doc-src/Nitpick/nitpick.tex
author blanchet
Tue Feb 23 14:11:36 2010 +0100 (2010-02-23)
changeset 35331 450ab945c451
parent 35312 99cd1f96b400
child 35335 f715cfde056a
permissions -rw-r--r--
document Quickcheck's "no_assms" option
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\documentclass[a4paper,12pt]{article}
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\usepackage[T1]{fontenc}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage[english,french]{babel}
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\usepackage{color}
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\usepackage{graphicx}
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%\usepackage{mathpazo}
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\usepackage{multicol}
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\usepackage{stmaryrd}
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%\usepackage[scaled=.85]{beramono}
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\usepackage{../iman,../pdfsetup}
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%\oddsidemargin=4.6mm
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%\evensidemargin=4.6mm
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%\textwidth=150mm
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%\topmargin=4.6mm
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%\headsep=0mm
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%\textheight=234mm
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\def\Colon{\mathord{:\mkern-1.5mu:}}
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%\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}}
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%\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}}
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\def\lparr{\mathopen{(\mkern-4mu\mid}}
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\def\rparr{\mathclose{\mid\mkern-4mu)}}
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\def\unk{{?}}
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\def\undef{(\lambda x.\; \unk)}
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%\def\unr{\textit{others}}
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\def\unr{\ldots}
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\def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
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\def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
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\hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
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counter-example counter-examples data-type data-types co-data-type 
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co-data-types in-duc-tive co-in-duc-tive}
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\urlstyle{tt}
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\begin{document}
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\selectlanguage{english}
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\title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
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Picking Nits \\[\smallskipamount]
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\Large A User's Guide to Nitpick for Isabelle/HOL}
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\author{\hbox{} \\
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Jasmin Christian Blanchette \\
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{\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\
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\hbox{}}
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\maketitle
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\tableofcontents
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\setlength{\parskip}{.7em plus .2em minus .1em}
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\setlength{\parindent}{0pt}
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\setlength{\abovedisplayskip}{\parskip}
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\setlength{\abovedisplayshortskip}{.9\parskip}
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\setlength{\belowdisplayskip}{\parskip}
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\setlength{\belowdisplayshortskip}{.9\parskip}
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% General-purpose enum environment with correct spacing
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\newenvironment{enum}%
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    {\begin{list}{}{%
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        \setlength{\topsep}{.1\parskip}%
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        \setlength{\partopsep}{.1\parskip}%
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        \setlength{\itemsep}{\parskip}%
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        \advance\itemsep by-\parsep}}
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    {\end{list}}
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\def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin
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\advance\rightskip by\leftmargin}
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\def\post{\vskip0pt plus1ex\endgroup}
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\def\prew{\pre\advance\rightskip by-\leftmargin}
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\def\postw{\post}
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\section{Introduction}
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\label{introduction}
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Nitpick \cite{blanchette-nipkow-2009} is a counterexample generator for
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Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
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combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
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quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
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first-order relational model finder developed by the Software Design Group at
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MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
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borrows many ideas and code fragments, but it benefits from Kodkod's
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optimizations and a new encoding scheme. The name Nitpick is shamelessly
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appropriated from a now retired Alloy precursor.
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Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
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theorem and wait a few seconds. Nonetheless, there are situations where knowing
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how it works under the hood and how it reacts to various options helps
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increase the test coverage. This manual also explains how to install the tool on
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your workstation. Should the motivation fail you, think of the many hours of
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hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.
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Another common use of Nitpick is to find out whether the axioms of a locale are
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satisfiable, while the locale is being developed. To check this, it suffices to
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write
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\prew
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\textbf{lemma}~``$\textit{False}$'' \\
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\textbf{nitpick}~[\textit{show\_all}]
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\postw
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after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
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must find a model for the axioms. If it finds no model, we have an indication
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that the axioms might be unsatisfiable.
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Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
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machine called \texttt{java}. The examples presented in this manual can be found
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in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.
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Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
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Nitpick also provides an automatic mode that can be enabled using the
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``Auto Nitpick'' option from the ``Isabelle'' menu in Proof General. In this
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mode, Nitpick is run on every newly entered theorem, much like Auto Quickcheck.
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The collective time limit for Auto Nitpick and Auto Quickcheck can be set using
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the ``Auto Counterexample Time Limit'' option.
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\newbox\boxA
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\setbox\boxA=\hbox{\texttt{nospam}}
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The known bugs and limitations at the time of writing are listed in
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\S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
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or this manual should be directed to
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\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
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in.\allowbreak tum.\allowbreak de}.
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\vskip2.5\smallskipamount
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\textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
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suggesting several textual improvements.
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% and Perry James for reporting a typo.
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\section{First Steps}
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\label{first-steps}
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This section introduces Nitpick by presenting small examples. If possible, you
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should try out the examples on your workstation. Your theory file should start
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as follows:
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\prew
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\textbf{theory}~\textit{Scratch} \\
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\textbf{imports}~\textit{Main~Coinductive\_List~Quotient\_Product~RealDef} \\
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\textbf{begin}
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\postw
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The results presented here were obtained using the JNI version of MiniSat and
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with multithreading disabled to reduce nondeterminism and a time limit of
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15~seconds (instead of 30~seconds). This was done by adding the line
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\prew
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\textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSat\_JNI}, \,\textit{max\_threads}~= 1, \,\textit{timeout} = 15$\,s$]
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\postw
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after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
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Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
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be installed, as explained in \S\ref{optimizations}. If you have already
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configured SAT solvers in Isabelle (e.g., for Refute), these will also be
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available to Nitpick.
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\subsection{Propositional Logic}
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\label{propositional-logic}
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Let's start with a trivial example from propositional logic:
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\prew
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\textbf{lemma}~``$P \longleftrightarrow Q$'' \\
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\textbf{nitpick}
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\postw
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You should get the following output:
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\prew
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\slshape
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Nitpick found a counterexample: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \textit{True}$ \\
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\hbox{}\qquad\qquad $Q = \textit{False}$
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\postw
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Nitpick can also be invoked on individual subgoals, as in the example below:
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\prew
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\textbf{apply}~\textit{auto} \\[2\smallskipamount]
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{\slshape goal (2 subgoals): \\
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\phantom{0}1. $P\,\Longrightarrow\, Q$ \\
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\phantom{0}2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
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\textbf{nitpick}~1 \\[2\smallskipamount]
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{\slshape Nitpick found a counterexample: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \textit{True}$ \\
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\hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
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\textbf{nitpick}~2 \\[2\smallskipamount]
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{\slshape Nitpick found a counterexample: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \textit{False}$ \\
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\hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
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\textbf{oops}
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\postw
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\subsection{Type Variables}
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\label{type-variables}
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If you are left unimpressed by the previous example, don't worry. The next
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one is more mind- and computer-boggling:
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\prew
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\textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
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\postw
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\pagebreak[2] %% TYPESETTING
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The putative lemma involves the definite description operator, {THE}, presented
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in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
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operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
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lemma is merely asserting the indefinite description operator axiom with {THE}
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substituted for {SOME}.
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The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
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containing type variables, Nitpick enumerates the possible domains for each type
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variable, up to a given cardinality (8 by default), looking for a finite
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countermodel:
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\prew
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\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
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\slshape
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Trying 8 scopes: \nopagebreak \\
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\hbox{}\qquad \textit{card}~$'a$~= 1; \\
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\hbox{}\qquad \textit{card}~$'a$~= 2; \\
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\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
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\hbox{}\qquad \textit{card}~$'a$~= 8. \\[2\smallskipamount]
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Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
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\hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
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Total time: 580 ms.
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\postw
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Nitpick found a counterexample in which $'a$ has cardinality 3. (For
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cardinalities 1 and 2, the formula holds.) In the counterexample, the three
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values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
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The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
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\textit{verbose} is enabled. You can specify \textit{verbose} each time you
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invoke \textbf{nitpick}, or you can set it globally using the command
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\prew
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\textbf{nitpick\_params} [\textit{verbose}]
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\postw
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This command also displays the current default values for all of the options
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supported by Nitpick. The options are listed in \S\ref{option-reference}.
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\subsection{Constants}
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\label{constants}
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By just looking at Nitpick's output, it might not be clear why the
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counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
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this time telling it to show the values of the constants that occur in the
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formula:
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\prew
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\textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
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\textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
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\slshape
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Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
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\hbox{}\qquad\qquad $x = a_3$ \\
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\hbox{}\qquad Constant: \nopagebreak \\
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\hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
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\postw
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We can see more clearly now. Since the predicate $P$ isn't true for a unique
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value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
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$a_1$. Since $P~a_1$ is false, the entire formula is falsified.
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As an optimization, Nitpick's preprocessor introduced the special constant
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``\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
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$\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
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satisfying $P~y$. We disable this optimization by passing the
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\textit{full\_descrs} option:
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\prew
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\textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
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\slshape
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Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
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\hbox{}\qquad\qquad $x = a_3$ \\
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\hbox{}\qquad Constant: \nopagebreak \\
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\hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
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\postw
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As the result of another optimization, Nitpick directly assigned a value to the
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subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
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disable this second optimization by using the command
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\prew
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\textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
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\textit{show\_consts}]
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\postw
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we finally get \textit{The}:
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\prew
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\slshape Constant: \nopagebreak \\
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\hbox{}\qquad $\mathit{The} = \undef{}
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    (\!\begin{aligned}[t]%
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    & \{a_1, a_2, a_3\} := a_3,\> \{a_1, a_2\} := a_3,\> \{a_1, a_3\} := a_3, \\[-2pt] %% TYPESETTING
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    & \{a_1\} := a_1,\> \{a_2, a_3\} := a_1,\> \{a_2\} := a_2, \\[-2pt]
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    & \{a_3\} := a_3,\> \{\} := a_3)\end{aligned}$
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\postw
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Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
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just like before.\footnote{The Isabelle/HOL notation $f(x :=
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y)$ denotes the function that maps $x$ to $y$ and that otherwise behaves like
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$f$.}
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Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
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unique $x$ such that'') at the front of our putative lemma's assumption:
blanchet@33191
   326
blanchet@33191
   327
\prew
blanchet@33191
   328
\textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
blanchet@33191
   329
\postw
blanchet@33191
   330
blanchet@33191
   331
The fix appears to work:
blanchet@33191
   332
blanchet@33191
   333
\prew
blanchet@33191
   334
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
   335
\slshape Nitpick found no counterexample.
blanchet@33191
   336
\postw
blanchet@33191
   337
blanchet@33191
   338
We can further increase our confidence in the formula by exhausting all
blanchet@33191
   339
cardinalities up to 50:
blanchet@33191
   340
blanchet@33191
   341
\prew
blanchet@33191
   342
\textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
blanchet@33191
   343
can be entered as \texttt{-} (hyphen) or
blanchet@33191
   344
\texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
blanchet@33191
   345
\slshape Nitpick found no counterexample.
blanchet@33191
   346
\postw
blanchet@33191
   347
blanchet@33191
   348
Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
blanchet@33191
   349
blanchet@33191
   350
\prew
blanchet@33191
   351
\textbf{sledgehammer} \\[2\smallskipamount]
blanchet@33191
   352
{\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
blanchet@33191
   353
$\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
blanchet@33191
   354
Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
blanchet@33191
   355
\textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
blanchet@33191
   356
{\slshape No subgoals!}% \\[2\smallskipamount]
blanchet@33191
   357
%\textbf{done}
blanchet@33191
   358
\postw
blanchet@33191
   359
blanchet@33191
   360
This must be our lucky day.
blanchet@33191
   361
blanchet@33191
   362
\subsection{Skolemization}
blanchet@33191
   363
\label{skolemization}
blanchet@33191
   364
blanchet@33191
   365
Are all invertible functions onto? Let's find out:
blanchet@33191
   366
blanchet@33191
   367
\prew
blanchet@33191
   368
\textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
blanchet@33191
   369
 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
blanchet@33191
   370
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
   371
\slshape
blanchet@33191
   372
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
blanchet@33191
   373
\hbox{}\qquad Free variable: \nopagebreak \\
blanchet@33191
   374
\hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
blanchet@33191
   375
\hbox{}\qquad Skolem constants: \nopagebreak \\
blanchet@33191
   376
\hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
blanchet@33191
   377
\hbox{}\qquad\qquad $y = a_2$
blanchet@33191
   378
\postw
blanchet@33191
   379
blanchet@33191
   380
Although $f$ is the only free variable occurring in the formula, Nitpick also
blanchet@33191
   381
displays values for the bound variables $g$ and $y$. These values are available
blanchet@33191
   382
to Nitpick because it performs skolemization as a preprocessing step.
blanchet@33191
   383
blanchet@33191
   384
In the previous example, skolemization only affected the outermost quantifiers.
blanchet@33191
   385
This is not always the case, as illustrated below:
blanchet@33191
   386
blanchet@33191
   387
\prew
blanchet@33191
   388
\textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
blanchet@33191
   389
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
   390
\slshape
blanchet@33191
   391
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
blanchet@33191
   392
\hbox{}\qquad Skolem constant: \nopagebreak \\
blanchet@33191
   393
\hbox{}\qquad\qquad $\lambda x.\; f =
blanchet@33191
   394
    \undef{}(\!\begin{aligned}[t]
blanchet@33191
   395
    & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
blanchet@33191
   396
    & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
blanchet@33191
   397
\postw
blanchet@33191
   398
blanchet@33191
   399
The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
blanchet@33191
   400
$x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
blanchet@33191
   401
function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
blanchet@33191
   402
maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
blanchet@33191
   403
blanchet@33191
   404
The source of the Skolem constants is sometimes more obscure:
blanchet@33191
   405
blanchet@33191
   406
\prew
blanchet@33191
   407
\textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
blanchet@33191
   408
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
   409
\slshape
blanchet@33191
   410
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
blanchet@33191
   411
\hbox{}\qquad Free variable: \nopagebreak \\
blanchet@33191
   412
\hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
blanchet@33191
   413
\hbox{}\qquad Skolem constants: \nopagebreak \\
blanchet@33191
   414
\hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
blanchet@33191
   415
\hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
blanchet@33191
   416
\postw
blanchet@33191
   417
blanchet@33191
   418
What happened here is that Nitpick expanded the \textit{sym} constant to its
blanchet@33191
   419
definition:
blanchet@33191
   420
blanchet@33191
   421
\prew
blanchet@33191
   422
$\mathit{sym}~r \,\equiv\,
blanchet@33191
   423
 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
blanchet@33191
   424
\postw
blanchet@33191
   425
blanchet@33191
   426
As their names suggest, the Skolem constants $\mathit{sym}.x$ and
blanchet@33191
   427
$\mathit{sym}.y$ are simply the bound variables $x$ and $y$
blanchet@33191
   428
from \textit{sym}'s definition.
blanchet@33191
   429
blanchet@33191
   430
Although skolemization is a useful optimization, you can disable it by invoking
blanchet@33191
   431
Nitpick with \textit{dont\_skolemize}. See \S\ref{optimizations} for details.
blanchet@33191
   432
blanchet@33191
   433
\subsection{Natural Numbers and Integers}
blanchet@33191
   434
\label{natural-numbers-and-integers}
blanchet@33191
   435
blanchet@33191
   436
Because of the axiom of infinity, the type \textit{nat} does not admit any
blanchet@34124
   437
finite models. To deal with this, Nitpick's approach is to consider finite
blanchet@34124
   438
subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined
blanchet@34124
   439
value (displayed as `$\unk$'). The type \textit{int} is handled similarly.
blanchet@34124
   440
Internally, undefined values lead to a three-valued logic.
blanchet@33191
   441
blanchet@35284
   442
Here is an example involving \textit{int\/}:
blanchet@33191
   443
blanchet@33191
   444
\prew
blanchet@33191
   445
\textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
blanchet@33191
   446
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
   447
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
   448
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@33191
   449
\hbox{}\qquad\qquad $i = 0$ \\
blanchet@33191
   450
\hbox{}\qquad\qquad $j = 1$ \\
blanchet@33191
   451
\hbox{}\qquad\qquad $m = 1$ \\
blanchet@33191
   452
\hbox{}\qquad\qquad $n = 0$
blanchet@33191
   453
\postw
blanchet@33191
   454
blanchet@34124
   455
Internally, Nitpick uses either a unary or a binary representation of numbers.
blanchet@34124
   456
The unary representation is more efficient but only suitable for numbers very
blanchet@34124
   457
close to zero. By default, Nitpick attempts to choose the more appropriate
blanchet@34124
   458
encoding by inspecting the formula at hand. This behavior can be overridden by
blanchet@34124
   459
passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For
blanchet@34124
   460
binary notation, the number of bits to use can be specified using
blanchet@34124
   461
the \textit{bits} option. For example:
blanchet@34124
   462
blanchet@34124
   463
\prew
blanchet@34124
   464
\textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$]
blanchet@34124
   465
\postw
blanchet@34124
   466
blanchet@33191
   467
With infinite types, we don't always have the luxury of a genuine counterexample
blanchet@33191
   468
and must often content ourselves with a potential one. The tedious task of
blanchet@33191
   469
finding out whether the potential counterexample is in fact genuine can be
blanchet@34124
   470
outsourced to \textit{auto} by passing \textit{check\_potential}. For example:
blanchet@33191
   471
blanchet@33191
   472
\prew
blanchet@33191
   473
\textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
blanchet@34124
   474
\textbf{nitpick} [\textit{card~nat}~= 100, \textit{check\_potential}] \\[2\smallskipamount]
blanchet@35220
   475
\slshape Warning: The conjecture either trivially holds for the given scopes or (more likely) lies outside Nitpick's supported
blanchet@35185
   476
fragment. Only potential counterexamples may be found. \\[2\smallskipamount]
blanchet@35185
   477
Nitpick found a potential counterexample: \\[2\smallskipamount]
blanchet@33191
   478
\hbox{}\qquad Free variable: \nopagebreak \\
blanchet@33191
   479
\hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
blanchet@33191
   480
Confirmation by ``\textit{auto}'': The above counterexample is genuine.
blanchet@33191
   481
\postw
blanchet@33191
   482
blanchet@33191
   483
You might wonder why the counterexample is first reported as potential. The root
blanchet@33191
   484
of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
blanchet@33191
   485
\mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
blanchet@33191
   486
that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
blanchet@33191
   487
\textit{False}; but otherwise, it does not know anything about values of $n \ge
blanchet@33191
   488
\textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
blanchet@33191
   489
\textit{True}. Since the assumption can never be satisfied, the putative lemma
blanchet@33191
   490
can never be falsified.
blanchet@33191
   491
blanchet@33191
   492
Incidentally, if you distrust the so-called genuine counterexamples, you can
blanchet@33191
   493
enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
blanchet@34124
   494
aware that \textit{auto} will usually fail to prove that the counterexample is
blanchet@33191
   495
genuine or spurious.
blanchet@33191
   496
blanchet@33191
   497
Some conjectures involving elementary number theory make Nitpick look like a
blanchet@33191
   498
giant with feet of clay:
blanchet@33191
   499
blanchet@33191
   500
\prew
blanchet@33191
   501
\textbf{lemma} ``$P~\textit{Suc}$'' \\
blanchet@35309
   502
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
   503
\slshape
blanchet@33191
   504
Nitpick found no counterexample.
blanchet@33191
   505
\postw
blanchet@33191
   506
blanchet@34124
   507
On any finite set $N$, \textit{Suc} is a partial function; for example, if $N =
blanchet@34124
   508
\{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\,
blanchet@34124
   509
\ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when passed as
blanchet@34124
   510
argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$. The next
blanchet@34124
   511
example is similar:
blanchet@33191
   512
blanchet@33191
   513
\prew
blanchet@33191
   514
\textbf{lemma} ``$P~(\textit{op}~{+}\Colon
blanchet@33191
   515
\textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
blanchet@33191
   516
\textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
blanchet@33191
   517
{\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
blanchet@33191
   518
\hbox{}\qquad Free variable: \nopagebreak \\
blanchet@33191
   519
\hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
blanchet@33191
   520
\textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
blanchet@33191
   521
{\slshape Nitpick found no counterexample.}
blanchet@33191
   522
\postw
blanchet@33191
   523
blanchet@33191
   524
The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
blanchet@33191
   525
$\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
blanchet@33191
   526
1\}$.
blanchet@33191
   527
blanchet@33191
   528
Because numbers are infinite and are approximated using a three-valued logic,
blanchet@33191
   529
there is usually no need to systematically enumerate domain sizes. If Nitpick
blanchet@33191
   530
cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
blanchet@33191
   531
unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
blanchet@33191
   532
example above is an exception to this principle.) Nitpick nonetheless enumerates
blanchet@33191
   533
all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
blanchet@33191
   534
cardinalities are fast to handle and give rise to simpler counterexamples. This
blanchet@33191
   535
is explained in more detail in \S\ref{scope-monotonicity}.
blanchet@33191
   536
blanchet@33191
   537
\subsection{Inductive Datatypes}
blanchet@33191
   538
\label{inductive-datatypes}
blanchet@33191
   539
blanchet@33191
   540
Like natural numbers and integers, inductive datatypes with recursive
blanchet@33191
   541
constructors admit no finite models and must be approximated by a subterm-closed
blanchet@33191
   542
subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
blanchet@33191
   543
Nitpick looks for all counterexamples that can be built using at most 10
blanchet@33191
   544
different lists.
blanchet@33191
   545
blanchet@33191
   546
Let's see with an example involving \textit{hd} (which returns the first element
blanchet@33191
   547
of a list) and $@$ (which concatenates two lists):
blanchet@33191
   548
blanchet@33191
   549
\prew
blanchet@33191
   550
\textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
blanchet@33191
   551
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
   552
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
blanchet@33191
   553
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@33191
   554
\hbox{}\qquad\qquad $\textit{xs} = []$ \\
blanchet@35078
   555
\hbox{}\qquad\qquad $\textit{y} = a_1$
blanchet@33191
   556
\postw
blanchet@33191
   557
blanchet@33191
   558
To see why the counterexample is genuine, we enable \textit{show\_consts}
blanchet@33191
   559
and \textit{show\_\allowbreak datatypes}:
blanchet@33191
   560
blanchet@33191
   561
\prew
blanchet@33191
   562
{\slshape Datatype:} \\
blanchet@35078
   563
\hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\
blanchet@33191
   564
{\slshape Constants:} \\
blanchet@35078
   565
\hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_1, a_1])$ \\
blanchet@35078
   566
\hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$
blanchet@33191
   567
\postw
blanchet@33191
   568
blanchet@33191
   569
Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
blanchet@33191
   570
including $a_2$.
blanchet@33191
   571
blanchet@33191
   572
The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
blanchet@35078
   573
append operator whose second argument is fixed to be $[y, y]$. Appending $[a_1,
blanchet@35078
   574
a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not
blanchet@33191
   575
representable in the subset of $'a$~\textit{list} considered by Nitpick, which
blanchet@33191
   576
is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
blanchet@35078
   577
appending $[a_1, a_1]$ to itself gives $\unk$.
blanchet@33191
   578
blanchet@33191
   579
Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
blanchet@33191
   580
considers the following subsets:
blanchet@33191
   581
blanchet@33191
   582
\kern-.5\smallskipamount %% TYPESETTING
blanchet@33191
   583
blanchet@33191
   584
\prew
blanchet@33191
   585
\begin{multicols}{3}
blanchet@33191
   586
$\{[],\, [a_1],\, [a_2]\}$; \\
blanchet@33191
   587
$\{[],\, [a_1],\, [a_3]\}$; \\
blanchet@33191
   588
$\{[],\, [a_2],\, [a_3]\}$; \\
blanchet@33191
   589
$\{[],\, [a_1],\, [a_1, a_1]\}$; \\
blanchet@33191
   590
$\{[],\, [a_1],\, [a_2, a_1]\}$; \\
blanchet@33191
   591
$\{[],\, [a_1],\, [a_3, a_1]\}$; \\
blanchet@33191
   592
$\{[],\, [a_2],\, [a_1, a_2]\}$; \\
blanchet@33191
   593
$\{[],\, [a_2],\, [a_2, a_2]\}$; \\
blanchet@33191
   594
$\{[],\, [a_2],\, [a_3, a_2]\}$; \\
blanchet@33191
   595
$\{[],\, [a_3],\, [a_1, a_3]\}$; \\
blanchet@33191
   596
$\{[],\, [a_3],\, [a_2, a_3]\}$; \\
blanchet@33191
   597
$\{[],\, [a_3],\, [a_3, a_3]\}$.
blanchet@33191
   598
\end{multicols}
blanchet@33191
   599
\postw
blanchet@33191
   600
blanchet@33191
   601
\kern-2\smallskipamount %% TYPESETTING
blanchet@33191
   602
blanchet@33191
   603
All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
blanchet@33191
   604
are listed and only those. As an example of a non-subterm-closed subset,
blanchet@35078
   605
consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe
blanchet@35078
   606
that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin
blanchet@33191
   607
\mathcal{S}$ as a subterm.
blanchet@33191
   608
blanchet@33191
   609
Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
blanchet@33191
   610
blanchet@33191
   611
\prew
blanchet@33191
   612
\textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
blanchet@33191
   613
\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
blanchet@33191
   614
\\
blanchet@33191
   615
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
blanchet@33191
   616
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
blanchet@33191
   617
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@35078
   618
\hbox{}\qquad\qquad $\textit{xs} = [a_1]$ \\
blanchet@35078
   619
\hbox{}\qquad\qquad $\textit{ys} = [a_2]$ \\
blanchet@33191
   620
\hbox{}\qquad Datatypes: \\
blanchet@33191
   621
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
blanchet@35078
   622
\hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$
blanchet@33191
   623
\postw
blanchet@33191
   624
blanchet@33191
   625
Because datatypes are approximated using a three-valued logic, there is usually
blanchet@33191
   626
no need to systematically enumerate cardinalities: If Nitpick cannot find a
blanchet@33191
   627
genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
blanchet@33191
   628
unlikely that one could be found for smaller cardinalities.
blanchet@33191
   629
blanchet@35284
   630
\subsection{Typedefs, Quotient Types, Records, Rationals, and Reals}
blanchet@33191
   631
\label{typedefs-records-rationals-and-reals}
blanchet@33191
   632
blanchet@33191
   633
Nitpick generally treats types declared using \textbf{typedef} as datatypes
blanchet@33191
   634
whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
blanchet@33191
   635
For example:
blanchet@33191
   636
blanchet@33191
   637
\prew
blanchet@33191
   638
\textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
blanchet@33191
   639
\textbf{by}~\textit{blast} \\[2\smallskipamount]
blanchet@33191
   640
\textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
blanchet@33191
   641
\textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
blanchet@33191
   642
\textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
blanchet@33191
   643
\textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
blanchet@33191
   644
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
blanchet@33191
   645
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
   646
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@35078
   647
\hbox{}\qquad\qquad $P = \{\Abs{0},\, \Abs{1}\}$ \\
blanchet@33191
   648
\hbox{}\qquad\qquad $x = \Abs{2}$ \\
blanchet@33191
   649
\hbox{}\qquad Datatypes: \\
blanchet@33191
   650
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
blanchet@35078
   651
\hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$
blanchet@33191
   652
\postw
blanchet@33191
   653
blanchet@33191
   654
In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
blanchet@33191
   655
blanchet@35284
   656
Quotient types are handled in much the same way. The following fragment defines
blanchet@35284
   657
the integer type \textit{my\_int} by encoding the integer $x$ by a pair of
blanchet@35284
   658
natural numbers $(m, n)$ such that $x + n = m$:
blanchet@35284
   659
blanchet@35284
   660
\prew
blanchet@35284
   661
\textbf{fun} \textit{my\_int\_rel} \textbf{where} \\
blanchet@35284
   662
``$\textit{my\_int\_rel}~(x,\, y)~(u,\, v) = (x + v = u + y)$'' \\[2\smallskipamount]
blanchet@35284
   663
%
blanchet@35284
   664
\textbf{quotient\_type}~\textit{my\_int} = ``$\textit{nat} \times \textit{nat\/}$''$\;{/}\;$\textit{my\_int\_rel} \\
blanchet@35284
   665
\textbf{by}~(\textit{auto simp add\/}:\ \textit{equivp\_def expand\_fun\_eq}) \\[2\smallskipamount]
blanchet@35284
   666
%
blanchet@35284
   667
\textbf{definition}~\textit{add\_raw}~\textbf{where} \\
blanchet@35284
   668
``$\textit{add\_raw} \,\equiv\, \lambda(x,\, y)~(u,\, v).\; (x + (u\Colon\textit{nat}), y + (v\Colon\textit{nat}))$'' \\[2\smallskipamount]
blanchet@35284
   669
%
blanchet@35284
   670
\textbf{quotient\_definition} ``$\textit{add\/}\Colon\textit{my\_int} \Rightarrow \textit{my\_int} \Rightarrow \textit{my\_int\/}$'' \textbf{is} \textit{add\_raw} \\[2\smallskipamount]
blanchet@35284
   671
%
blanchet@35284
   672
\textbf{lemma} ``$\textit{add}~x~y = \textit{add}~x~x$'' \\
blanchet@35284
   673
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
blanchet@35284
   674
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@35284
   675
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@35284
   676
\hbox{}\qquad\qquad $x = \Abs{(0,\, 0)}$ \\
blanchet@35284
   677
\hbox{}\qquad\qquad $y = \Abs{(1,\, 0)}$ \\
blanchet@35284
   678
\hbox{}\qquad Datatypes: \\
blanchet@35284
   679
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, \unr\}$ \\
blanchet@35284
   680
\hbox{}\qquad\qquad $\textit{nat} \times \textit{nat} = \{(0,\, 0),\> (1,\, 0),\> \unr\}$ \\
blanchet@35284
   681
\hbox{}\qquad\qquad $\textit{my\_int} = \{\Abs{(0,\, 0)},\> \Abs{(1,\, 0)},\> \unr\}$
blanchet@35284
   682
\postw
blanchet@35284
   683
blanchet@35284
   684
In the counterexample, $\Abs{(0,\, 0)}$ and $\Abs{(1,\, 0)}$ represent the
blanchet@35284
   685
integers $0$ and $1$, respectively. Other representants would have been
blanchet@35284
   686
possible---e.g., $\Abs{(5,\, 5)}$ and $\Abs{(12,\, 11)}$.
blanchet@35284
   687
blanchet@35284
   688
Records are also handled as datatypes with a single constructor:
blanchet@33191
   689
blanchet@33191
   690
\prew
blanchet@33191
   691
\textbf{record} \textit{point} = \\
blanchet@33191
   692
\hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
blanchet@33191
   693
\hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
blanchet@33191
   694
\textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
blanchet@33191
   695
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
blanchet@33191
   696
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
   697
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@35078
   698
\hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
blanchet@35078
   699
\hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
blanchet@33191
   700
\hbox{}\qquad Datatypes: \\
blanchet@33191
   701
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
blanchet@35078
   702
\hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
blanchet@35078
   703
& \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
blanchet@35078
   704
& \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
blanchet@33191
   705
\postw
blanchet@33191
   706
blanchet@35284
   707
blanchet@35284
   708
blanchet@33191
   709
Finally, Nitpick provides rudimentary support for rationals and reals using a
blanchet@33191
   710
similar approach:
blanchet@33191
   711
blanchet@33191
   712
\prew
blanchet@33191
   713
\textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
blanchet@33191
   714
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
blanchet@33191
   715
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
   716
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@33191
   717
\hbox{}\qquad\qquad $x = 1/2$ \\
blanchet@33191
   718
\hbox{}\qquad\qquad $y = -1/2$ \\
blanchet@33191
   719
\hbox{}\qquad Datatypes: \\
blanchet@33191
   720
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
blanchet@33191
   721
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
blanchet@33191
   722
\hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
blanchet@33191
   723
\postw
blanchet@33191
   724
blanchet@33191
   725
\subsection{Inductive and Coinductive Predicates}
blanchet@33191
   726
\label{inductive-and-coinductive-predicates}
blanchet@33191
   727
blanchet@33191
   728
Inductively defined predicates (and sets) are particularly problematic for
blanchet@33191
   729
counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
blanchet@33191
   730
loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
blanchet@33191
   731
the problem is that they are defined using a least fixed point construction.
blanchet@33191
   732
blanchet@33191
   733
Nitpick's philosophy is that not all inductive predicates are equal. Consider
blanchet@33191
   734
the \textit{even} predicate below:
blanchet@33191
   735
blanchet@33191
   736
\prew
blanchet@33191
   737
\textbf{inductive}~\textit{even}~\textbf{where} \\
blanchet@33191
   738
``\textit{even}~0'' $\,\mid$ \\
blanchet@33191
   739
``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
blanchet@33191
   740
\postw
blanchet@33191
   741
blanchet@33191
   742
This predicate enjoys the desirable property of being well-founded, which means
blanchet@33191
   743
that the introduction rules don't give rise to infinite chains of the form
blanchet@33191
   744
blanchet@33191
   745
\prew
blanchet@33191
   746
$\cdots\,\Longrightarrow\, \textit{even}~k''
blanchet@33191
   747
       \,\Longrightarrow\, \textit{even}~k'
blanchet@33191
   748
       \,\Longrightarrow\, \textit{even}~k.$
blanchet@33191
   749
\postw
blanchet@33191
   750
blanchet@33191
   751
For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
blanchet@33191
   752
$k/2 + 1$:
blanchet@33191
   753
blanchet@33191
   754
\prew
blanchet@33191
   755
$\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
blanchet@33191
   756
       \,\Longrightarrow\, \textit{even}~(k - 2)
blanchet@33191
   757
       \,\Longrightarrow\, \textit{even}~k.$
blanchet@33191
   758
\postw
blanchet@33191
   759
blanchet@33191
   760
Wellfoundedness is desirable because it enables Nitpick to use a very efficient
blanchet@33191
   761
fixed point computation.%
blanchet@33191
   762
\footnote{If an inductive predicate is
blanchet@33191
   763
well-founded, then it has exactly one fixed point, which is simultaneously the
blanchet@33191
   764
least and the greatest fixed point. In these circumstances, the computation of
blanchet@33191
   765
the least fixed point amounts to the computation of an arbitrary fixed point,
blanchet@33191
   766
which can be performed using a straightforward recursive equation.}
blanchet@33191
   767
Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
blanchet@33191
   768
just as Isabelle's \textbf{function} package usually discharges termination
blanchet@33191
   769
proof obligations automatically.
blanchet@33191
   770
blanchet@33191
   771
Let's try an example:
blanchet@33191
   772
blanchet@33191
   773
\prew
blanchet@33191
   774
\textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
blanchet@34126
   775
\textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
blanchet@33191
   776
\slshape The inductive predicate ``\textit{even}'' was proved well-founded.
blanchet@33191
   777
Nitpick can compute it efficiently. \\[2\smallskipamount]
blanchet@33191
   778
Trying 1 scope: \\
blanchet@33191
   779
\hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
blanchet@33191
   780
Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
blanchet@33191
   781
\hbox{}\qquad Empty assignment \\[2\smallskipamount]
blanchet@33191
   782
Nitpick could not find a better counterexample. \\[2\smallskipamount]
blanchet@33191
   783
Total time: 2274 ms.
blanchet@33191
   784
\postw
blanchet@33191
   785
blanchet@33191
   786
No genuine counterexample is possible because Nitpick cannot rule out the
blanchet@33191
   787
existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
blanchet@33191
   788
$\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
blanchet@33191
   789
existential quantifier:
blanchet@33191
   790
blanchet@33191
   791
\prew
blanchet@33191
   792
\textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
blanchet@34126
   793
\textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}] \\[2\smallskipamount]
blanchet@33191
   794
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
   795
\hbox{}\qquad Empty assignment
blanchet@33191
   796
\postw
blanchet@33191
   797
blanchet@33191
   798
So far we were blessed by the wellfoundedness of \textit{even}. What happens if
blanchet@33191
   799
we use the following definition instead?
blanchet@33191
   800
blanchet@33191
   801
\prew
blanchet@33191
   802
\textbf{inductive} $\textit{even}'$ \textbf{where} \\
blanchet@33191
   803
``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
blanchet@33191
   804
``$\textit{even}'~2$'' $\,\mid$ \\
blanchet@33191
   805
``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
blanchet@33191
   806
\postw
blanchet@33191
   807
blanchet@33191
   808
This definition is not well-founded: From $\textit{even}'~0$ and
blanchet@33191
   809
$\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
blanchet@33191
   810
predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
blanchet@33191
   811
blanchet@33191
   812
Let's check a property involving $\textit{even}'$. To make up for the
blanchet@33191
   813
foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
blanchet@33191
   814
\textit{nat}'s cardinality to a mere 10:
blanchet@33191
   815
blanchet@33191
   816
\prew
blanchet@33191
   817
\textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
blanchet@33191
   818
\lnot\;\textit{even}'~n$'' \\
blanchet@33191
   819
\textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
blanchet@33191
   820
\slshape
blanchet@33191
   821
The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
blanchet@33191
   822
Nitpick might need to unroll it. \\[2\smallskipamount]
blanchet@33191
   823
Trying 6 scopes: \\
blanchet@33191
   824
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
blanchet@33191
   825
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
blanchet@33191
   826
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
blanchet@33191
   827
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
blanchet@33191
   828
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
blanchet@33191
   829
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
blanchet@33191
   830
Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
blanchet@33191
   831
\hbox{}\qquad Constant: \nopagebreak \\
blanchet@33191
   832
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
blanchet@33191
   833
& 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
blanchet@33191
   834
& 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
blanchet@33191
   835
& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
blanchet@33191
   836
Total time: 1140 ms.
blanchet@33191
   837
\postw
blanchet@33191
   838
blanchet@33191
   839
Nitpick's output is very instructive. First, it tells us that the predicate is
blanchet@33191
   840
unrolled, meaning that it is computed iteratively from the empty set. Then it
blanchet@33191
   841
lists six scopes specifying different bounds on the numbers of iterations:\ 0,
blanchet@33191
   842
1, 2, 4, 8, and~9.
blanchet@33191
   843
blanchet@33191
   844
The output also shows how each iteration contributes to $\textit{even}'$. The
blanchet@33191
   845
notation $\lambda i.\; \textit{even}'$ indicates that the value of the
blanchet@33191
   846
predicate depends on an iteration counter. Iteration 0 provides the basis
blanchet@33191
   847
elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
blanchet@33191
   848
throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
blanchet@33191
   849
iterations would not contribute any new elements.
blanchet@33191
   850
blanchet@33191
   851
Some values are marked with superscripted question
blanchet@33191
   852
marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
blanchet@33191
   853
predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
blanchet@33191
   854
\textit{True} or $\unk$, never \textit{False}.
blanchet@33191
   855
blanchet@33191
   856
When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
blanchet@33191
   857
iterations. However, these numbers are bounded by the cardinality of the
blanchet@33191
   858
predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
blanchet@33191
   859
ever needed to compute the value of a \textit{nat} predicate. You can specify
blanchet@33191
   860
the number of iterations using the \textit{iter} option, as explained in
blanchet@33191
   861
\S\ref{scope-of-search}.
blanchet@33191
   862
blanchet@33191
   863
In the next formula, $\textit{even}'$ occurs both positively and negatively:
blanchet@33191
   864
blanchet@33191
   865
\prew
blanchet@33191
   866
\textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
blanchet@34124
   867
\textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
blanchet@33191
   868
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
   869
\hbox{}\qquad Free variable: \nopagebreak \\
blanchet@33191
   870
\hbox{}\qquad\qquad $n = 1$ \\
blanchet@33191
   871
\hbox{}\qquad Constants: \nopagebreak \\
blanchet@33191
   872
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
blanchet@33191
   873
& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$  \\
blanchet@33191
   874
\hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
blanchet@33191
   875
\postw
blanchet@33191
   876
blanchet@33191
   877
Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
blanchet@33191
   878
8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
blanchet@33191
   879
fixed point (not necessarily the least one). It is used to falsify
blanchet@33191
   880
$\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
blanchet@33191
   881
$\textit{even}'~(n - 2)$.
blanchet@33191
   882
blanchet@33191
   883
Coinductive predicates are handled dually. For example:
blanchet@33191
   884
blanchet@33191
   885
\prew
blanchet@33191
   886
\textbf{coinductive} \textit{nats} \textbf{where} \\
blanchet@33191
   887
``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
blanchet@33191
   888
\textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
blanchet@33191
   889
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
blanchet@33191
   890
\slshape Nitpick found a counterexample:
blanchet@33191
   891
\\[2\smallskipamount]
blanchet@33191
   892
\hbox{}\qquad Constants: \nopagebreak \\
blanchet@33191
   893
\hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
blanchet@33191
   894
& 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
blanchet@33191
   895
& \unr\})\end{aligned}$ \\
blanchet@33191
   896
\hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
blanchet@33191
   897
\postw
blanchet@33191
   898
blanchet@33191
   899
As a special case, Nitpick uses Kodkod's transitive closure operator to encode
blanchet@33191
   900
negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
blanchet@33191
   901
inductive predicates for which each the predicate occurs in at most one
blanchet@33191
   902
assumption of each introduction rule. For example:
blanchet@33191
   903
blanchet@33191
   904
\prew
blanchet@33191
   905
\textbf{inductive} \textit{odd} \textbf{where} \\
blanchet@33191
   906
``$\textit{odd}~1$'' $\,\mid$ \\
blanchet@33191
   907
``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
blanchet@33191
   908
\textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
blanchet@33191
   909
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
blanchet@33191
   910
\slshape Nitpick found a counterexample:
blanchet@33191
   911
\\[2\smallskipamount]
blanchet@33191
   912
\hbox{}\qquad Free variable: \nopagebreak \\
blanchet@33191
   913
\hbox{}\qquad\qquad $n = 1$ \\
blanchet@33191
   914
\hbox{}\qquad Constants: \nopagebreak \\
blanchet@33191
   915
\hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
blanchet@33191
   916
\hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
blanchet@33191
   917
\hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
blanchet@33191
   918
\!\begin{aligned}[t]
blanchet@33191
   919
  & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
blanchet@33191
   920
  & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
blanchet@33191
   921
       (3, 5), \\[-2pt]
blanchet@33191
   922
  & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
blanchet@33191
   923
  & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
blanchet@33191
   924
\hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
blanchet@33191
   925
\postw
blanchet@33191
   926
blanchet@33191
   927
\noindent
blanchet@33191
   928
In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
blanchet@33191
   929
$\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
blanchet@33191
   930
elements from known ones. The set $\textit{odd}$ consists of all the values
blanchet@33191
   931
reachable through the reflexive transitive closure of
blanchet@33191
   932
$\textit{odd}_{\textrm{step}}$ starting with any element from
blanchet@33191
   933
$\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
blanchet@33191
   934
transitive closure to encode linear predicates is normally either more thorough
blanchet@33191
   935
or more efficient than unrolling (depending on the value of \textit{iter}), but
blanchet@33191
   936
for those cases where it isn't you can disable it by passing the
blanchet@33191
   937
\textit{dont\_star\_linear\_preds} option.
blanchet@33191
   938
blanchet@33191
   939
\subsection{Coinductive Datatypes}
blanchet@33191
   940
\label{coinductive-datatypes}
blanchet@33191
   941
blanchet@33191
   942
While Isabelle regrettably lacks a high-level mechanism for defining coinductive
blanchet@33191
   943
datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
blanchet@33191
   944
list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
blanchet@33191
   945
these lazy lists seamlessly and provides a hook, described in
blanchet@33191
   946
\S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
blanchet@33191
   947
datatypes.
blanchet@33191
   948
blanchet@33191
   949
(Co)intuitively, a coinductive datatype is similar to an inductive datatype but
blanchet@33191
   950
allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
blanchet@33191
   951
\ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
blanchet@33191
   952
1, 2, 3, \ldots]$ can be defined as lazy lists using the
blanchet@33191
   953
$\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
blanchet@33191
   954
$\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
blanchet@33191
   955
\mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
blanchet@33191
   956
blanchet@33191
   957
Although it is otherwise no friend of infinity, Nitpick can find counterexamples
blanchet@33191
   958
involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
blanchet@33191
   959
finite lists:
blanchet@33191
   960
blanchet@33191
   961
\prew
blanchet@33191
   962
\textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
blanchet@33191
   963
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
   964
\slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
blanchet@33191
   965
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@33191
   966
\hbox{}\qquad\qquad $\textit{a} = a_1$ \\
blanchet@33191
   967
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
blanchet@33191
   968
\postw
blanchet@33191
   969
blanchet@33191
   970
The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
blanchet@33191
   971
for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
blanchet@33191
   972
infinite list $[a_1, a_1, a_1, \ldots]$.
blanchet@33191
   973
blanchet@33191
   974
The next example is more interesting:
blanchet@33191
   975
blanchet@33191
   976
\prew
blanchet@33191
   977
\textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
blanchet@33191
   978
\textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
blanchet@33191
   979
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
blanchet@33191
   980
\slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
blanchet@33191
   981
some scopes. \\[2\smallskipamount]
blanchet@33191
   982
Trying 8 scopes: \\
blanchet@35284
   983
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 1,
blanchet@33191
   984
and \textit{bisim\_depth}~= 0. \\
blanchet@33191
   985
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
blanchet@35284
   986
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 8,
blanchet@33191
   987
and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
blanchet@33191
   988
Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
blanchet@35284
   989
\textit{card}~``\kern1pt$'a~\textit{list\/}$''~= 2, and \textit{bisim\_\allowbreak
blanchet@33191
   990
depth}~= 1:
blanchet@33191
   991
\\[2\smallskipamount]
blanchet@33191
   992
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@35078
   993
\hbox{}\qquad\qquad $\textit{a} = a_1$ \\
blanchet@35078
   994
\hbox{}\qquad\qquad $\textit{b} = a_2$ \\
blanchet@35078
   995
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
blanchet@35078
   996
\hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
blanchet@33191
   997
Total time: 726 ms.
blanchet@33191
   998
\postw
blanchet@33191
   999
blanchet@35078
  1000
The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
blanchet@35078
  1001
$\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
blanchet@35078
  1002
$[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment
blanchet@33191
  1003
within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
blanchet@33191
  1004
the segment leading to the binder is the stem.
blanchet@33191
  1005
blanchet@33191
  1006
A salient property of coinductive datatypes is that two objects are considered
blanchet@33191
  1007
equal if and only if they lead to the same observations. For example, the lazy
blanchet@33191
  1008
lists $\textrm{THE}~\omega.\; \omega =
blanchet@33191
  1009
\textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
blanchet@33191
  1010
$\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
blanchet@33191
  1011
\textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
blanchet@33191
  1012
to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
blanchet@33191
  1013
equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
blanchet@33191
  1014
concept of equality for coinductive datatypes is called bisimulation and is
blanchet@33191
  1015
defined coinductively.
blanchet@33191
  1016
blanchet@33191
  1017
Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
blanchet@33191
  1018
the Kodkod problem to ensure that distinct objects lead to different
blanchet@33191
  1019
observations. This precaution is somewhat expensive and often unnecessary, so it
blanchet@33191
  1020
can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
blanchet@33191
  1021
bisimilarity check is then performed \textsl{after} the counterexample has been
blanchet@33191
  1022
found to ensure correctness. If this after-the-fact check fails, the
blanchet@33191
  1023
counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
blanchet@33191
  1024
again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
blanchet@33191
  1025
check for the previous example saves approximately 150~milli\-seconds; the speed
blanchet@33191
  1026
gains can be more significant for larger scopes.
blanchet@33191
  1027
blanchet@33191
  1028
The next formula illustrates the need for bisimilarity (either as a Kodkod
blanchet@33191
  1029
predicate or as an after-the-fact check) to prevent spurious counterexamples:
blanchet@33191
  1030
blanchet@33191
  1031
\prew
blanchet@33191
  1032
\textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
blanchet@33191
  1033
\,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
blanchet@34124
  1034
\textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
blanchet@33191
  1035
\slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
blanchet@33191
  1036
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@35078
  1037
\hbox{}\qquad\qquad $a = a_1$ \\
blanchet@33191
  1038
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
blanchet@35078
  1039
\textit{LCons}~a_1~\omega$ \\
blanchet@35078
  1040
\hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
blanchet@33191
  1041
\hbox{}\qquad Codatatype:\strut \nopagebreak \\
blanchet@33191
  1042
\hbox{}\qquad\qquad $'a~\textit{llist} =
blanchet@33191
  1043
\{\!\begin{aligned}[t]
blanchet@35078
  1044
  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
blanchet@35078
  1045
  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
blanchet@33191
  1046
\\[2\smallskipamount]
blanchet@33191
  1047
Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
blanchet@33191
  1048
that the counterexample is genuine. \\[2\smallskipamount]
blanchet@33191
  1049
{\upshape\textbf{nitpick}} \\[2\smallskipamount]
blanchet@33191
  1050
\slshape Nitpick found no counterexample.
blanchet@33191
  1051
\postw
blanchet@33191
  1052
blanchet@33191
  1053
In the first \textbf{nitpick} invocation, the after-the-fact check discovered 
blanchet@33191
  1054
that the two known elements of type $'a~\textit{llist}$ are bisimilar.
blanchet@33191
  1055
blanchet@33191
  1056
A compromise between leaving out the bisimilarity predicate from the Kodkod
blanchet@33191
  1057
problem and performing the after-the-fact check is to specify a lower
blanchet@33191
  1058
nonnegative \textit{bisim\_depth} value than the default one provided by
blanchet@33191
  1059
Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
blanchet@33191
  1060
be distinguished from each other by their prefixes of length $K$. Be aware that
blanchet@33191
  1061
setting $K$ to a too low value can overconstrain Nitpick, preventing it from
blanchet@33191
  1062
finding any counterexamples.
blanchet@33191
  1063
blanchet@33191
  1064
\subsection{Boxing}
blanchet@33191
  1065
\label{boxing}
blanchet@33191
  1066
blanchet@33191
  1067
Nitpick normally maps function and product types directly to the corresponding
blanchet@33191
  1068
Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
blanchet@33191
  1069
cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
blanchet@33191
  1070
\Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
blanchet@33191
  1071
off to treat these types in the same way as plain datatypes, by approximating
blanchet@33191
  1072
them by a subset of a given cardinality. This technique is called ``boxing'' and
blanchet@33191
  1073
is particularly useful for functions passed as arguments to other functions, for
blanchet@33191
  1074
high-arity functions, and for large tuples. Under the hood, boxing involves
blanchet@33191
  1075
wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
blanchet@33191
  1076
isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
blanchet@33191
  1077
blanchet@33191
  1078
To illustrate boxing, we consider a formalization of $\lambda$-terms represented
blanchet@33191
  1079
using de Bruijn's notation:
blanchet@33191
  1080
blanchet@33191
  1081
\prew
blanchet@33191
  1082
\textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
blanchet@33191
  1083
\postw
blanchet@33191
  1084
blanchet@33191
  1085
The $\textit{lift}~t~k$ function increments all variables with indices greater
blanchet@33191
  1086
than or equal to $k$ by one:
blanchet@33191
  1087
blanchet@33191
  1088
\prew
blanchet@33191
  1089
\textbf{primrec} \textit{lift} \textbf{where} \\
blanchet@33191
  1090
``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
blanchet@33191
  1091
``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
blanchet@33191
  1092
``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
blanchet@33191
  1093
\postw
blanchet@33191
  1094
blanchet@33191
  1095
The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
blanchet@33191
  1096
term $t$ has a loose variable with index $k$ or more:
blanchet@33191
  1097
blanchet@33191
  1098
\prew
blanchet@33191
  1099
\textbf{primrec}~\textit{loose} \textbf{where} \\
blanchet@33191
  1100
``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
blanchet@33191
  1101
``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
blanchet@33191
  1102
``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
blanchet@33191
  1103
\postw
blanchet@33191
  1104
blanchet@33191
  1105
Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
blanchet@33191
  1106
on $t$:
blanchet@33191
  1107
blanchet@33191
  1108
\prew
blanchet@33191
  1109
\textbf{primrec}~\textit{subst} \textbf{where} \\
blanchet@33191
  1110
``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
blanchet@33191
  1111
``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
blanchet@33191
  1112
\phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
blanchet@33191
  1113
``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
blanchet@33191
  1114
\postw
blanchet@33191
  1115
blanchet@33191
  1116
A substitution is a function that maps variable indices to terms. Observe that
blanchet@33191
  1117
$\sigma$ is a function passed as argument and that Nitpick can't optimize it
blanchet@33191
  1118
away, because the recursive call for the \textit{Lam} case involves an altered
blanchet@33191
  1119
version. Also notice the \textit{lift} call, which increments the variable
blanchet@33191
  1120
indices when moving under a \textit{Lam}.
blanchet@33191
  1121
blanchet@33191
  1122
A reasonable property to expect of substitution is that it should leave closed
blanchet@33191
  1123
terms unchanged. Alas, even this simple property does not hold:
blanchet@33191
  1124
blanchet@33191
  1125
\pre
blanchet@33191
  1126
\textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
blanchet@33191
  1127
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
blanchet@33191
  1128
\slshape
blanchet@33191
  1129
Trying 8 scopes: \nopagebreak \\
blanchet@33191
  1130
\hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
blanchet@33191
  1131
\hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
blanchet@33191
  1132
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
blanchet@33191
  1133
\hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
blanchet@33191
  1134
Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
blanchet@33191
  1135
and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
blanchet@33191
  1136
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@33191
  1137
\hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
blanchet@33191
  1138
& 0 := \textit{Var}~0,\>
blanchet@33191
  1139
  1 := \textit{Var}~0,\>
blanchet@33191
  1140
  2 := \textit{Var}~0, \\[-2pt]
blanchet@33191
  1141
& 3 := \textit{Var}~0,\>
blanchet@33191
  1142
  4 := \textit{Var}~0,\>
blanchet@33191
  1143
  5 := \textit{Var}~0)\end{aligned}$ \\
blanchet@33191
  1144
\hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
blanchet@33191
  1145
Total time: $4679$ ms.
blanchet@33191
  1146
\postw
blanchet@33191
  1147
blanchet@33191
  1148
Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
blanchet@33191
  1149
\textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
blanchet@33191
  1150
$\lambda$-term notation, $t$~is
blanchet@33191
  1151
$\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
blanchet@35284
  1152
The bug is in \textit{subst\/}: The $\textit{lift}~(\sigma~m)~1$ call should be
blanchet@33191
  1153
replaced with $\textit{lift}~(\sigma~m)~0$.
blanchet@33191
  1154
blanchet@33191
  1155
An interesting aspect of Nitpick's verbose output is that it assigned inceasing
blanchet@33191
  1156
cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
blanchet@33191
  1157
For the formula of interest, knowing 6 values of that type was enough to find
blanchet@33191
  1158
the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
blanchet@33191
  1159
considered, a hopeless undertaking:
blanchet@33191
  1160
blanchet@33191
  1161
\prew
blanchet@33191
  1162
\textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
blanchet@33191
  1163
{\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
blanchet@33191
  1164
\postw
blanchet@33191
  1165
blanchet@33191
  1166
{\looseness=-1
blanchet@33191
  1167
Boxing can be enabled or disabled globally or on a per-type basis using the
blanchet@33191
  1168
\textit{box} option. Moreover, setting the cardinality of a function or
blanchet@33191
  1169
product type implicitly enables boxing for that type. Nitpick usually performs
blanchet@33191
  1170
reasonable choices about which types should be boxed, but option tweaking
blanchet@33191
  1171
sometimes helps.
blanchet@33191
  1172
blanchet@33191
  1173
}
blanchet@33191
  1174
blanchet@33191
  1175
\subsection{Scope Monotonicity}
blanchet@33191
  1176
\label{scope-monotonicity}
blanchet@33191
  1177
blanchet@33191
  1178
The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
blanchet@33191
  1179
and \textit{max}) controls which scopes are actually tested. In general, to
blanchet@33191
  1180
exhaust all models below a certain cardinality bound, the number of scopes that
blanchet@33191
  1181
Nitpick must consider increases exponentially with the number of type variables
blanchet@33191
  1182
(and \textbf{typedecl}'d types) occurring in the formula. Given the default
blanchet@33191
  1183
cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
blanchet@33191
  1184
considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
blanchet@33191
  1185
blanchet@33191
  1186
Fortunately, many formulas exhibit a property called \textsl{scope
blanchet@33191
  1187
monotonicity}, meaning that if the formula is falsifiable for a given scope,
blanchet@33191
  1188
it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
blanchet@33191
  1189
blanchet@33191
  1190
Consider the formula
blanchet@33191
  1191
blanchet@33191
  1192
\prew
blanchet@33191
  1193
\textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
blanchet@33191
  1194
\postw
blanchet@33191
  1195
blanchet@33191
  1196
where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
blanchet@33191
  1197
$'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
blanchet@33191
  1198
exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
blanchet@33191
  1199
that any counterexample found with a small scope would still be a counterexample
blanchet@33191
  1200
in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
blanchet@33191
  1201
by the larger scope. Nitpick comes to the same conclusion after a careful
blanchet@33191
  1202
inspection of the formula and the relevant definitions:
blanchet@33191
  1203
blanchet@33191
  1204
\prew
blanchet@33191
  1205
\textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
blanchet@33191
  1206
\slshape
blanchet@33191
  1207
The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
blanchet@33191
  1208
Nitpick might be able to skip some scopes.
blanchet@33191
  1209
 \\[2\smallskipamount]
blanchet@33191
  1210
Trying 8 scopes: \\
blanchet@33191
  1211
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
blanchet@33191
  1212
\textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
blanchet@33191
  1213
\textit{list}''~= 1, \\
blanchet@33191
  1214
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
blanchet@33191
  1215
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
blanchet@33191
  1216
\hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
blanchet@33191
  1217
\textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
blanchet@33191
  1218
\textit{list}''~= 2, \\
blanchet@33191
  1219
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
blanchet@33191
  1220
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
blanchet@33191
  1221
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
blanchet@33191
  1222
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
blanchet@33191
  1223
\textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
blanchet@33191
  1224
\textit{list}''~= 8, \\
blanchet@33191
  1225
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
blanchet@33191
  1226
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
blanchet@33191
  1227
\\[2\smallskipamount]
blanchet@33191
  1228
Nitpick found a counterexample for
blanchet@33191
  1229
\textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
blanchet@33191
  1230
\textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
blanchet@33191
  1231
\textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
blanchet@33191
  1232
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
blanchet@33191
  1233
\\[2\smallskipamount]
blanchet@33191
  1234
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@35078
  1235
\hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
blanchet@35078
  1236
\hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
blanchet@33191
  1237
Total time: 1636 ms.
blanchet@33191
  1238
\postw
blanchet@33191
  1239
blanchet@33191
  1240
In theory, it should be sufficient to test a single scope:
blanchet@33191
  1241
blanchet@33191
  1242
\prew
blanchet@33191
  1243
\textbf{nitpick}~[\textit{card}~= 8]
blanchet@33191
  1244
\postw
blanchet@33191
  1245
blanchet@33191
  1246
However, this is often less efficient in practice and may lead to overly complex
blanchet@33191
  1247
counterexamples.
blanchet@33191
  1248
blanchet@33191
  1249
If the monotonicity check fails but we believe that the formula is monotonic (or
blanchet@33191
  1250
we don't mind missing some counterexamples), we can pass the
blanchet@33191
  1251
\textit{mono} option. To convince yourself that this option is risky,
blanchet@33191
  1252
simply consider this example from \S\ref{skolemization}:
blanchet@33191
  1253
blanchet@33191
  1254
\prew
blanchet@33191
  1255
\textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
blanchet@33191
  1256
 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
blanchet@33191
  1257
\textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
blanchet@33191
  1258
{\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
blanchet@33191
  1259
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
  1260
\slshape
blanchet@33191
  1261
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
blanchet@33191
  1262
\hbox{}\qquad $\vdots$
blanchet@33191
  1263
\postw
blanchet@33191
  1264
blanchet@33191
  1265
(It turns out the formula holds if and only if $\textit{card}~'a \le
blanchet@33191
  1266
\textit{card}~'b$.) Although this is rarely advisable, the automatic
blanchet@33191
  1267
monotonicity checks can be disabled by passing \textit{non\_mono}
blanchet@33191
  1268
(\S\ref{optimizations}).
blanchet@33191
  1269
blanchet@33191
  1270
As insinuated in \S\ref{natural-numbers-and-integers} and
blanchet@33191
  1271
\S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
blanchet@33191
  1272
are normally monotonic and treated as such. The same is true for record types,
blanchet@33191
  1273
\textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
blanchet@33191
  1274
cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
blanchet@33191
  1275
\textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
blanchet@33191
  1276
consider only 8~scopes instead of $32\,768$.
blanchet@33191
  1277
blanchet@34982
  1278
\subsection{Inductive Properties}
blanchet@34982
  1279
\label{inductive-properties}
blanchet@34982
  1280
blanchet@34982
  1281
Inductive properties are a particular pain to prove, because the failure to
blanchet@34982
  1282
establish an induction step can mean several things:
blanchet@34982
  1283
%
blanchet@34982
  1284
\begin{enumerate}
blanchet@34982
  1285
\item The property is invalid.
blanchet@34982
  1286
\item The property is valid but is too weak to support the induction step.
blanchet@34982
  1287
\item The property is valid and strong enough; it's just that we haven't found
blanchet@34982
  1288
the proof yet.
blanchet@34982
  1289
\end{enumerate}
blanchet@34982
  1290
%
blanchet@34982
  1291
Depending on which scenario applies, we would take the appropriate course of
blanchet@34982
  1292
action:
blanchet@34982
  1293
%
blanchet@34982
  1294
\begin{enumerate}
blanchet@34982
  1295
\item Repair the statement of the property so that it becomes valid.
blanchet@34982
  1296
\item Generalize the property and/or prove auxiliary properties.
blanchet@34982
  1297
\item Work harder on a proof.
blanchet@34982
  1298
\end{enumerate}
blanchet@34982
  1299
%
blanchet@34982
  1300
How can we distinguish between the three scenarios? Nitpick's normal mode of
blanchet@34982
  1301
operation can often detect scenario 1, and Isabelle's automatic tactics help with
blanchet@34982
  1302
scenario 3. Using appropriate techniques, it is also often possible to use
blanchet@34982
  1303
Nitpick to identify scenario 2. Consider the following transition system,
blanchet@34982
  1304
in which natural numbers represent states:
blanchet@34982
  1305
blanchet@34982
  1306
\prew
blanchet@34982
  1307
\textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
blanchet@34982
  1308
``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
blanchet@34982
  1309
``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
blanchet@34982
  1310
``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
blanchet@34982
  1311
\postw
blanchet@34982
  1312
blanchet@34982
  1313
We will try to prove that only even numbers are reachable:
blanchet@34982
  1314
blanchet@34982
  1315
\prew
blanchet@34982
  1316
\textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
blanchet@34982
  1317
\postw
blanchet@34982
  1318
blanchet@34982
  1319
Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
blanchet@34982
  1320
so let's attempt a proof by induction:
blanchet@34982
  1321
blanchet@34982
  1322
\prew
blanchet@34982
  1323
\textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
blanchet@34982
  1324
\textbf{apply}~\textit{auto}
blanchet@34982
  1325
\postw
blanchet@34982
  1326
blanchet@34982
  1327
This leaves us in the following proof state:
blanchet@34982
  1328
blanchet@34982
  1329
\prew
blanchet@34982
  1330
{\slshape goal (2 subgoals): \\
blanchet@34982
  1331
\phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
blanchet@34982
  1332
\phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
blanchet@34982
  1333
}
blanchet@34982
  1334
\postw
blanchet@34982
  1335
blanchet@34982
  1336
If we run Nitpick on the first subgoal, it still won't find any
blanchet@34982
  1337
counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
blanchet@34982
  1338
is helpless. However, notice the $n \in \textit{reach}$ assumption, which
blanchet@34982
  1339
strengthens the induction hypothesis but is not immediately usable in the proof.
blanchet@34982
  1340
If we remove it and invoke Nitpick, this time we get a counterexample:
blanchet@34982
  1341
blanchet@34982
  1342
\prew
blanchet@34982
  1343
\textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
blanchet@34982
  1344
\textbf{nitpick} \\[2\smallskipamount]
blanchet@34982
  1345
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@34982
  1346
\hbox{}\qquad Skolem constant: \nopagebreak \\
blanchet@34982
  1347
\hbox{}\qquad\qquad $n = 0$
blanchet@34982
  1348
\postw
blanchet@34982
  1349
blanchet@34982
  1350
Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
blanchet@34982
  1351
to strength the lemma:
blanchet@34982
  1352
blanchet@34982
  1353
\prew
blanchet@34982
  1354
\textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
blanchet@34982
  1355
\postw
blanchet@34982
  1356
blanchet@34982
  1357
Unfortunately, the proof by induction still gets stuck, except that Nitpick now
blanchet@34982
  1358
finds the counterexample $n = 2$. We generalize the lemma further to
blanchet@34982
  1359
blanchet@34982
  1360
\prew
blanchet@34982
  1361
\textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
blanchet@34982
  1362
\postw
blanchet@34982
  1363
blanchet@34982
  1364
and this time \textit{arith} can finish off the subgoals.
blanchet@34982
  1365
blanchet@34982
  1366
A similar technique can be employed for structural induction. The
blanchet@35180
  1367
following mini formalization of full binary trees will serve as illustration:
blanchet@34982
  1368
blanchet@34982
  1369
\prew
blanchet@34982
  1370
\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]
blanchet@34982
  1371
\textbf{primrec}~\textit{labels}~\textbf{where} \\
blanchet@34982
  1372
``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
blanchet@34982
  1373
``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
blanchet@34982
  1374
\textbf{primrec}~\textit{swap}~\textbf{where} \\
blanchet@34982
  1375
``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
blanchet@34982
  1376
\phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
blanchet@34982
  1377
``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
blanchet@34982
  1378
\postw
blanchet@34982
  1379
blanchet@34982
  1380
The \textit{labels} function returns the set of labels occurring on leaves of a
blanchet@34982
  1381
tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
blanchet@34982
  1382
labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
blanchet@34982
  1383
obtained by swapping $a$ and $b$:
blanchet@34982
  1384
blanchet@34982
  1385
\prew
blanchet@35180
  1386
\textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
blanchet@34982
  1387
\postw
blanchet@34982
  1388
blanchet@34982
  1389
Nitpick can't find any counterexample, so we proceed with induction
blanchet@34982
  1390
(this time favoring a more structured style):
blanchet@34982
  1391
blanchet@34982
  1392
\prew
blanchet@34982
  1393
\textbf{proof}~(\textit{induct}~$t$) \\
blanchet@34982
  1394
\hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
blanchet@34982
  1395
\textbf{next} \\
blanchet@34982
  1396
\hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
blanchet@34982
  1397
\postw
blanchet@34982
  1398
blanchet@34982
  1399
Nitpick can't find any counterexample at this point either, but it makes the
blanchet@34982
  1400
following suggestion:
blanchet@34982
  1401
blanchet@34982
  1402
\prew
blanchet@34982
  1403
\slshape
blanchet@35178
  1404
Hint: To check that the induction hypothesis is general enough, try this command:
blanchet@35183
  1405
\textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
blanchet@34982
  1406
\postw
blanchet@34982
  1407
blanchet@34982
  1408
If we follow the hint, we get a ``nonstandard'' counterexample for the step:
blanchet@34982
  1409
blanchet@34982
  1410
\prew
blanchet@35180
  1411
\slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
blanchet@34982
  1412
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@35078
  1413
\hbox{}\qquad\qquad $a = a_1$ \\
blanchet@35078
  1414
\hbox{}\qquad\qquad $b = a_2$ \\
blanchet@35078
  1415
\hbox{}\qquad\qquad $t = \xi_1$ \\
blanchet@35078
  1416
\hbox{}\qquad\qquad $u = \xi_2$ \\
blanchet@35180
  1417
\hbox{}\qquad Datatype: \nopagebreak \\
blanchet@35180
  1418
\hbox{}\qquad\qquad $\alpha~\textit{btree} = \{\xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \textit{Branch}~\xi_1~\xi_2\}$ \\
blanchet@34982
  1419
\hbox{}\qquad {\slshape Constants:} \nopagebreak \\
blanchet@34982
  1420
\hbox{}\qquad\qquad $\textit{labels} = \undef
blanchet@34982
  1421
    (\!\begin{aligned}[t]%
blanchet@35180
  1422
    & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
blanchet@35180
  1423
    & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
blanchet@34982
  1424
\hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \undef
blanchet@34982
  1425
    (\!\begin{aligned}[t]%
blanchet@35078
  1426
    & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
blanchet@35180
  1427
    & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount]
blanchet@34982
  1428
The existence of a nonstandard model suggests that the induction hypothesis is not general enough or perhaps
blanchet@34982
  1429
even wrong. See the ``Inductive Properties'' section of the Nitpick manual for details (``\textit{isabelle doc nitpick}'').
blanchet@34982
  1430
\postw
blanchet@34982
  1431
blanchet@34982
  1432
Reading the Nitpick manual is a most excellent idea.
blanchet@35183
  1433
But what's going on? The \textit{non\_std} option told the tool to look for
blanchet@35183
  1434
nonstandard models of binary trees, which means that new ``nonstandard'' trees
blanchet@35183
  1435
$\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees
blanchet@35183
  1436
generated by the \textit{Leaf} and \textit{Branch} constructors.%
blanchet@34982
  1437
\footnote{Notice the similarity between allowing nonstandard trees here and
blanchet@34982
  1438
allowing unreachable states in the preceding example (by removing the ``$n \in
blanchet@34982
  1439
\textit{reach\/}$'' assumption). In both cases, we effectively enlarge the
blanchet@34982
  1440
set of objects over which the induction is performed while doing the step
blanchet@35078
  1441
in order to test the induction hypothesis's strength.}
blanchet@35180
  1442
Unlike standard trees, these new trees contain cycles. We will see later that
blanchet@35180
  1443
every property of acyclic trees that can be proved without using induction also
blanchet@35180
  1444
holds for cyclic trees. Hence,
blanchet@34982
  1445
%
blanchet@34982
  1446
\begin{quote}
blanchet@34982
  1447
\textsl{If the induction
blanchet@34982
  1448
hypothesis is strong enough, the induction step will hold even for nonstandard
blanchet@34982
  1449
objects, and Nitpick won't find any nonstandard counterexample.}
blanchet@34982
  1450
\end{quote}
blanchet@34982
  1451
%
blanchet@35180
  1452
But here the tool find some nonstandard trees $t = \xi_1$
blanchet@35180
  1453
and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in
blanchet@35180
  1454
\textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$.
blanchet@34982
  1455
Because neither tree contains both $a$ and $b$, the induction hypothesis tells
blanchet@34982
  1456
us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$,
blanchet@34982
  1457
and as a result we know nothing about the labels of the tree
blanchet@34982
  1458
$\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals
blanchet@34982
  1459
$\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose
blanchet@34982
  1460
labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup}
blanchet@34982
  1461
\textit{labels}$ $(\textit{swap}~u~a~b)$.
blanchet@34982
  1462
blanchet@34982
  1463
The solution is to ensure that we always know what the labels of the subtrees
blanchet@34982
  1464
are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in
blanchet@34982
  1465
$t$ in the statement of the lemma:
blanchet@34982
  1466
blanchet@34982
  1467
\prew
blanchet@34982
  1468
\textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\
blanchet@34982
  1469
\phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\
blanchet@34982
  1470
\phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\
blanchet@34982
  1471
\phantom{\textbf{lemma} ``(}$\textrm{else}$ \\
blanchet@34982
  1472
\phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$''
blanchet@34982
  1473
\postw
blanchet@34982
  1474
blanchet@34982
  1475
This time, Nitpick won't find any nonstandard counterexample, and we can perform
blanchet@35078
  1476
the induction step using \textit{auto}.
blanchet@34982
  1477
blanchet@33191
  1478
\section{Case Studies}
blanchet@33191
  1479
\label{case-studies}
blanchet@33191
  1480
blanchet@33191
  1481
As a didactic device, the previous section focused mostly on toy formulas whose
blanchet@33191
  1482
validity can easily be assessed just by looking at the formula. We will now
blanchet@33191
  1483
review two somewhat more realistic case studies that are within Nitpick's
blanchet@33191
  1484
reach:\ a context-free grammar modeled by mutually inductive sets and a
blanchet@33191
  1485
functional implementation of AA trees. The results presented in this
blanchet@33191
  1486
section were produced with the following settings:
blanchet@33191
  1487
blanchet@33191
  1488
\prew
blanchet@33191
  1489
\textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
blanchet@33191
  1490
\postw
blanchet@33191
  1491
blanchet@33191
  1492
\subsection{A Context-Free Grammar}
blanchet@33191
  1493
\label{a-context-free-grammar}
blanchet@33191
  1494
blanchet@33191
  1495
Our first case study is taken from section 7.4 in the Isabelle tutorial
blanchet@33191
  1496
\cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
blanchet@33191
  1497
Ullman, produces all strings with an equal number of $a$'s and $b$'s:
blanchet@33191
  1498
blanchet@33191
  1499
\prew
blanchet@33191
  1500
\begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
blanchet@33191
  1501
$S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
blanchet@33191
  1502
$A$ & $::=$ & $aS \mid bAA$ \\
blanchet@33191
  1503
$B$ & $::=$ & $bS \mid aBB$
blanchet@33191
  1504
\end{tabular}
blanchet@33191
  1505
\postw
blanchet@33191
  1506
blanchet@33191
  1507
The intuition behind the grammar is that $A$ generates all string with one more
blanchet@33191
  1508
$a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
blanchet@33191
  1509
blanchet@33191
  1510
The alphabet consists exclusively of $a$'s and $b$'s:
blanchet@33191
  1511
blanchet@33191
  1512
\prew
blanchet@33191
  1513
\textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
blanchet@33191
  1514
\postw
blanchet@33191
  1515
blanchet@33191
  1516
Strings over the alphabet are represented by \textit{alphabet list}s.
blanchet@33191
  1517
Nonterminals in the grammar become sets of strings. The production rules
blanchet@33191
  1518
presented above can be expressed as a mutually inductive definition:
blanchet@33191
  1519
blanchet@33191
  1520
\prew
blanchet@33191
  1521
\textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
blanchet@33191
  1522
\textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
blanchet@33191
  1523
\textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
blanchet@33191
  1524
\textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
blanchet@33191
  1525
\textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
blanchet@33191
  1526
\textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
blanchet@33191
  1527
\textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
blanchet@33191
  1528
\postw
blanchet@33191
  1529
blanchet@33191
  1530
The conversion of the grammar into the inductive definition was done manually by
blanchet@33191
  1531
Joe Blow, an underpaid undergraduate student. As a result, some errors might
blanchet@33191
  1532
have sneaked in.
blanchet@33191
  1533
blanchet@33191
  1534
Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
blanchet@33191
  1535
d'\^etre}. A good approach is to state desirable properties of the specification
blanchet@33191
  1536
(here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
blanchet@33191
  1537
as $b$'s) and check them with Nitpick. If the properties are correctly stated,
blanchet@33191
  1538
counterexamples will point to bugs in the specification. For our grammar
blanchet@33191
  1539
example, we will proceed in two steps, separating the soundness and the
blanchet@33191
  1540
completeness of the set $S$. First, soundness:
blanchet@33191
  1541
blanchet@33191
  1542
\prew
blanchet@35284
  1543
\textbf{theorem}~\textit{S\_sound\/}: \\
blanchet@33191
  1544
``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
blanchet@33191
  1545
  \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
blanchet@33191
  1546
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
  1547
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
  1548
\hbox{}\qquad Free variable: \nopagebreak \\
blanchet@33191
  1549
\hbox{}\qquad\qquad $w = [b]$
blanchet@33191
  1550
\postw
blanchet@33191
  1551
blanchet@33191
  1552
It would seem that $[b] \in S$. How could this be? An inspection of the
blanchet@33191
  1553
introduction rules reveals that the only rule with a right-hand side of the form
blanchet@33191
  1554
$b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
blanchet@33191
  1555
\textit{R5}:
blanchet@33191
  1556
blanchet@33191
  1557
\prew
blanchet@33191
  1558
``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
blanchet@33191
  1559
\postw
blanchet@33191
  1560
blanchet@33191
  1561
On closer inspection, we can see that this rule is wrong. To match the
blanchet@33191
  1562
production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
blanchet@33191
  1563
again:
blanchet@33191
  1564
blanchet@33191
  1565
\prew
blanchet@33191
  1566
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
  1567
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
  1568
\hbox{}\qquad Free variable: \nopagebreak \\
blanchet@33191
  1569
\hbox{}\qquad\qquad $w = [a, a, b]$
blanchet@33191
  1570
\postw
blanchet@33191
  1571
blanchet@33191
  1572
Some detective work is necessary to find out what went wrong here. To get $[a,
blanchet@33191
  1573
a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
blanchet@33191
  1574
from \textit{R6}:
blanchet@33191
  1575
blanchet@33191
  1576
\prew
blanchet@33191
  1577
``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
blanchet@33191
  1578
\postw
blanchet@33191
  1579
blanchet@33191
  1580
Now, this formula must be wrong: The same assumption occurs twice, and the
blanchet@33191
  1581
variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
blanchet@33191
  1582
the assumptions should have been a $w$.
blanchet@33191
  1583
blanchet@33191
  1584
With the correction made, we don't get any counterexample from Nitpick. Let's
blanchet@33191
  1585
move on and check completeness:
blanchet@33191
  1586
blanchet@33191
  1587
\prew
blanchet@33191
  1588
\textbf{theorem}~\textit{S\_complete}: \\
blanchet@33191
  1589
``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
blanchet@33191
  1590
   \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
blanchet@33191
  1591
  \longrightarrow w \in S$'' \\
blanchet@33191
  1592
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
  1593
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
  1594
\hbox{}\qquad Free variable: \nopagebreak \\
blanchet@33191
  1595
\hbox{}\qquad\qquad $w = [b, b, a, a]$
blanchet@33191
  1596
\postw
blanchet@33191
  1597
blanchet@33191
  1598
Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
blanchet@33191
  1599
$a$'s and $b$'s. But since our inductive definition passed the soundness check,
blanchet@33191
  1600
the introduction rules we have are probably correct. Perhaps we simply lack an
blanchet@33191
  1601
introduction rule. Comparing the grammar with the inductive definition, our
blanchet@33191
  1602
suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
blanchet@33191
  1603
without which the grammar cannot generate two or more $b$'s in a row. So we add
blanchet@33191
  1604
the rule
blanchet@33191
  1605
blanchet@33191
  1606
\prew
blanchet@33191
  1607
``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
blanchet@33191
  1608
\postw
blanchet@33191
  1609
blanchet@33191
  1610
With this last change, we don't get any counterexamples from Nitpick for either
blanchet@33191
  1611
soundness or completeness. We can even generalize our result to cover $A$ and
blanchet@33191
  1612
$B$ as well:
blanchet@33191
  1613
blanchet@33191
  1614
\prew
blanchet@33191
  1615
\textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
blanchet@33191
  1616
``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
blanchet@33191
  1617
``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
blanchet@33191
  1618
``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
blanchet@33191
  1619
\textbf{nitpick} \\[2\smallskipamount]
blanchet@35309
  1620
\slshape Nitpick ran out of time after checking 7 of 8 scopes.
blanchet@33191
  1621
\postw
blanchet@33191
  1622
blanchet@33191
  1623
\subsection{AA Trees}
blanchet@33191
  1624
\label{aa-trees}
blanchet@33191
  1625
blanchet@33191
  1626
AA trees are a kind of balanced trees discovered by Arne Andersson that provide
blanchet@33191
  1627
similar performance to red-black trees, but with a simpler implementation
blanchet@33191
  1628
\cite{andersson-1993}. They can be used to store sets of elements equipped with
blanchet@33191
  1629
a total order $<$. We start by defining the datatype and some basic extractor
blanchet@33191
  1630
functions:
blanchet@33191
  1631
blanchet@33191
  1632
\prew
blanchet@34982
  1633
\textbf{datatype} $'a$~\textit{aa\_tree} = \\
blanchet@34982
  1634
\hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}''  \\[2\smallskipamount]
blanchet@33191
  1635
\textbf{primrec} \textit{data} \textbf{where} \\
blanchet@33191
  1636
``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
blanchet@33191
  1637
``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
blanchet@33191
  1638
\textbf{primrec} \textit{dataset} \textbf{where} \\
blanchet@33191
  1639
``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
blanchet@33191
  1640
``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
blanchet@33191
  1641
\textbf{primrec} \textit{level} \textbf{where} \\
blanchet@33191
  1642
``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
blanchet@33191
  1643
``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
blanchet@33191
  1644
\textbf{primrec} \textit{left} \textbf{where} \\
blanchet@33191
  1645
``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
blanchet@33191
  1646
``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
blanchet@33191
  1647
\textbf{primrec} \textit{right} \textbf{where} \\
blanchet@33191
  1648
``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
blanchet@33191
  1649
``$\textit{right}~(N~\_~\_~\_~u) = u$''
blanchet@33191
  1650
\postw
blanchet@33191
  1651
blanchet@33191
  1652
The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
blanchet@33191
  1653
as follows \cite{wikipedia-2009-aa-trees}:
blanchet@33191
  1654
blanchet@33191
  1655
\kern.2\parskip %% TYPESETTING
blanchet@33191
  1656
blanchet@33191
  1657
\pre
blanchet@33191
  1658
Each node has a level field, and the following invariants must remain true for
blanchet@33191
  1659
the tree to be valid:
blanchet@33191
  1660
blanchet@33191
  1661
\raggedright
blanchet@33191
  1662
blanchet@33191
  1663
\kern-.4\parskip %% TYPESETTING
blanchet@33191
  1664
blanchet@33191
  1665
\begin{enum}
blanchet@33191
  1666
\item[]
blanchet@33191
  1667
\begin{enum}
blanchet@33191
  1668
\item[1.] The level of a leaf node is one.
blanchet@33191
  1669
\item[2.] The level of a left child is strictly less than that of its parent.
blanchet@33191
  1670
\item[3.] The level of a right child is less than or equal to that of its parent.
blanchet@33191
  1671
\item[4.] The level of a right grandchild is strictly less than that of its grandparent.
blanchet@33191
  1672
\item[5.] Every node of level greater than one must have two children.
blanchet@33191
  1673
\end{enum}
blanchet@33191
  1674
\end{enum}
blanchet@33191
  1675
\post
blanchet@33191
  1676
blanchet@33191
  1677
\kern.4\parskip %% TYPESETTING
blanchet@33191
  1678
blanchet@33191
  1679
The \textit{wf} predicate formalizes this description:
blanchet@33191
  1680
blanchet@33191
  1681
\prew
blanchet@33191
  1682
\textbf{primrec} \textit{wf} \textbf{where} \\
blanchet@33191
  1683
``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
blanchet@33191
  1684
``$\textit{wf}~(N~\_~k~t~u) =$ \\
blanchet@33191
  1685
\phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
blanchet@33191
  1686
\phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
blanchet@33191
  1687
\phantom{``$($}$\textrm{else}$ \\
blanchet@33193
  1688
\hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
blanchet@33191
  1689
\mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
blanchet@33193
  1690
\mathrel{\land} \textit{level}~u \le k$ \\
blanchet@33193
  1691
\hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
blanchet@33191
  1692
\postw
blanchet@33191
  1693
blanchet@33191
  1694
Rebalancing the tree upon insertion and removal of elements is performed by two
blanchet@33191
  1695
auxiliary functions called \textit{skew} and \textit{split}, defined below:
blanchet@33191
  1696
blanchet@33191
  1697
\prew
blanchet@33191
  1698
\textbf{primrec} \textit{skew} \textbf{where} \\
blanchet@33191
  1699
``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
blanchet@33191
  1700
``$\textit{skew}~(N~x~k~t~u) = {}$ \\
blanchet@33191
  1701
\phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
blanchet@33191
  1702
\textit{level}~t~\textrm{then}$ \\
blanchet@33191
  1703
\phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
blanchet@33191
  1704
(\textit{right}~t)~u)$ \\
blanchet@33191
  1705
\phantom{``(}$\textrm{else}$ \\
blanchet@33191
  1706
\phantom{``(\quad}$N~x~k~t~u)$''
blanchet@33191
  1707
\postw
blanchet@33191
  1708
blanchet@33191
  1709
\prew
blanchet@33191
  1710
\textbf{primrec} \textit{split} \textbf{where} \\
blanchet@33191
  1711
``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
blanchet@33191
  1712
``$\textit{split}~(N~x~k~t~u) = {}$ \\
blanchet@33191
  1713
\phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
blanchet@33191
  1714
\textit{level}~(\textit{right}~u)~\textrm{then}$ \\
blanchet@33191
  1715
\phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
blanchet@33191
  1716
(N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
blanchet@33191
  1717
\phantom{``(}$\textrm{else}$ \\
blanchet@33191
  1718
\phantom{``(\quad}$N~x~k~t~u)$''
blanchet@33191
  1719
\postw
blanchet@33191
  1720
blanchet@33191
  1721
Performing a \textit{skew} or a \textit{split} should have no impact on the set
blanchet@33191
  1722
of elements stored in the tree:
blanchet@33191
  1723
blanchet@33191
  1724
\prew
blanchet@35284
  1725
\textbf{theorem}~\textit{dataset\_skew\_split\/}:\\
blanchet@33191
  1726
``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
blanchet@33191
  1727
``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
blanchet@33191
  1728
\textbf{nitpick} \\[2\smallskipamount]
blanchet@35309
  1729
{\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
blanchet@33191
  1730
\postw
blanchet@33191
  1731
blanchet@33191
  1732
Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
blanchet@33191
  1733
should not alter the tree:
blanchet@33191
  1734
blanchet@33191
  1735
\prew
blanchet@35284
  1736
\textbf{theorem}~\textit{wf\_skew\_split\/}:\\
blanchet@33191
  1737
``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
blanchet@33191
  1738
``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
blanchet@33191
  1739
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
  1740
{\slshape Nitpick found no counterexample.}
blanchet@33191
  1741
\postw
blanchet@33191
  1742
blanchet@33191
  1743
Insertion is implemented recursively. It preserves the sort order:
blanchet@33191
  1744
blanchet@33191
  1745
\prew
blanchet@33191
  1746
\textbf{primrec}~\textit{insort} \textbf{where} \\
blanchet@33191
  1747
``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
blanchet@33191
  1748
``$\textit{insort}~(N~y~k~t~u)~x =$ \\
blanchet@33191
  1749
\phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
blanchet@33191
  1750
\phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
blanchet@33191
  1751
\postw
blanchet@33191
  1752
blanchet@33191
  1753
Notice that we deliberately commented out the application of \textit{skew} and
blanchet@33191
  1754
\textit{split}. Let's see if this causes any problems:
blanchet@33191
  1755
blanchet@33191
  1756
\prew
blanchet@35284
  1757
\textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
blanchet@33191
  1758
\textbf{nitpick} \\[2\smallskipamount]
blanchet@33191
  1759
\slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
blanchet@33191
  1760
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@35078
  1761
\hbox{}\qquad\qquad $t = N~a_1~1~\Lambda~\Lambda$ \\
blanchet@35078
  1762
\hbox{}\qquad\qquad $x = a_2$
blanchet@33191
  1763
\postw
blanchet@33191
  1764
blanchet@34038
  1765
It's hard to see why this is a counterexample. To improve readability, we will
blanchet@34038
  1766
restrict the theorem to \textit{nat}, so that we don't need to look up the value
blanchet@34038
  1767
of the $\textit{op}~{<}$ constant to find out which element is smaller than the
blanchet@34038
  1768
other. In addition, we will tell Nitpick to display the value of
blanchet@34038
  1769
$\textit{insort}~t~x$ using the \textit{eval} option. This gives
blanchet@33191
  1770
blanchet@33191
  1771
\prew
blanchet@35284
  1772
\textbf{theorem} \textit{wf\_insort\_nat\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
blanchet@33191
  1773
\textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
blanchet@33191
  1774
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
blanchet@33191
  1775
\hbox{}\qquad Free variables: \nopagebreak \\
blanchet@33191
  1776
\hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
blanchet@33191
  1777
\hbox{}\qquad\qquad $x = 0$ \\
blanchet@33191
  1778
\hbox{}\qquad Evaluated term: \\
blanchet@33191
  1779
\hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
blanchet@33191
  1780
\postw
blanchet@33191
  1781
blanchet@33191
  1782
Nitpick's output reveals that the element $0$ was added as a left child of $1$,
blanchet@33191
  1783
where both have a level of 1. This violates the second AA tree invariant, which
blanchet@33191
  1784
states that a left child's level must be less than its parent's. This shouldn't
blanchet@33191
  1785
come as a surprise, considering that we commented out the tree rebalancing code.
blanchet@33191
  1786
Reintroducing the code seems to solve the problem:
blanchet@33191
  1787
blanchet@33191
  1788
\prew
blanchet@35284
  1789
\textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
blanchet@33191
  1790
\textbf{nitpick} \\[2\smallskipamount]
blanchet@35072
  1791
{\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
blanchet@33191
  1792
\postw
blanchet@33191
  1793
blanchet@33191
  1794
Insertion should transform the set of elements represented by the tree in the
blanchet@33191
  1795
obvious way:
blanchet@33191
  1796
blanchet@33191
  1797
\prew
blanchet@35284
  1798
\textbf{theorem} \textit{dataset\_insort\/}:\kern.4em
blanchet@33191
  1799
``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
blanchet@33191
  1800
\textbf{nitpick} \\[2\smallskipamount]
blanchet@35072
  1801
{\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
blanchet@33191
  1802
\postw
blanchet@33191
  1803
blanchet@35072
  1804
We could continue like this and sketch a complete theory of AA trees. Once the
blanchet@35072
  1805
definitions and main theorems are in place and have been thoroughly tested using
blanchet@35072
  1806
Nitpick, we could start working on the proofs. Developing theories this way
blanchet@35072
  1807
usually saves time, because faulty theorems and definitions are discovered much
blanchet@35072
  1808
earlier in the process.
blanchet@33191
  1809
blanchet@33191
  1810
\section{Option Reference}
blanchet@33191
  1811
\label{option-reference}
blanchet@33191
  1812
blanchet@33191
  1813
\def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
blanchet@33191
  1814
\def\qty#1{$\left<\textit{#1}\right>$}
blanchet@33191
  1815
\def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
blanchet@33191
  1816
\def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
blanchet@33191
  1817
\def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
blanchet@33191
  1818
\def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
blanchet@34982
  1819
\def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
blanchet@34982
  1820
\def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
blanchet@34982
  1821
\def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
blanchet@34982
  1822
\def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
blanchet@34982
  1823
\def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
blanchet@33191
  1824
blanchet@33191
  1825
Nitpick's behavior can be influenced by various options, which can be specified
blanchet@33191
  1826
in brackets after the \textbf{nitpick} command. Default values can be set
blanchet@33191
  1827
using \textbf{nitpick\_\allowbreak params}. For example:
blanchet@33191
  1828
blanchet@33191
  1829
\prew
blanchet@33191
  1830
\textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
blanchet@33191
  1831
\postw
blanchet@33191
  1832
blanchet@33191
  1833
The options are categorized as follows:\ mode of operation
blanchet@33191
  1834
(\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
blanchet@33191
  1835
format (\S\ref{output-format}), automatic counterexample checks
blanchet@33191
  1836
(\S\ref{authentication}), optimizations
blanchet@33191
  1837
(\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
blanchet@33191
  1838
blanchet@33561
  1839
You can instruct Nitpick to run automatically on newly entered theorems by
blanchet@33561
  1840
enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof
blanchet@33561
  1841
General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation})
blanchet@33561
  1842
and \textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
blanchet@33561
  1843
\textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
blanchet@33561
  1844
(\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
blanchet@33561
  1845
disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
blanchet@33561
  1846
\textit{timeout} (\S\ref{timeouts}) is superseded by the ``Auto Counterexample
blanchet@33561
  1847
Time Limit'' in Proof General's ``Isabelle'' menu. Nitpick's output is also more
blanchet@33561
  1848
concise.
blanchet@33561
  1849
blanchet@33191
  1850
The number of options can be overwhelming at first glance. Do not let that worry
blanchet@33191
  1851
you: Nitpick's defaults have been chosen so that it almost always does the right
blanchet@33191
  1852
thing, and the most important options have been covered in context in
blanchet@33191
  1853
\S\ref{first-steps}.
blanchet@33191
  1854
blanchet@33191
  1855
The descriptions below refer to the following syntactic quantities:
blanchet@33191
  1856
blanchet@33191
  1857
\begin{enum}
blanchet@33191
  1858
\item[$\bullet$] \qtybf{string}: A string.
blanchet@35284
  1859
\item[$\bullet$] \qtybf{bool\/}: \textit{true} or \textit{false}.
blanchet@35284
  1860
\item[$\bullet$] \qtybf{bool\_or\_smart\/}: \textit{true}, \textit{false}, or \textit{smart}.
blanchet@35284
  1861
\item[$\bullet$] \qtybf{int\/}: An integer. Negative integers are prefixed with a hyphen.
blanchet@35284
  1862
\item[$\bullet$] \qtybf{int\_or\_smart\/}: An integer or \textit{smart}.
blanchet@33191
  1863
\item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
blanchet@33191
  1864
of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
blanchet@33191
  1865
blanchet@33191
  1866
\item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
blanchet@33191
  1867
\item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
blanchet@33191
  1868
(milliseconds), or the keyword \textit{none} ($\infty$ years).
blanchet@35284
  1869
\item[$\bullet$] \qtybf{const\/}: The name of a HOL constant.
blanchet@33191
  1870
\item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
blanchet@35284
  1871
\item[$\bullet$] \qtybf{term\_list\/}: A space-separated list of HOL terms (e.g.,
blanchet@33191
  1872
``$f~x$''~``$g~y$'').
blanchet@33191
  1873
\item[$\bullet$] \qtybf{type}: A HOL type.
blanchet@33191
  1874
\end{enum}
blanchet@33191
  1875
blanchet@33191
  1876
Default values are indicated in square brackets. Boolean options have a negated
blanchet@33561
  1877
counterpart (e.g., \textit{blocking} vs.\ \textit{no\_blocking}). When setting
blanchet@33561
  1878
Boolean options, ``= \textit{true}'' may be omitted.
blanchet@33191
  1879
blanchet@33191
  1880
\subsection{Mode of Operation}
blanchet@33191
  1881
\label{mode-of-operation}
blanchet@33191
  1882
blanchet@33191
  1883
\begin{enum}
blanchet@33191
  1884
\optrue{blocking}{non\_blocking}
blanchet@33191
  1885
Specifies whether the \textbf{nitpick} command should operate synchronously.
blanchet@33191
  1886
The asynchronous (non-blocking) mode lets the user start proving the putative
blanchet@33191
  1887
theorem while Nitpick looks for a counterexample, but it can also be more
blanchet@33191
  1888
confusing. For technical reasons, automatic runs currently always block.
blanchet@33191
  1889
blanchet@33191
  1890
\optrue{falsify}{satisfy}
blanchet@33191
  1891
Specifies whether Nitpick should look for falsifying examples (countermodels) or
blanchet@33191
  1892
satisfying examples (models). This manual assumes throughout that
blanchet@33191
  1893
\textit{falsify} is enabled.
blanchet@33191
  1894
blanchet@33191
  1895
\opsmart{user\_axioms}{no\_user\_axioms}
blanchet@33191
  1896
Specifies whether the user-defined axioms (specified using 
blanchet@33191
  1897
\textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
blanchet@33191
  1898
is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
blanchet@33191
  1899
the constants that occur in the formula to falsify. The option is implicitly set
blanchet@33191
  1900
to \textit{true} for automatic runs.
blanchet@33191
  1901
blanchet@33191
  1902
\textbf{Warning:} If the option is set to \textit{true}, Nitpick might
blanchet@33191
  1903
nonetheless ignore some polymorphic axioms. Counterexamples generated under
blanchet@33191
  1904
these conditions are tagged as ``likely genuine.'' The \textit{debug}
blanchet@33191
  1905
(\S\ref{output-format}) option can be used to find out which axioms were
blanchet@33191
  1906
considered.
blanchet@33191
  1907
blanchet@33191
  1908
\nopagebreak
blanchet@33561
  1909
{\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug}
blanchet@33561
  1910
(\S\ref{output-format}).}
blanchet@33191
  1911
blanchet@33191
  1912
\optrue{assms}{no\_assms}
blanchet@35331
  1913
Specifies whether the relevant assumptions in structured proofs should be
blanchet@33191
  1914
considered. The option is implicitly enabled for automatic runs.
blanchet@33191
  1915
blanchet@33191
  1916
\nopagebreak
blanchet@33561
  1917
{\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).}
blanchet@33191
  1918
blanchet@33191
  1919
\opfalse{overlord}{no\_overlord}
blanchet@33191
  1920
Specifies whether Nitpick should put its temporary files in
blanchet@33191
  1921
\texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
blanchet@33191
  1922
debugging Nitpick but also unsafe if several instances of the tool are run
blanchet@34998
  1923
simultaneously. The files are identified by the extensions
blanchet@34998
  1924
\texttt{.kki}, \texttt{.cnf}, \texttt{.out}, and
blanchet@34998
  1925
\texttt{.err}; you may safely remove them after Nitpick has run.
blanchet@33191
  1926
blanchet@33191
  1927
\nopagebreak
blanchet@33191
  1928
{\small See also \textit{debug} (\S\ref{output-format}).}
blanchet@33191
  1929
\end{enum}
blanchet@33191
  1930
blanchet@33191
  1931
\subsection{Scope of Search}
blanchet@33191
  1932
\label{scope-of-search}
blanchet@33191
  1933
blanchet@33191
  1934
\begin{enum}
blanchet@34982
  1935
\oparg{card}{type}{int\_seq}
blanchet@34124
  1936
Specifies the sequence of cardinalities to use for a given type.
blanchet@33191
  1937
For free types, and often also for \textbf{typedecl}'d types, it usually makes
blanchet@33191
  1938
sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
blanchet@33191
  1939
Although function and product types are normally mapped directly to the
blanchet@33191
  1940
corresponding Kodkod concepts, setting
blanchet@33191
  1941
the cardinality of such types is also allowed and implicitly enables ``boxing''
blanchet@33191
  1942
for them, as explained in the description of the \textit{box}~\qty{type}
blanchet@33191
  1943
and \textit{box} (\S\ref{scope-of-search}) options.
blanchet@33191
  1944
blanchet@33191
  1945
\nopagebreak
blanchet@33191
  1946
{\small See also \textit{mono} (\S\ref{scope-of-search}).}
blanchet@33191
  1947
blanchet@34982
  1948
\opdefault{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
blanchet@33191
  1949
Specifies the default sequence of cardinalities to use. This can be overridden
blanchet@33191
  1950
on a per-type basis using the \textit{card}~\qty{type} option described above.
blanchet@33191
  1951
blanchet@34982
  1952
\oparg{max}{const}{int\_seq}
blanchet@33191
  1953
Specifies the sequence of maximum multiplicities to use for a given
blanchet@33191
  1954
(co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
blanchet@33191
  1955
number of distinct values that it can construct. Nonsensical values (e.g.,
blanchet@33191
  1956
\textit{max}~[]~$=$~2) are silently repaired. This option is only available for
blanchet@33191
  1957
datatypes equipped with several constructors.
blanchet@33191
  1958
blanchet@34982
  1959
\opnodefault{max}{int\_seq}
blanchet@33191
  1960
Specifies the default sequence of maximum multiplicities to use for
blanchet@33191
  1961
(co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
blanchet@33191
  1962
basis using the \textit{max}~\qty{const} option described above.
blanchet@33191
  1963
blanchet@34124
  1964
\opsmart{binary\_ints}{unary\_ints}
blanchet@34124
  1965
Specifies whether natural numbers and integers should be encoded using a unary
blanchet@34124
  1966
or binary notation. In unary mode, the cardinality fully specifies the subset
blanchet@34124
  1967
used to approximate the type. For example:
blanchet@34124
  1968
%
blanchet@34124
  1969
$$\hbox{\begin{tabular}{@{}rll@{}}%
blanchet@34124
  1970
\textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
blanchet@34124
  1971
\textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
blanchet@34124
  1972
\textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
blanchet@34124
  1973
\end{tabular}}$$
blanchet@34124
  1974
%
blanchet@34124
  1975
In general:
blanchet@34124
  1976
%
blanchet@34124
  1977
$$\hbox{\begin{tabular}{@{}rll@{}}%
blanchet@34124
  1978
\textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
blanchet@34124
  1979
\textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
blanchet@34124
  1980
\end{tabular}}$$
blanchet@34124
  1981
%
blanchet@34124
  1982
In binary mode, the cardinality specifies the number of distinct values that can
blanchet@34124
  1983
be constructed. Each of these value is represented by a bit pattern whose length
blanchet@34124
  1984
is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default,
blanchet@34124
  1985
Nitpick attempts to choose the more appropriate encoding by inspecting the
blanchet@34124
  1986
formula at hand, preferring the binary notation for problems involving
blanchet@34124
  1987
multiplicative operators or large constants.
blanchet@34124
  1988
blanchet@34124
  1989
\textbf{Warning:} For technical reasons, Nitpick always reverts to unary for
blanchet@34124
  1990
problems that refer to the types \textit{rat} or \textit{real} or the constants
blanchet@34126
  1991
\textit{Suc}, \textit{gcd}, or \textit{lcm}.
blanchet@34124
  1992
blanchet@34124
  1993
{\small See also \textit{bits} (\S\ref{scope-of-search}) and
blanchet@34124
  1994
\textit{show\_datatypes} (\S\ref{output-format}).}
blanchet@34124
  1995
blanchet@34982
  1996
\opdefault{bits}{int\_seq}{$\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{8},\mathbf{10},\mathbf{12}$}
blanchet@34124
  1997
Specifies the number of bits to use to represent natural numbers and integers in
blanchet@34124
  1998
binary, excluding the sign bit. The minimum is 1 and the maximum is 31.
blanchet@34124
  1999
blanchet@34124
  2000
{\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).}
blanchet@34124
  2001
blanchet@34982
  2002
\opargboolorsmart{wf}{const}{non\_wf}
blanchet@33191
  2003
Specifies whether the specified (co)in\-duc\-tively defined predicate is
blanchet@33191
  2004
well-founded. The option can take the following values:
blanchet@33191
  2005
blanchet@33191
  2006
\begin{enum}
blanchet@33191
  2007
\item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
blanchet@33191
  2008
predicate as if it were well-founded. Since this is generally not sound when the
blanchet@33191
  2009
predicate is not well-founded, the counterexamples are tagged as ``likely
blanchet@33191
  2010
genuine.''
blanchet@33191
  2011
blanchet@33191
  2012
\item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
blanchet@33191
  2013
as if it were not well-founded. The predicate is then unrolled as prescribed by
blanchet@33191
  2014
the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
blanchet@33191
  2015
options.
blanchet@33191
  2016
blanchet@33191
  2017
\item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
blanchet@33191
  2018
predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
blanchet@34982
  2019
\textit{size\_change} tactics. If this succeeds (or the predicate occurs with an
blanchet@33191
  2020
appropriate polarity in the formula to falsify), use an efficient fixed point
blanchet@33191
  2021
equation as specification of the predicate; otherwise, unroll the predicates
blanchet@33191
  2022
according to the \textit{iter}~\qty{const} and \textit{iter} options.
blanchet@33191
  2023
\end{enum}
blanchet@33191
  2024
blanchet@33191
  2025
\nopagebreak
blanchet@33191
  2026
{\small See also \textit{iter} (\S\ref{scope-of-search}),
blanchet@33191
  2027
\textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
blanchet@33191
  2028
(\S\ref{timeouts}).}
blanchet@33191
  2029
blanchet@33191
  2030
\opsmart{wf}{non\_wf}
blanchet@33191
  2031
Specifies the default wellfoundedness setting to use. This can be overridden on
blanchet@33191
  2032
a per-predicate basis using the \textit{wf}~\qty{const} option above.
blanchet@33191
  2033
blanchet@34982
  2034
\oparg{iter}{const}{int\_seq}
blanchet@33191
  2035
Specifies the sequence of iteration counts to use when unrolling a given
blanchet@33191
  2036
(co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
blanchet@33191
  2037
predicates that occur negatively and coinductive predicates that occur
blanchet@33191
  2038
positively in the formula to falsify and that cannot be proved to be
blanchet@33191
  2039
well-founded, but this behavior is influenced by the \textit{wf} option. The
blanchet@33191
  2040
iteration counts are automatically bounded by the cardinality of the predicate's
blanchet@33191
  2041
domain.
blanchet@33191
  2042
blanchet@33191
  2043
{\small See also \textit{wf} (\S\ref{scope-of-search}) and
blanchet@33191
  2044
\textit{star\_linear\_preds} (\S\ref{optimizations}).}
blanchet@33191
  2045
blanchet@34982
  2046
\opdefault{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
blanchet@33191
  2047
Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
blanchet@33191
  2048
predicates. This can be overridden on a per-predicate basis using the
blanchet@33191
  2049
\textit{iter} \qty{const} option above.
blanchet@33191
  2050
blanchet@34982
  2051
\opdefault{bisim\_depth}{int\_seq}{$\mathbf{7}$}
blanchet@33191
  2052
Specifies the sequence of iteration counts to use when unrolling the
blanchet@33191
  2053
bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
blanchet@33191
  2054
of $-1$ means that no predicate is generated, in which case Nitpick performs an
blanchet@33191
  2055
after-the-fact check to see if the known coinductive datatype values are
blanchet@33191
  2056
bidissimilar. If two values are found to be bisimilar, the counterexample is
blanchet@33191
  2057
tagged as ``likely genuine.'' The iteration counts are automatically bounded by
blanchet@33191
  2058
the sum of the cardinalities of the coinductive datatypes occurring in the
blanchet@33191
  2059
formula to falsify.
blanchet@33191
  2060
blanchet@34982
  2061
\opargboolorsmart{box}{type}{dont\_box}
blanchet@33191
  2062
Specifies whether Nitpick should attempt to wrap (``box'') a given function or
blanchet@33191
  2063
product type in an isomorphic datatype internally. Boxing is an effective mean
blanchet@33191
  2064
to reduce the search space and speed up Nitpick, because the isomorphic datatype
blanchet@33191
  2065
is approximated by a subset of the possible function or pair values;
blanchet@33191
  2066
like other drastic optimizations, it can also prevent the discovery of
blanchet@33191
  2067
counterexamples. The option can take the following values:
blanchet@33191
  2068
blanchet@33191
  2069
\begin{enum}
blanchet@33191
  2070
\item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
blanchet@33191
  2071
practicable.
blanchet@33191
  2072
\item[$\bullet$] \textbf{\textit{false}}: Never box the type.
blanchet@33191
  2073
\item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
blanchet@33191
  2074
is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
blanchet@33191
  2075
higher-order functions are good candidates for boxing.
blanchet@33191
  2076
\end{enum}
blanchet@33191
  2077
blanchet@33191
  2078
Setting the \textit{card}~\qty{type} option for a function or product type
blanchet@33191
  2079
implicitly enables boxing for that type.
blanchet@33191
  2080
blanchet@33191
  2081
\nopagebreak
blanchet@33191
  2082
{\small See also \textit{verbose} (\S\ref{output-format})
blanchet@33191
  2083
and \textit{debug} (\S\ref{output-format}).}
blanchet@33191
  2084
blanchet@33191
  2085
\opsmart{box}{dont\_box}
blanchet@33191
  2086
Specifies the default boxing setting to use. This can be overridden on a
blanchet@33191
  2087
per-type basis using the \textit{box}~\qty{type} option described above.
blanchet@33191
  2088
blanchet@34982
  2089
\opargboolorsmart{mono}{type}{non\_mono}
blanchet@34982
  2090
Specifies whether the given type should be considered monotonic when
blanchet@33191
  2091
enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a
blanchet@33191
  2092
monotonicity check on the type. Setting this option to \textit{true} can reduce
blanchet@33191
  2093
the number of scopes tried, but it also diminishes the theoretical chance of
blanchet@33191
  2094
finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
blanchet@33191
  2095
blanchet@33191
  2096
\nopagebreak
blanchet@33191
  2097
{\small See also \textit{card} (\S\ref{scope-of-search}),
blanchet@33556
  2098
\textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
blanchet@33191
  2099
(\S\ref{output-format}).}
blanchet@33191
  2100
blanchet@33191
  2101
\opsmart{mono}{non\_box}
blanchet@33191
  2102
Specifies the default monotonicity setting to use. This can be overridden on a
blanchet@33191
  2103
per-type basis using the \textit{mono}~\qty{type} option described above.
blanchet@33191
  2104
blanchet@33556
  2105
\opfalse{merge\_type\_vars}{dont\_merge\_type\_vars}
blanchet@33191
  2106
Specifies whether type variables with the same sort constraints should be
blanchet@33191
  2107
merged. Setting this option to \textit{true} can reduce the number of scopes
blanchet@33191
  2108
tried and the size of the generated Kodkod formulas, but it also diminishes the
blanchet@33191
  2109
theoretical chance of finding a counterexample.
blanchet@33191
  2110
blanchet@33191
  2111
{\small See also \textit{mono} (\S\ref{scope-of-search}).}
blanchet@34982
  2112
blanchet@34982
  2113
\opargbool{std}{type}{non\_std}
blanchet@35189
  2114
Specifies whether the given (recursive) datatype should be given standard
blanchet@35189
  2115
models. Nonstandard models are unsound but can help debug structural induction
blanchet@35189
  2116
proofs, as explained in \S\ref{inductive-properties}.
blanchet@34982
  2117
blanchet@34982
  2118
\optrue{std}{non\_std}
blanchet@34982
  2119
Specifies the default standardness to use. This can be overridden on a per-type
blanchet@34982
  2120
basis using the \textit{std}~\qty{type} option described above.
blanchet@33191
  2121
\end{enum}
blanchet@33191
  2122
blanchet@33191
  2123
\subsection{Output Format}
blanchet@33191
  2124
\label{output-format}
blanchet@33191
  2125
blanchet@33191
  2126
\begin{enum}
blanchet@33191
  2127
\opfalse{verbose}{quiet}
blanchet@33191
  2128
Specifies whether the \textbf{nitpick} command should explain what it does. This
blanchet@33191
  2129
option is useful to determine which scopes are tried or which SAT solver is
blanchet@33191
  2130
used. This option is implicitly disabled for automatic runs.
blanchet@33191
  2131
blanchet@33191
  2132
\opfalse{debug}{no\_debug}
blanchet@33191
  2133
Specifies whether Nitpick should display additional debugging information beyond
blanchet@33191
  2134
what \textit{verbose} already displays. Enabling \textit{debug} also enables
blanchet@33191
  2135
\textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
blanchet@33191
  2136
option is implicitly disabled for automatic runs.
blanchet@33191
  2137
blanchet@33191
  2138
\nopagebreak
blanchet@33561
  2139
{\small See also \textit{overlord} (\S\ref{mode-of-operation}) and
blanchet@33561
  2140
\textit{batch\_size} (\S\ref{optimizations}).}
blanchet@33191
  2141
blanchet@33191
  2142
\optrue{show\_skolems}{hide\_skolem}
blanchet@33191
  2143
Specifies whether the values of Skolem constants should be displayed as part of
blanchet@33191
  2144
counterexamples. Skolem constants correspond to bound variables in the original
blanchet@33191
  2145
formula and usually help us to understand why the counterexample falsifies the
blanchet@33191
  2146
formula.
blanchet@33191
  2147
blanchet@33191
  2148
\nopagebreak
blanchet@33191
  2149
{\small See also \textit{skolemize} (\S\ref{optimizations}).}
blanchet@33191
  2150
blanchet@33191
  2151
\opfalse{show\_datatypes}{hide\_datatypes}
blanchet@33191
  2152
Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
blanchet@33191
  2153
be displayed as part of counterexamples. Such subsets are sometimes helpful when
blanchet@33191
  2154
investigating whether a potential counterexample is genuine or spurious, but
blanchet@33191
  2155
their potential for clutter is real.
blanchet@33191
  2156
blanchet@33191
  2157
\opfalse{show\_consts}{hide\_consts}
blanchet@33191
  2158
Specifies whether the values of constants occurring in the formula (including
blanchet@33191
  2159
its axioms) should be displayed along with any counterexample. These values are
blanchet@33191
  2160
sometimes helpful when investigating why a counterexample is
blanchet@33191
  2161
genuine, but they can clutter the output.
blanchet@33191
  2162
blanchet@33191
  2163
\opfalse{show\_all}{dont\_show\_all}
blanchet@33191
  2164
Enabling this option effectively enables \textit{show\_skolems},
blanchet@33191
  2165
\textit{show\_datatypes}, and \textit{show\_consts}.
blanchet@33191
  2166
blanchet@34982
  2167
\opdefault{max\_potential}{int}{$\mathbf{1}$}
blanchet@33191
  2168
Specifies the maximum number of potential counterexamples to display. Setting
blanchet@33191
  2169
this option to 0 speeds up the search for a genuine counterexample. This option
blanchet@33191
  2170
is implicitly set to 0 for automatic runs. If you set this option to a value
blanchet@33191
  2171
greater than 1, you will need an incremental SAT solver: For efficiency, it is
blanchet@33191
  2172
recommended to install the JNI version of MiniSat and set \textit{sat\_solver} =
blanchet@35078
  2173
\textit{MiniSat\_JNI}. Also be aware that many of the counterexamples may look
blanchet@33191
  2174
identical, unless the \textit{show\_all} (\S\ref{output-format}) option is
blanchet@33191
  2175
enabled.
blanchet@33191
  2176
blanchet@33191
  2177
\nopagebreak
blanchet@33561
  2178
{\small See also \textit{check\_potential} (\S\ref{authentication}) and
blanchet@33191
  2179
\textit{sat\_solver} (\S\ref{optimizations}).}
blanchet@33191
  2180
blanchet@34982
  2181
\opdefault{max\_genuine}{int}{$\mathbf{1}$}
blanchet@33191
  2182
Specifies the maximum number of genuine counterexamples to display. If you set
blanchet@33191
  2183
this option to a value greater than 1, you will need an incremental SAT solver:
blanchet@33191
  2184
For efficiency, it is recommended to install the JNI version of MiniSat and set
blanchet@35078
  2185
\textit{sat\_solver} = \textit{MiniSat\_JNI}. Also be aware that many of the
blanchet@33191
  2186
counterexamples may look identical, unless the \textit{show\_all}
blanchet@33191
  2187
(\S\ref{output-format}) option is enabled.
blanchet@33191
  2188
blanchet@33191
  2189
\nopagebreak
blanchet@33191
  2190
{\small See also \textit{check\_genuine} (\S\ref{authentication}) and
blanchet@33191
  2191
\textit{sat\_solver} (\S\ref{optimizations}).}
blanchet@33191
  2192
blanchet@34982
  2193
\opnodefault{eval}{term\_list}
blanchet@33191
  2194
Specifies the list of terms whose values should be displayed along with
blanchet@33191
  2195
counterexamples. This option suffers from an ``observer effect'': Nitpick might
blanchet@33191
  2196
find different counterexamples for different values of this option.
blanchet@33191
  2197
blanchet@34982
  2198
\oparg{format}{term}{int\_seq}
blanchet@33191
  2199
Specifies how to uncurry the value displayed for a variable or constant.
blanchet@33191
  2200
Uncurrying sometimes increases the readability of the output for high-arity
blanchet@33191
  2201
functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
blanchet@33191
  2202
{'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
blanchet@33191
  2203
{'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
blanchet@33191
  2204
arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
blanchet@33191
  2205
{'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
blanchet@33191
  2206
of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
blanchet@33191
  2207
$n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
blanchet@33191
  2208
arguments that are not accounted for are left alone, as if the specification had
blanchet@33191
  2209
been $1,\ldots,1,n_1,\ldots,n_k$.
blanchet@33191
  2210
blanchet@33191
  2211
\nopagebreak
blanchet@33191
  2212
{\small See also \textit{uncurry} (\S\ref{optimizations}).}
blanchet@33191
  2213
blanchet@34982
  2214
\opdefault{format}{int\_seq}{$\mathbf{1}$}
blanchet@33191
  2215
Specifies the default format to use. Irrespective of the default format, the
blanchet@33191
  2216
extra arguments to a Skolem constant corresponding to the outer bound variables
blanchet@33191
  2217
are kept separated from the remaining arguments, the \textbf{for} arguments of
blanchet@33191
  2218
an inductive definitions are kept separated from the remaining arguments, and
blanchet@33191
  2219
the iteration counter of an unrolled inductive definition is shown alone. The
blanchet@33191
  2220
default format can be overridden on a per-variable or per-constant basis using
blanchet@33191
  2221
the \textit{format}~\qty{term} option described above.
blanchet@33191
  2222
\end{enum}
blanchet@33191
  2223
blanchet@33191
  2224
\subsection{Authentication}
blanchet@33191
  2225
\label{authentication}
blanchet@33191
  2226
blanchet@33191
  2227
\begin{enum}
blanchet@33191
  2228
\opfalse{check\_potential}{trust\_potential}
blanchet@33191
  2229
Specifies whether potential counterexamples should be given to Isabelle's
blanchet@33191
  2230
\textit{auto} tactic to assess their validity. If a potential counterexample is
blanchet@33191
  2231
shown to be genuine, Nitpick displays a message to this effect and terminates.
blanchet@33191
  2232
blanchet@33191
  2233
\nopagebreak
blanchet@33561
  2234
{\small See also \textit{max\_potential} (\S\ref{output-format}).}
blanchet@33191
  2235
blanchet@33191
  2236
\opfalse{check\_genuine}{trust\_genuine}
blanchet@33191
  2237
Specifies whether genuine and likely genuine counterexamples should be given to
blanchet@33191
  2238
Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
blanchet@33191
  2239
counterexample is shown to be spurious, the user is kindly asked to send a bug
blanchet@33191
  2240
report to the author at
blanchet@33191
  2241
\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
blanchet@33191
  2242
blanchet@33191
  2243
\nopagebreak
blanchet@33561
  2244
{\small See also \textit{max\_genuine} (\S\ref{output-format}).}
blanchet@33191
  2245
blanchet@34982
  2246
\opnodefault{expect}{string}
blanchet@33191
  2247
Specifies the expected outcome, which must be one of the following:
blanchet@33191
  2248
blanchet@33191
  2249
\begin{enum}
blanchet@33191
  2250
\item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
blanchet@33191
  2251
\item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a ``likely
blanchet@33191
  2252
genuine'' counterexample (i.e., a counterexample that is genuine unless
blanchet@33191
  2253
it contradicts a missing axiom or a dangerous option was used inappropriately).
blanchet@33191
  2254
\item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
blanchet@33191
  2255
\item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
blanchet@33191
  2256
\item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
blanchet@33191
  2257
Kodkod ran out of memory).
blanchet@33191
  2258
\end{enum}
blanchet@33191
  2259
blanchet@33191
  2260
Nitpick emits an error if the actual outcome differs from the expected outcome.
blanchet@33191
  2261
This option is useful for regression testing.
blanchet@33191
  2262
\end{enum}
blanchet@33191
  2263
blanchet@33191
  2264
\subsection{Optimizations}
blanchet@33191
  2265
\label{optimizations}
blanchet@33191
  2266
blanchet@33191
  2267
\def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
blanchet@33191
  2268
blanchet@33191
  2269
\sloppy
blanchet@33191
  2270
blanchet@33191
  2271
\begin{enum}
blanchet@34982
  2272
\opdefault{sat\_solver}{string}{smart}
blanchet@33191
  2273
Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
blanchet@33191
  2274
to be faster than their Java counterparts, but they can be more difficult to
blanchet@33191
  2275
install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
blanchet@33191
  2276
\textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
blanchet@35078
  2277
you will need an incremental SAT solver, such as \textit{MiniSat\_JNI}
blanchet@33191
  2278
(recommended) or \textit{SAT4J}.
blanchet@33191
  2279
blanchet@33191
  2280
The supported solvers are listed below:
blanchet@33191
  2281
blanchet@33191
  2282
\begin{enum}
blanchet@33191
  2283
blanchet@33191
  2284
\item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
blanchet@33191
  2285
written in \cpp{}. To use MiniSat, set the environment variable
blanchet@33191
  2286
\texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
blanchet@33191
  2287
executable. The \cpp{} sources and executables for MiniSat are available at
blanchet@33191
  2288
\url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
blanchet@33191
  2289
and 2.0 beta (2007-07-21).
blanchet@33191
  2290
blanchet@35078
  2291
\item[$\bullet$] \textbf{\textit{MiniSat\_JNI}}: The JNI (Java Native Interface)
blanchet@33191
  2292
version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which
blanchet@33191
  2293
you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
blanchet@33191
  2294
version of MiniSat, the JNI version can be used incrementally.
blanchet@33191
  2295
blanchet@33731
  2296
%%% No longer true:
blanchet@33731
  2297
%%% "It is bundled with Kodkodi and requires no further installation or
blanchet@33731
  2298
%%% configuration steps. Alternatively,"
blanchet@33191
  2299
\item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
blanchet@33731
  2300
written in C. You can install a standard version of
blanchet@33191
  2301
PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
blanchet@33191
  2302
that contains the \texttt{picosat} executable. The C sources for PicoSAT are
blanchet@33191
  2303
available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
blanchet@33191
  2304
Nitpick has been tested with version 913.
blanchet@33191
  2305
blanchet@33191
  2306
\item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
blanchet@33191
  2307
in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
blanchet@33191
  2308
the directory that contains the \texttt{zchaff} executable. The \cpp{} sources
blanchet@33191
  2309
and executables for zChaff are available at
blanchet@33191
  2310
\url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
blanchet@33191
  2311
versions 2004-05-13, 2004-11-15, and 2007-03-12.
blanchet@33191
  2312
blanchet@35078
  2313
\item[$\bullet$] \textbf{\textit{zChaff\_JNI}}: The JNI version of zChaff is
blanchet@33191
  2314
bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on
blanchet@33191
  2315
Kodkod's web site \cite{kodkod-2009}.
blanchet@33191
  2316
blanchet@33191
  2317
\item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
blanchet@33191
  2318
\cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
blanchet@33191
  2319
directory that contains the \texttt{rsat} executable. The \cpp{} sources for
blanchet@33191
  2320
RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been
blanchet@33191
  2321
tested with version 2.01.
blanchet@33191
  2322
blanchet@33191
  2323
\item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
blanchet@33191
  2324
written in C. To use BerkMin, set the environment variable
blanchet@33191
  2325
\texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
blanchet@33191
  2326
executable. The BerkMin executables are available at
blanchet@33191
  2327
\url{http://eigold.tripod.com/BerkMin.html}.
blanchet@33191
  2328
blanchet@35078
  2329
\item[$\bullet$] \textbf{\textit{BerkMin\_Alloy}}: Variant of BerkMin that is
blanchet@33191
  2330
included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
blanchet@33191
  2331
version of BerkMin, set the environment variable
blanchet@33191
  2332
\texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
blanchet@33191
  2333
executable.
blanchet@33191
  2334
blanchet@33191
  2335
\item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
blanchet@33191
  2336
written in C. To use Jerusat, set the environment variable
blanchet@33191
  2337
\texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
blanchet@33191
  2338
executable. The C sources for Jerusat are available at
blanchet@33191
  2339
\url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
blanchet@33191
  2340
blanchet@33191
  2341
\item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
blanchet@33191
  2342
written in Java that can be used incrementally. It is bundled with Kodkodi and
blanchet@33191
  2343
requires no further installation or configuration steps. Do not attempt to
blanchet@33191
  2344
install the official SAT4J packages, because their API is incompatible with
blanchet@33191
  2345
Kodkod.
blanchet@33191
  2346
blanchet@35078
  2347
\item[$\bullet$] \textbf{\textit{SAT4J\_Light}}: Variant of SAT4J that is
blanchet@33191
  2348
optimized for small problems. It can also be used incrementally.
blanchet@33191
  2349
blanchet@33191
  2350
\item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
blanchet@33191
  2351
experimental solver written in \cpp. To use HaifaSat, set the environment
blanchet@33191
  2352
variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
blanchet@33191
  2353
\texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at
blanchet@33191
  2354
\url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
blanchet@33191
  2355
blanchet@33191
  2356
\item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
blanchet@33731
  2357
\textit{smart}, Nitpick selects the first solver among MiniSat,
blanchet@35078
  2358
PicoSAT, zChaff, RSat, BerkMin, BerkMin\_Alloy, Jerusat, MiniSat\_JNI, and zChaff\_JNI
blanchet@33731
  2359
that is recognized by Isabelle. If none is found, it falls back on SAT4J, which
blanchet@33731
  2360
should always be available. If \textit{verbose} (\S\ref{output-format}) is
blanchet@33731
  2361
enabled, Nitpick displays which SAT solver was chosen.
blanchet@33191
  2362
\end{enum}
blanchet@33191
  2363
\fussy
blanchet@33191
  2364
blanchet@34982
  2365
\opdefault{batch\_size}{int\_or\_smart}{smart}
blanchet@33191
  2366
Specifies the maximum number of Kodkod problems that should be lumped together
blanchet@33191
  2367
when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
blanchet@33191
  2368
together ensures that Kodkodi is launched less often, but it makes the verbose
blanchet@33191
  2369
output less readable and is sometimes detrimental to performance. If
blanchet@33191
  2370
\textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
blanchet@33191
  2371
\textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
blanchet@33191
  2372
blanchet@33191
  2373
\optrue{destroy\_constrs}{dont\_destroy\_constrs}
blanchet@33191
  2374
Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
blanchet@33191
  2375
be rewritten to use (automatically generated) discriminators and destructors.
blanchet@33191
  2376
This optimization can drastically reduce the size of the Boolean formulas given
blanchet@33191
  2377
to the SAT solver.
blanchet@33191
  2378
blanchet@33191
  2379
\nopagebreak
blanchet@33191
  2380
{\small See also \textit{debug} (\S\ref{output-format}).}
blanchet@33191
  2381
blanchet@33191
  2382
\optrue{specialize}{dont\_specialize}
blanchet@33191
  2383
Specifies whether functions invoked with static arguments should be specialized.
blanchet@33191
  2384
This optimization can drastically reduce the search space, especially for
blanchet@33191
  2385
higher-order functions.
blanchet@33191
  2386
blanchet@33191
  2387
\nopagebreak
blanchet@33191
  2388
{\small See also \textit{debug} (\S\ref{output-format}) and
blanchet@33191
  2389
\textit{show\_consts} (\S\ref{output-format}).}
blanchet@33191
  2390
blanchet@33191
  2391
\optrue{skolemize}{dont\_skolemize}
blanchet@33191
  2392
Specifies whether the formula should be skolemized. For performance reasons,
blanchet@33191
  2393
(positive) $\forall$-quanti\-fiers that occur in the scope of a higher-order
blanchet@33191
  2394
(positive) $\exists$-quanti\-fier are left unchanged.
blanchet@33191
  2395
blanchet@33191
  2396
\nopagebreak
blanchet@33191
  2397
{\small See also \textit{debug} (\S\ref{output-format}) and
blanchet@33191
  2398
\textit{show\_skolems} (\S\ref{output-format}).}
blanchet@33191
  2399
blanchet@33191
  2400
\optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
blanchet@33191
  2401
Specifies whether Nitpick should use Kodkod's transitive closure operator to
blanchet@33191
  2402
encode non-well-founded ``linear inductive predicates,'' i.e., inductive
blanchet@33191
  2403
predicates for which each the predicate occurs in at most one assumption of each
blanchet@33191
  2404
introduction rule. Using the reflexive transitive closure is in principle
blanchet@33191
  2405
equivalent to setting \textit{iter} to the cardinality of the predicate's
blanchet@33191
  2406
domain, but it is usually more efficient.
blanchet@33191
  2407
blanchet@33191
  2408
{\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
blanchet@33191
  2409
(\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
blanchet@33191
  2410
blanchet@33191
  2411
\optrue{uncurry}{dont\_uncurry}
blanchet@33191
  2412
Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no
blanchet@33191
  2413
tangible effect on efficiency, but it creates opportunities for the boxing 
blanchet@33191
  2414
optimization.
blanchet@33191
  2415
blanchet@33191
  2416
\nopagebreak
blanchet@33191
  2417
{\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug}
blanchet@33191
  2418
(\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).}
blanchet@33191
  2419
blanchet@33191
  2420
\optrue{fast\_descrs}{full\_descrs}
blanchet@33191
  2421
Specifies whether Nitpick should optimize the definite and indefinite
blanchet@33191
  2422
description operators (THE and SOME). The optimized versions usually help
blanchet@33191
  2423
Nitpick generate more counterexamples or at least find them faster, but only the
blanchet@33191
  2424
unoptimized versions are complete when all types occurring in the formula are
blanchet@33191
  2425
finite.
blanchet@33191
  2426
blanchet@33191
  2427
{\small See also \textit{debug} (\S\ref{output-format}).}
blanchet@33191
  2428
blanchet@33191
  2429
\optrue{peephole\_optim}{no\_peephole\_optim}
blanchet@33191
  2430
Specifies whether Nitpick should simplify the generated Kodkod formulas using a
blanchet@33191
  2431
peephole optimizer. These optimizations can make a significant difference.
blanchet@33191
  2432
Unless you are tracking down a bug in Nitpick or distrust the peephole
blanchet@33191
  2433
optimizer, you should leave this option enabled.
blanchet@33191
  2434
blanchet@34982
  2435
\opdefault{sym\_break}{int}{20}
blanchet@33191
  2436
Specifies an upper bound on the number of relations for which Kodkod generates
blanchet@33191
  2437
symmetry breaking predicates. According to the Kodkod documentation
blanchet@33191
  2438
\cite{kodkod-2009-options}, ``in general, the higher this value, the more
blanchet@33191
  2439
symmetries will be broken, and the faster the formula will be solved. But,
blanchet@33191
  2440
setting the value too high may have the opposite effect and slow down the
blanchet@33191
  2441
solving.''
blanchet@33191
  2442
blanchet@34982
  2443
\opdefault{sharing\_depth}{int}{3}
blanchet@33191
  2444
Specifies the depth to which Kodkod should check circuits for equivalence during
blanchet@33191
  2445
the translation to SAT. The default of 3 is the same as in Alloy. The minimum
blanchet@33191
  2446
allowed depth is 1. Increasing the sharing may result in a smaller SAT problem,
blanchet@33191
  2447
but can also slow down Kodkod.
blanchet@33191
  2448
blanchet@33191
  2449
\opfalse{flatten\_props}{dont\_flatten\_props}
blanchet@33191
  2450
Specifies whether Kodkod should try to eliminate intermediate Boolean variables.
blanchet@33191
  2451
Although this might sound like a good idea, in practice it can drastically slow
blanchet@33191
  2452
down Kodkod.
blanchet@33191
  2453
blanchet@34982
  2454
\opdefault{max\_threads}{int}{0}
blanchet@33191
  2455
Specifies the maximum number of threads to use in Kodkod. If this option is set
blanchet@33191
  2456
to 0, Kodkod will compute an appropriate value based on the number of processor
blanchet@33191
  2457
cores available.
blanchet@33191
  2458
blanchet@33191
  2459
\nopagebreak
blanchet@33191
  2460
{\small See also \textit{batch\_size} (\S\ref{optimizations}) and
blanchet@33191
  2461
\textit{timeout} (\S\ref{timeouts}).}
blanchet@33191
  2462
\end{enum}
blanchet@33191
  2463
blanchet@33191
  2464
\subsection{Timeouts}
blanchet@33191
  2465
\label{timeouts}
blanchet@33191
  2466
blanchet@33191
  2467
\begin{enum}
blanchet@34982
  2468
\opdefault{timeout}{time}{$\mathbf{30}$ s}
blanchet@33191
  2469
Specifies the maximum amount of time that the \textbf{nitpick} command should
blanchet@33191
  2470
spend looking for a counterexample. Nitpick tries to honor this constraint as
blanchet@33191
  2471
well as it can but offers no guarantees. For automatic runs,
blanchet@33561
  2472
\textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share
blanchet@33561
  2473
a time slot whose length is specified by the ``Auto Counterexample Time
blanchet@33561
  2474
Limit'' option in Proof General.
blanchet@33191
  2475
blanchet@33191
  2476
\nopagebreak
blanchet@33561
  2477
{\small See also \textit{max\_threads} (\S\ref{optimizations}).}
blanchet@33191
  2478
blanchet@34982
  2479
\opdefault{tac\_timeout}{time}{$\mathbf{500}$\,ms}
blanchet@33191
  2480
Specifies the maximum amount of time that the \textit{auto} tactic should use
blanchet@33191
  2481
when checking a counterexample, and similarly that \textit{lexicographic\_order}
blanchet@34982
  2482
and \textit{size\_change} should use when checking whether a (co)in\-duc\-tive
blanchet@33191
  2483
predicate is well-founded. Nitpick tries to honor this constraint as well as it
blanchet@33191
  2484
can but offers no guarantees.
blanchet@33191
  2485
blanchet@33191
  2486
\nopagebreak
blanchet@33191
  2487
{\small See also \textit{wf} (\S\ref{scope-of-search}),
blanchet@33191
  2488
\textit{check\_potential} (\S\ref{authentication}),
blanchet@33191
  2489
and \textit{check\_genuine} (\S\ref{authentication}).}
blanchet@33191
  2490
\end{enum}
blanchet@33191
  2491
blanchet@33191
  2492
\section{Attribute Reference}
blanchet@33191
  2493
\label{attribute-reference}
blanchet@33191
  2494
blanchet@33191
  2495
Nitpick needs to consider the definitions of all constants occurring in a
blanchet@33191
  2496
formula in order to falsify it. For constants introduced using the
blanchet@33191
  2497
\textbf{definition} command, the definition is simply the associated
blanchet@33191
  2498
\textit{\_def} axiom. In contrast, instead of using the internal representation
blanchet@33191
  2499
of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
blanchet@33191
  2500
\textbf{nominal\_primrec} packages, Nitpick relies on the more natural
blanchet@33191
  2501
equational specification entered by the user.
blanchet@33191
  2502
blanchet@33191
  2503
Behind the scenes, Isabelle's built-in packages and theories rely on the
blanchet@33191
  2504
following attributes to affect Nitpick's behavior:
blanchet@33191
  2505
blanchet@33191
  2506
\begin{itemize}
blanchet@33191
  2507
\flushitem{\textit{nitpick\_def}}
blanchet@33191
  2508
blanchet@33191
  2509
\nopagebreak
blanchet@33191
  2510
This attribute specifies an alternative definition of a constant. The
blanchet@33191
  2511
alternative definition should be logically equivalent to the constant's actual
blanchet@33191
  2512
axiomatic definition and should be of the form
blanchet@33191
  2513
blanchet@33191
  2514
\qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
blanchet@33191
  2515
blanchet@33191
  2516
where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
blanchet@33191
  2517
$t$.
blanchet@33191
  2518
blanchet@33191
  2519
\flushitem{\textit{nitpick\_simp}}
blanchet@33191
  2520
blanchet@33191
  2521
\nopagebreak
blanchet@33191
  2522
This attribute specifies the equations that constitute the specification of a
blanchet@33191
  2523
constant. For functions defined using the \textbf{primrec}, \textbf{function},
blanchet@33191
  2524
and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
blanchet@33191
  2525
\textit{simps} rules. The equations must be of the form
blanchet@33191
  2526
blanchet@33191
  2527
\qquad $c~t_1~\ldots\ t_n \,=\, u.$
blanchet@33191
  2528
blanchet@33191
  2529
\flushitem{\textit{nitpick\_psimp}}
blanchet@33191
  2530
blanchet@33191
  2531
\nopagebreak
blanchet@33191
  2532
This attribute specifies the equations that constitute the partial specification
blanchet@33191
  2533
of a constant. For functions defined using the \textbf{function} package, this
blanchet@33191
  2534
corresponds to the \textit{psimps} rules. The conditional equations must be of
blanchet@33191
  2535
the form
blanchet@33191
  2536
blanchet@33191
  2537
\qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
blanchet@33191
  2538
blanchet@33191
  2539
\flushitem{\textit{nitpick\_intro}}
blanchet@33191
  2540
blanchet@33191
  2541
\nopagebreak
blanchet@33191
  2542
This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
blanchet@33191
  2543
For predicates defined using the \textbf{inductive} or \textbf{coinductive}
blanchet@33191
  2544
command, this corresponds to the \textit{intros} rules. The introduction rules
blanchet@33191
  2545
must be of the form
blanchet@33191
  2546
blanchet@33191
  2547
\qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
blanchet@33191
  2548
\ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\