle (e.g., for Refute), these will
-also be available to Nitpick.
-
-\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 `--'
-can be entered as \texttt{-} (hyphen) or
-\texttt{\char`\\\char`\<emdash\char`\>}.} \\[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. The tedious
-task of finding out whether the potentially spurious counterexample is in fact
-genuine can be delegated to \textit{auto} by passing \textit{check\_potential}.
-For example:
-
-\prew
-\textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
-\textbf{nitpick} [\textit{card~nat}~= 50, \textit{check\_potential}] \\[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 $P = \textit{False}$ \\[2\smallskipamount]
-Confirmation by ``\textit{auto}'': The above counterexample is genuine.
-\postw
-
-You might wonder why the counterexample is first reported as potentially
-spurious. The root of the problem is that the bound variable in $\forall n.\;
-\textit{Suc}~n \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds
-an $n$ such that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
-\textit{False}; but otherwise, it does not know anything about values of $n \ge
-\textit{card~nat}$ and must therefore evaluate the assumption to~$\unk$, not
-\textit{True}. Since the assumption can never be satisfied, the putative lemma
-can never be falsified.
-
-Incidentally, if you distrust the so-called genuine counterexamples, you can
-enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
-aware that \textit{auto} will usually fail to prove that the counterexample is
-genuine or spurious.
-
-Some conjectures involving elementary number theory make Nitpick look like a
-giant with feet of clay:
-
-\prew
-\textbf{lemma} ``$P~\textit{Suc\/}$'' \\
-\textbf{nitpick} \\[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 Datatype:} \\
-\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 ``Datatype'' 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\_datatypes}] \\[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 Datatypes: \\
-\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\_datatypes}] \\[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 Datatypes: \\
-\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\_datatypes}] \\[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 Datatypes: \\
-\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\_datatypes}] \\[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 Datatypes: \\
-\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\_datatypes}] \\[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 Datatypes: \\
-\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 $\textit{even} = (λx. ?)(0 := True, 1 := False, 2 := True, 3 := False)$ \\
-\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}
-
-While Isabelle regrettably lacks a high-level mechanism for defining coinductive
-datatypes, the \textit{Coinductive\_List} theory from Andreas Lochbihler's
-\textit{Coinductive} AFP entry \cite{lochbihler-2010} provides a coinductive
-``lazy list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick
-supports these lazy lists seamlessly and provides a hook, described in
-\S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
-datatypes.
-
-(Co)intuitively, a coinductive datatype is similar to an inductive datatype but
-allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
-\ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
-1, 2, 3, \ldots]$ can be defined as lazy lists using the
-$\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
-$\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
-\mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
-
-Although it is otherwise no friend of infinity, Nitpick can find counterexamples
-involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
-finite lists:
-
-\prew
-\textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs\/}$'' \\
-\textbf{nitpick} \\[2\smallskipamount]
-\slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
-\hbox{}\qquad 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{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\_datatypes}] \\[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 Codatatype:\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 ``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 lower
-nonnegative \textit{bisim\_depth} value than the default one provided by
-Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
-be distinguished from each other by their prefixes of length $K$. Be aware that
-setting $K$ to a too low value can overconstrain Nitpick, preventing it from
-finding any counterexamples.
-
-\subsection{Boxing}
-\label{boxing}
-
-Nitpick normally maps function and product types directly to the corresponding
-Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
-cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
-\Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
-off to treat these types in the same way as plain datatypes, by approximating
-them by a subset of a given cardinality. This technique is called ``boxing'' and
-is particularly useful for functions passed as arguments to other functions, for
-high-arity functions, and for large tuples. Under the hood, boxing involves
-wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
-isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
-
-To illustrate boxing, we consider a formalization of $\lambda$-terms represented
-using de Bruijn's notation:
-
-\prew
-\textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
-\postw
-
-The $\textit{lift}~t~k$ function increments all variables with indices greater
-than or equal to $k$ by one:
-
-\prew
-\textbf{primrec} \textit{lift} \textbf{where} \\
-``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
-``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
-``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
-\postw
-
-The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
-term $t$ has a loose variable with index $k$ or more:
-
-\prew
-\textbf{primrec}~\textit{loose} \textbf{where} \\
-``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
-``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
-``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
-\postw
-
-Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
-on $t$:
-
-\prew
-\textbf{primrec}~\textit{subst} \textbf{where} \\
-``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
-``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
-\phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
-``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
-\postw
-
-A substitution is a function that maps variable indices to terms. Observe that
-$\sigma$ is a function passed as argument and that Nitpick can't optimize it
-away, because the recursive call for the \textit{Lam} case involves an altered
-version. Also notice the \textit{lift} call, which increments the variable
-indices when moving under a \textit{Lam}.
-
-A reasonable property to expect of substitution is that it should leave closed
-terms unchanged. Alas, even this simple property does not hold:
-
-\pre
-\textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
-\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
-\slshape
-Trying 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.
-
-A similar technique can be employed for structural induction. The
-following mini formalization of full binary trees will serve as illustration:
-
-\prew
-\textbf{datatype} $\kern1pt'a$~\textit{bin\_tree} = $\textit{Leaf}~{\kern1pt'a}$ $\mid$ $\textit{Branch}$ ``\kern1pt$'a$ \textit{bin\_tree}'' ``\kern1pt$'a$ \textit{bin\_tree}'' \\[2\smallskipamount]
-\textbf{primrec}~\textit{labels}~\textbf{where} \\
-``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
-``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
-\textbf{primrec}~\textit{swap}~\textbf{where} \\
-``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
-\phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
-``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
-\postw
-
-The \textit{labels} function returns the set of labels occurring on leaves of a
-tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
-labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
-obtained by swapping $a$ and $b$:
-
-\prew
-\textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
-\postw
-
-Nitpick can't find any counterexample, so we proceed with induction
-(this time favoring a more structured style):
-
-\prew
-\textbf{proof}~(\textit{induct}~$t$) \\
-\hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
-\textbf{next} \\
-\hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
-\postw
-
-Nitpick can't find any counterexample at this point either, but it makes the
-following suggestion:
-
-\prew
-\slshape
-Hint: To check that the induction hypothesis is general enough, try this command:
-\textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
-\postw
-
-If we follow the hint, we get a ``nonstandard'' counterexample for the step:
-
-\prew
-\slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
-\hbox{}\qquad Free variables: \nopagebreak \\
-\hbox{}\qquad\qquad $a = a_1$ \\
-\hbox{}\qquad\qquad $b = a_2$ \\
-\hbox{}\qquad\qquad $t = \xi_1$ \\
-\hbox{}\qquad\qquad $u = \xi_2$ \\
-\hbox{}\qquad Datatype: \nopagebreak \\
-\hbox{}\qquad\qquad $'a~\textit{bin\_tree} =
-\{\!\begin{aligned}[t]
-& \xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \\[-2pt]
-& \textit{Branch}~\xi_1~\xi_2,\> \unr\}\end{aligned}$ \\
-\hbox{}\qquad {\slshape Constants:} \nopagebreak \\
-\hbox{}\qquad\qquad $\textit{labels} = \unkef
- (\!\begin{aligned}[t]%
- & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
- & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
-\hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \unkef
- (\!\begin{aligned}[t]%
- & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
- & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount]
-The existence of a nonstandard model suggests that the induction hypothesis is not general enough or may even
-be wrong. See the Nitpick manual's ``Inductive Properties'' section for details (``\textit{isabelle doc nitpick}'').
-\postw
-
-Reading the Nitpick manual is a most excellent idea.
-But what's going on? The \textit{non\_std} option told the tool to look for
-nonstandard models of binary trees, which means that new ``nonstandard'' trees
-$\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees
-generated by the \textit{Leaf} and \textit{Branch} constructors.%
-\footnote{Notice the similarity between allowing nonstandard trees here and
-allowing unreachable states in the preceding example (by removing the ``$n \in
-\textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
-set of objects over which the induction is performed while doing the step
-in order to test the induction hypothesis's strength.}
-Unlike standard trees, these new trees contain cycles. We will see later that
-every property of acyclic trees that can be proved without using induction also
-holds for cyclic trees. Hence,
-%
-\begin{quote}
-\textsl{If the induction
-hypothesis is strong enough, the induction step will hold even for nonstandard
-objects, and Nitpick won't find any nonstandard counterexample.}
-\end{quote}
-%
-But here the tool find some nonstandard trees $t = \xi_1$
-and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in
-\textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$.
-Because neither tree contains both $a$ and $b$, the induction hypothesis tells
-us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
-and as a result we know nothing about the labels of the tree
-$\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
-$\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
-labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
-\textit{labels}$ $(\textit{swap}~u~a~b)$.
-
-The solution is to ensure that we always know what the labels of the subtrees
-are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
-$t$ in the statement of the lemma:
-
-\prew
-\textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
-\phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
-\phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
-\phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
-\phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
-\postw
-
-This time, Nitpick won't find any nonstandard counterexample, and we can perform
-the induction step using \textit{auto}.
-
-\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}), automatic counterexample checks
-(\S\ref{authentication}), optimizations
-(\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
-
-You can instruct Nitpick to run automatically on newly entered theorems by
-enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
-General. 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{blocking}
-(\S\ref{mode-of-operation}), \textit{verbose} (\S\ref{output-format}), and
-\textit{debug} (\S\ref{output-format}) are disabled, \textit{max\_threads}
-(\S\ref{optimizations}) is taken to be 1, \textit{max\_potential}
-(\S\ref{output-format}) is taken to be 0, and \textit{timeout}
-(\S\ref{timeouts}) is superseded by the ``Auto Tools Time Limit'' in
-Proof General's ``Isabelle'' menu. 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 `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<emdash\char`\>}.
-\item[\labelitemi] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
-\item[\labelitemi] \qtybf{float\_or\_none}: An integer (e.g., 60) or floating-point number
-(e.g., 0.5) expressing a number of seconds, or the keyword \textit{none}
-($\infty$ 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{blocking} vs.\
-\textit{non\_blocking}). When setting them, ``= \textit{true}'' may be omitted.
-
-\subsection{Mode of Operation}
-\label{mode-of-operation}
-
-\begin{enum}
-\optrue{blocking}{non\_blocking}
-Specifies whether the \textbf{nitpick} command should operate synchronously.
-The asynchronous (non-blocking) mode lets the user start proving the putative
-theorem while Nitpick looks for a counterexample, but it can also be more
-confusing. For technical reasons, automatic runs currently always block.
-
-\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{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.
-
-\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\_datatypes} (\S\ref{output-format}).}
-
-\opdefault{bits}{int\_seq}{\upshape 1,2,3,4,6,8,10,12,14,16}
-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}).}
-
-\opargbool{std}{type}{non\_std}
-Specifies whether the given (recursive) datatype should be given standard
-models. Nonstandard models are unsound but can help debug structural induction
-proofs, as explained in \S\ref{inductive-properties}.
-
-\optrue{std}{non\_std}
-Specifies the default standardness to use. This can be overridden on a per-type
-basis using the \textit{std}~\qty{type} option described above.
-\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{overlord} (\S\ref{mode-of-operation}) and
-\textit{batch\_size} (\S\ref{optimizations}).}
-
-\opfalse{show\_datatypes}{hide\_datatypes}
-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\_datatypes}, \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{check\_potential} (\S\ref{authentication}) and
-\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{check\_genuine} (\S\ref{authentication}) and
-\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{Authentication}
-\label{authentication}
-
-\begin{enum}
-\opfalse{check\_potential}{trust\_potential}
-Specifies whether potentially spurious counterexamples should be given to
-Isabelle's \textit{auto} tactic to assess their validity. If a potentially
-spurious counterexample is shown to be genuine, Nitpick displays a message to
-this effect and terminates.
-
-\nopagebreak
-{\small See also \textit{max\_potential} (\S\ref{output-format}).}
-
-\opfalse{check\_genuine}{trust\_genuine}
-Specifies whether genuine and quasi genuine counterexamples should be given to
-Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
-counterexample is shown to be spurious, the user is kindly asked to send a bug
-report to the author at \authoremail.
-
-\nopagebreak
-{\small See also \textit{max\_genuine} (\S\ref{output-format}).}
-
-\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{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{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{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{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}
-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\_or\_none}{\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,
-\textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
-a time slot whose length is specified by the ``Auto Counterexample Time
-Limit'' option in Proof General.
-
-\nopagebreak
-{\small See also \textit{max\_threads} (\S\ref{optimizations}).}
-
-\opdefault{tac\_timeout}{float\_or\_none}{\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, \textit{auto} tactic when
-checking a counterexample, 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}),
-\textit{mono} (\S\ref{scope-of-search}),
-\textit{check\_potential} (\S\ref{authentication}),
-and \textit{check\_genuine} (\S\ref{authentication}).}
-\end{enum}
-
-\section{Attribute Reference}
-\label{attribute-reference}
-
-Nitpick needs to consider the definitions of all constants occurring in a
-formula in order to falsify it. For constants introduced using the
-\textbf{definition} command, the definition is simply the associated
-\textit{\_def} axiom. In contrast, instead of using the internal representation
-of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
-\textbf{nominal\_primrec} packages, Nitpick relies on the more natural
-equational specification entered by the user.
-
-Behind the scenes, Isabelle's built-in packages and theories rely on the
-following attributes to affect Nitpick's behavior:
-
-\begin{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 \hbox{(\textbf{ax\_})}\allowbreak\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
-\hbox{(\textbf{ax\_})}\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 coinductive datatypes as well as term
-postprocessors.
-
-\subsection{Invocation of Nitpick}
-\label{invocation-of-nitpick}
-
-The \textit{Nitpick} structure offers the following functions for invoking your
-favorite counterexample generator:
-
-\prew
-$\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
-\hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{mode}
-\rightarrow \textit{int} \rightarrow \textit{int} \rightarrow \textit{int}$ \\
-$\hbox{}\quad{\rightarrow}\; (\textit{term} * \textit{term})~\textit{list}
-\rightarrow \textit{term~list} \rightarrow \textit{term} \rightarrow \textit{string} * \textit{Proof.state}$ \\
-$\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
-\hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{mode} \rightarrow \textit{int} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
-\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{Registration of Coinductive Datatypes}
-\label{registration-of-coinductive-datatypes}
-
-If you have defined a custom coinductive datatype, you can tell Nitpick about
-it, so that it can use an efficient Kodkod axiomatization similar to the one it
-uses for lazy lists. The interface for registering and unregistering coinductive
-datatypes consists of the following pair of functions defined in the
-\textit{Nitpick\_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.
-
-\subsection{Registration of Term Postprocessors}
-\label{registration-of-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.
-
-\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 maintains a global cache of wellfoundedness conditions,
-which can become invalid if you change the definition of an inductive predicate
-that is registered in the cache. To clear the cache,
-run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g.,
-$0.51$).
-
-\item[\labelitemi] Nitpick produces spurious counterexamples when invoked after a
-\textbf{guess} command in a structured proof.
-
-\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.
-\end{enum}
-
-\let\em=\sl
-\bibliography{../manual}{}
-\bibliographystyle{abbrv}
-
-\end{document}
--- a/doc-src/ROOT Tue Aug 28 13:04:15 2012 +0200
+++ b/doc-src/ROOT Tue Aug 28 13:09:01 2012 +0200
@@ -194,6 +194,16 @@
"document/build"
"document/root.tex"
+session Nitpick (doc) in "Nitpick" = Pure +
+ options [document_variants = "nitpick"]
+ theories
+ files
+ "../pdfsetup.sty"
+ "../iman.sty"
+ "../manual.bib"
+ "document/build"
+ "document/root.tex"
+
session ProgProve (doc) in "ProgProve" = HOL +
options [document_variants = "prog-prove", show_question_marks = false]
theories