doc-src/Nitpick/nitpick.tex
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clarify attribute documentation
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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{footmisc}
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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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%\headheight=0mm
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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-2010} 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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You can also invoke Nitpick from the ``Commands'' submenu of the
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``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c C-a
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C-n. This is equivalent to entering the \textbf{nitpick} command with no
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arguments in the theory text.
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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{Installation}
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%\label{installation}
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%
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%MISSING:
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%
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%  * Nitpick is part of Isabelle/HOL
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%  * but it relies on an external tool called Kodkodi (Kodkod wrapper)
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%  * Two options:
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%    * if you use a prebuilt Isabelle package, Kodkodi is automatically there
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%    * if you work from sources, the latest Kodkodi can be obtained from ...
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%      download it, install it in some directory of your choice (e.g.,
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%      $ISABELLE_HOME/contrib/kodkodi), and add the absolute path to Kodkodi
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%      in your .isabelle/etc/components file
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%
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%  * If you're not sure, just try the example in the next section
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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~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 (Java Native Interface)
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version of MiniSat and with multithreading disabled to reduce nondeterminism.
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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]
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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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%FIXME: If you get the output:...
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%Then do such-and-such.
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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 $\mathit{The} = \undef{}
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    (\!\begin{aligned}[t]%
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    & \{a_1, a_2, a_3\} := a_3,\> \{a_1, a_2\} := a_3,\> \{a_1, a_3\} := a_3, \\[-2pt] %% TYPESETTING
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   340
    & \{a_1\} := a_1,\> \{a_2, a_3\} := a_1,\> \{a_2\} := a_2, \\[-2pt]
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   341
    & \{a_3\} := a_3,\> \{\} := a_3)\end{aligned}$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   342
\postw
fe3c65d9c577 Added Nitpick manual.
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parents:
diff changeset
   343
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   344
Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
   345
just like before.\footnote{The Isabelle/HOL notation $f(x :=
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
   346
y)$ denotes the function that maps $x$ to $y$ and that otherwise behaves like
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
   347
$f$.}
33191
fe3c65d9c577 Added Nitpick manual.
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parents:
diff changeset
   348
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   349
Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   350
unique $x$ such that'') at the front of our putative lemma's assumption:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   351
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   352
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   353
\textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   354
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   355
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   356
The fix appears to work:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   357
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   358
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   359
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   360
\slshape Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   361
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   362
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   363
We can further increase our confidence in the formula by exhausting all
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   364
cardinalities up to 50:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   365
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   366
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   367
\textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   368
can be entered as \texttt{-} (hyphen) or
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   369
\texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   370
\slshape Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   371
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   372
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   373
Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   374
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   375
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   376
\textbf{sledgehammer} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   377
{\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   378
$\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   379
Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   380
\textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   381
{\slshape No subgoals!}% \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   382
%\textbf{done}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   383
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   384
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   385
This must be our lucky day.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   386
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   387
\subsection{Skolemization}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   388
\label{skolemization}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   389
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   390
Are all invertible functions onto? Let's find out:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   391
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   392
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   393
\textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   394
 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   395
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   396
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   397
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   398
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   399
\hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   400
\hbox{}\qquad Skolem constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   401
\hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   402
\hbox{}\qquad\qquad $y = a_2$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   403
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   404
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   405
Although $f$ is the only free variable occurring in the formula, Nitpick also
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   406
displays values for the bound variables $g$ and $y$. These values are available
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   407
to Nitpick because it performs skolemization as a preprocessing step.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   408
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   409
In the previous example, skolemization only affected the outermost quantifiers.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   410
This is not always the case, as illustrated below:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   411
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   412
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   413
\textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   414
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   415
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   416
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   417
\hbox{}\qquad Skolem constant: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   418
\hbox{}\qquad\qquad $\lambda x.\; f =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   419
    \undef{}(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   420
    & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   421
    & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   422
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   423
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   424
The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   425
$x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   426
function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   427
maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   428
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   429
The source of the Skolem constants is sometimes more obscure:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   430
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   431
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   432
\textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   433
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   434
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   435
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   436
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   437
\hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   438
\hbox{}\qquad Skolem constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   439
\hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   440
\hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   441
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   442
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   443
What happened here is that Nitpick expanded the \textit{sym} constant to its
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   444
definition:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   445
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   446
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   447
$\mathit{sym}~r \,\equiv\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   448
 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   449
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   450
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   451
As their names suggest, the Skolem constants $\mathit{sym}.x$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   452
$\mathit{sym}.y$ are simply the bound variables $x$ and $y$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   453
from \textit{sym}'s definition.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   454
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   455
\subsection{Natural Numbers and Integers}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   456
\label{natural-numbers-and-integers}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   457
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   458
Because of the axiom of infinity, the type \textit{nat} does not admit any
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   459
finite models. To deal with this, Nitpick's approach is to consider finite
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   460
subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   461
value (displayed as `$\unk$'). The type \textit{int} is handled similarly.
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   462
Internally, undefined values lead to a three-valued logic.
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   463
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   464
Here is an example involving \textit{int\/}:
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   465
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   466
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   467
\textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   468
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   469
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   470
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   471
\hbox{}\qquad\qquad $i = 0$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   472
\hbox{}\qquad\qquad $j = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   473
\hbox{}\qquad\qquad $m = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   474
\hbox{}\qquad\qquad $n = 0$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   475
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   476
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   477
Internally, Nitpick uses either a unary or a binary representation of numbers.
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   478
The unary representation is more efficient but only suitable for numbers very
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   479
close to zero. By default, Nitpick attempts to choose the more appropriate
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   480
encoding by inspecting the formula at hand. This behavior can be overridden by
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   481
passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   482
binary notation, the number of bits to use can be specified using
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   483
the \textit{bits} option. For example:
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   484
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   485
\prew
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   486
\textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$]
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   487
\postw
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   488
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   489
With infinite types, we don't always have the luxury of a genuine counterexample
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   490
and must often content ourselves with a potential one. The tedious task of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   491
finding out whether the potential counterexample is in fact genuine can be
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   492
outsourced to \textit{auto} by passing \textit{check\_potential}. For example:
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   493
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   494
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   495
\textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   496
\textbf{nitpick} [\textit{card~nat}~= 50, \textit{check\_potential}] \\[2\smallskipamount]
35385
29f81babefd7 improved precision of infinite "shallow" datatypes in Nitpick;
blanchet
parents: 35335
diff changeset
   497
\slshape Warning: The conjecture either trivially holds for the given scopes or lies outside Nitpick's supported
35185
9b8f351cced6 added yet another hint to Nitpick's output, this time warning about problems for which nothing was effectively tested
blanchet
parents: 35183
diff changeset
   498
fragment. Only potential counterexamples may be found. \\[2\smallskipamount]
9b8f351cced6 added yet another hint to Nitpick's output, this time warning about problems for which nothing was effectively tested
blanchet
parents: 35183
diff changeset
   499
Nitpick found a potential counterexample: \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   500
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   501
\hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   502
Confirmation by ``\textit{auto}'': The above counterexample is genuine.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   503
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   504
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   505
You might wonder why the counterexample is first reported as potential. The root
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   506
of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   507
\mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   508
that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   509
\textit{False}; but otherwise, it does not know anything about values of $n \ge
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   510
\textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   511
\textit{True}. Since the assumption can never be satisfied, the putative lemma
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   512
can never be falsified.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   513
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   514
Incidentally, if you distrust the so-called genuine counterexamples, you can
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   515
enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   516
aware that \textit{auto} will usually fail to prove that the counterexample is
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   517
genuine or spurious.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   518
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   519
Some conjectures involving elementary number theory make Nitpick look like a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   520
giant with feet of clay:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   521
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   522
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   523
\textbf{lemma} ``$P~\textit{Suc}$'' \\
35309
997aa3a3e4bb catch IO errors in Nitpick's "kodkodi" invocation + shorten execution time of "Manual_Nits" example
blanchet
parents: 35284
diff changeset
   524
\textbf{nitpick} \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   525
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   526
Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   527
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   528
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   529
On any finite set $N$, \textit{Suc} is a partial function; for example, if $N =
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   530
\{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\,
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   531
\ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when passed as
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   532
argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$. The next
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   533
example is similar:
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   534
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   535
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   536
\textbf{lemma} ``$P~(\textit{op}~{+}\Colon
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   537
\textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   538
\textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   539
{\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   540
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   541
\hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   542
\textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   543
{\slshape Nitpick found no counterexample.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   544
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   545
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   546
The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   547
$\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   548
1\}$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   549
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   550
Because numbers are infinite and are approximated using a three-valued logic,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   551
there is usually no need to systematically enumerate domain sizes. If Nitpick
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   552
cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   553
unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   554
example above is an exception to this principle.) Nitpick nonetheless enumerates
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   555
all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   556
cardinalities are fast to handle and give rise to simpler counterexamples. This
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   557
is explained in more detail in \S\ref{scope-monotonicity}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   558
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   559
\subsection{Inductive Datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   560
\label{inductive-datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   561
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   562
Like natural numbers and integers, inductive datatypes with recursive
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   563
constructors admit no finite models and must be approximated by a subterm-closed
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   564
subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   565
Nitpick looks for all counterexamples that can be built using at most 10
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   566
different lists.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   567
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   568
Let's see with an example involving \textit{hd} (which returns the first element
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   569
of a list) and $@$ (which concatenates two lists):
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   570
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   571
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   572
\textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   573
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   574
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   575
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   576
\hbox{}\qquad\qquad $\textit{xs} = []$ \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   577
\hbox{}\qquad\qquad $\textit{y} = a_1$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   578
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   579
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   580
To see why the counterexample is genuine, we enable \textit{show\_consts}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   581
and \textit{show\_\allowbreak datatypes}:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   582
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   583
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   584
{\slshape Datatype:} \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   585
\hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   586
{\slshape Constants:} \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   587
\hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_1, a_1])$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   588
\hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   589
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   590
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   591
Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   592
including $a_2$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   593
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   594
The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   595
append operator whose second argument is fixed to be $[y, y]$. Appending $[a_1,
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   596
a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   597
representable in the subset of $'a$~\textit{list} considered by Nitpick, which
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   598
is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   599
appending $[a_1, a_1]$ to itself gives $\unk$.
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   600
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   601
Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   602
considers the following subsets:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   603
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   604
\kern-.5\smallskipamount %% TYPESETTING
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   605
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   606
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   607
\begin{multicols}{3}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   608
$\{[],\, [a_1],\, [a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   609
$\{[],\, [a_1],\, [a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   610
$\{[],\, [a_2],\, [a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   611
$\{[],\, [a_1],\, [a_1, a_1]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   612
$\{[],\, [a_1],\, [a_2, a_1]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   613
$\{[],\, [a_1],\, [a_3, a_1]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   614
$\{[],\, [a_2],\, [a_1, a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   615
$\{[],\, [a_2],\, [a_2, a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   616
$\{[],\, [a_2],\, [a_3, a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   617
$\{[],\, [a_3],\, [a_1, a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   618
$\{[],\, [a_3],\, [a_2, a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   619
$\{[],\, [a_3],\, [a_3, a_3]\}$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   620
\end{multicols}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   621
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   622
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   623
\kern-2\smallskipamount %% TYPESETTING
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   624
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   625
All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   626
are listed and only those. As an example of a non-subterm-closed subset,
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   627
consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   628
that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   629
\mathcal{S}$ as a subterm.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   630
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   631
Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   632
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   633
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   634
\textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   635
\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   636
\\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   637
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   638
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   639
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   640
\hbox{}\qquad\qquad $\textit{xs} = [a_1]$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   641
\hbox{}\qquad\qquad $\textit{ys} = [a_2]$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   642
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   643
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   644
\hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   645
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   646
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   647
Because datatypes are approximated using a three-valued logic, there is usually
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   648
no need to systematically enumerate cardinalities: If Nitpick cannot find a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   649
genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   650
unlikely that one could be found for smaller cardinalities.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   651
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   652
\subsection{Typedefs, Quotient Types, Records, Rationals, and Reals}
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   653
\label{typedefs-quotient-types-records-rationals-and-reals}
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   654
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   655
Nitpick generally treats types declared using \textbf{typedef} as datatypes
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   656
whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   657
For example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   658
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   659
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   660
\textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   661
\textbf{by}~\textit{blast} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   662
\textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   663
\textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   664
\textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   665
\textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   666
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   667
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   668
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   669
\hbox{}\qquad\qquad $P = \{\Abs{0},\, \Abs{1}\}$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   670
\hbox{}\qquad\qquad $x = \Abs{2}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   671
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   672
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   673
\hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   674
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   675
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   676
In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   677
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   678
Quotient types are handled in much the same way. The following fragment defines
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   679
the integer type \textit{my\_int} by encoding the integer $x$ by a pair of
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   680
natural numbers $(m, n)$ such that $x + n = m$:
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   681
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   682
\prew
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   683
\textbf{fun} \textit{my\_int\_rel} \textbf{where} \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   684
``$\textit{my\_int\_rel}~(x,\, y)~(u,\, v) = (x + v = u + y)$'' \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   685
%
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   686
\textbf{quotient\_type}~\textit{my\_int} = ``$\textit{nat} \times \textit{nat\/}$''$\;{/}\;$\textit{my\_int\_rel} \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   687
\textbf{by}~(\textit{auto simp add\/}:\ \textit{equivp\_def expand\_fun\_eq}) \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   688
%
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   689
\textbf{definition}~\textit{add\_raw}~\textbf{where} \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   690
``$\textit{add\_raw} \,\equiv\, \lambda(x,\, y)~(u,\, v).\; (x + (u\Colon\textit{nat}), y + (v\Colon\textit{nat}))$'' \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   691
%
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   692
\textbf{quotient\_definition} ``$\textit{add\/}\Colon\textit{my\_int} \Rightarrow \textit{my\_int} \Rightarrow \textit{my\_int\/}$'' \textbf{is} \textit{add\_raw} \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   693
%
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   694
\textbf{lemma} ``$\textit{add}~x~y = \textit{add}~x~x$'' \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   695
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   696
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   697
\hbox{}\qquad Free variables: \nopagebreak \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   698
\hbox{}\qquad\qquad $x = \Abs{(0,\, 0)}$ \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   699
\hbox{}\qquad\qquad $y = \Abs{(1,\, 0)}$ \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   700
\hbox{}\qquad Datatypes: \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   701
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, \unr\}$ \\
35665
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
   702
\hbox{}\qquad\qquad $\textit{nat} \times \textit{nat}~[\textsl{boxed\/}] = \{(0,\, 0),\> (1,\, 0),\> \unr\}$ \\
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   703
\hbox{}\qquad\qquad $\textit{my\_int} = \{\Abs{(0,\, 0)},\> \Abs{(1,\, 0)},\> \unr\}$
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   704
\postw
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   705
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   706
In the counterexample, $\Abs{(0,\, 0)}$ and $\Abs{(1,\, 0)}$ represent the
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   707
integers $0$ and $1$, respectively. Other representants would have been
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   708
possible---e.g., $\Abs{(5,\, 5)}$ and $\Abs{(12,\, 11)}$. If we are going to
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   709
use \textit{my\_int} extensively, it pays off to install a term postprocessor
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   710
that converts the pair notation to the standard mathematical notation:
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   711
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   712
\prew
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   713
$\textbf{ML}~\,\{{*} \\
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   714
\!\begin{aligned}[t]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   715
%& ({*}~\,\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \\[-2pt]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   716
%& \phantom{(*}~\,{\rightarrow}\;\textit{term}~\,{*}) \\[-2pt]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   717
& \textbf{fun}\,~\textit{my\_int\_postproc}~\_~\_~\_~T~(\textit{Const}~\_~\$~(\textit{Const}~\_~\$~\textit{t1}~\$~\textit{t2\/})) = {} \\[-2pt]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   718
& \phantom{fun}\,~\textit{HOLogic.mk\_number}~T~(\textit{snd}~(\textit{HOLogic.dest\_number~t1}) \\[-2pt]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   719
& \phantom{fun\,~\textit{HOLogic.mk\_number}~T~(}{-}~\textit{snd}~(\textit{HOLogic.dest\_number~t2\/})) \\[-2pt]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   720
& \phantom{fun}\!{\mid}\,~\textit{my\_int\_postproc}~\_~\_~\_~\_~t = t \\[-2pt]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   721
{*}\}\end{aligned}$ \\[2\smallskipamount]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   722
$\textbf{setup}~\,\{{*} \\
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   723
\!\begin{aligned}[t]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   724
& \textit{Nitpick.register\_term\_postprocessor}~\!\begin{aligned}[t]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   725
  & @\{\textrm{typ}~\textit{my\_int}\}~\textit{my\_int\_postproc}\end{aligned} \\[-2pt]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   726
{*}\}\end{aligned}$
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   727
\postw
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   728
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   729
Records are also handled as datatypes with a single constructor:
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   730
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   731
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   732
\textbf{record} \textit{point} = \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   733
\hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   734
\hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   735
\textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   736
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   737
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   738
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   739
\hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   740
\hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   741
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   742
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   743
\hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   744
& \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   745
& \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   746
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   747
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   748
Finally, Nitpick provides rudimentary support for rationals and reals using a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   749
similar approach:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   750
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   751
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   752
\textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   753
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   754
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   755
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   756
\hbox{}\qquad\qquad $x = 1/2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   757
\hbox{}\qquad\qquad $y = -1/2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   758
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   759
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   760
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   761
\hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   762
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   763
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   764
\subsection{Inductive and Coinductive Predicates}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   765
\label{inductive-and-coinductive-predicates}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   766
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   767
Inductively defined predicates (and sets) are particularly problematic for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   768
counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   769
loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
38176
bc2f9383fd59 clarify attribute documentation
blanchet
parents: 38175
diff changeset
   770
the problem is that they are defined using a least fixed-point construction.
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   771
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   772
Nitpick's philosophy is that not all inductive predicates are equal. Consider
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   773
the \textit{even} predicate below:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   774
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   775
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   776
\textbf{inductive}~\textit{even}~\textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   777
``\textit{even}~0'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   778
``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   779
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   780
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   781
This predicate enjoys the desirable property of being well-founded, which means
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   782
that the introduction rules don't give rise to infinite chains of the form
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   783
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   784
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   785
$\cdots\,\Longrightarrow\, \textit{even}~k''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   786
       \,\Longrightarrow\, \textit{even}~k'
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   787
       \,\Longrightarrow\, \textit{even}~k.$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   788
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   789
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   790
For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   791
$k/2 + 1$:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   792
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   793
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   794
$\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   795
       \,\Longrightarrow\, \textit{even}~(k - 2)
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   796
       \,\Longrightarrow\, \textit{even}~k.$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   797
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   798
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   799
Wellfoundedness is desirable because it enables Nitpick to use a very efficient
38176
bc2f9383fd59 clarify attribute documentation
blanchet
parents: 38175
diff changeset
   800
fixed-point computation.%
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   801
\footnote{If an inductive predicate is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   802
well-founded, then it has exactly one fixed point, which is simultaneously the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   803
least and the greatest fixed point. In these circumstances, the computation of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   804
the least fixed point amounts to the computation of an arbitrary fixed point,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   805
which can be performed using a straightforward recursive equation.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   806
Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   807
just as Isabelle's \textbf{function} package usually discharges termination
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   808
proof obligations automatically.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   809
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   810
Let's try an example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   811
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   812
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   813
\textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   814
\textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   815
\slshape The inductive predicate ``\textit{even}'' was proved well-founded.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   816
Nitpick can compute it efficiently. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   817
Trying 1 scope: \\
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   818
\hbox{}\qquad \textit{card nat}~= 50. \\[2\smallskipamount]
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   819
Nitpick found a potential counterexample for \textit{card nat}~= 50: \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   820
\hbox{}\qquad Empty assignment \\[2\smallskipamount]
38123
36f649db4a6c clarify Nitpick's output in case of a potential counterexample
blanchet
parents: 38122
diff changeset
   821
Nitpick could not find a better counterexample. It checked 1 of 1 scope. \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   822
Total time: 2274 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   823
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   824
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   825
No genuine counterexample is possible because Nitpick cannot rule out the
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   826
existence of a natural number $n \ge 50$ such that both $\textit{even}~n$ and
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   827
$\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   828
existential quantifier:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   829
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   830
\prew
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   831
\textbf{lemma} ``$\exists n \mathbin{\le} 49.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
   832
\textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}] \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   833
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   834
\hbox{}\qquad Empty assignment
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   835
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   836
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   837
So far we were blessed by the wellfoundedness of \textit{even}. What happens if
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   838
we use the following definition instead?
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   839
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   840
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   841
\textbf{inductive} $\textit{even}'$ \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   842
``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   843
``$\textit{even}'~2$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   844
``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   845
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   846
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   847
This definition is not well-founded: From $\textit{even}'~0$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   848
$\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   849
predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   850
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   851
Let's check a property involving $\textit{even}'$. To make up for the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   852
foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   853
\textit{nat}'s cardinality to a mere 10:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   854
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   855
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   856
\textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   857
\lnot\;\textit{even}'~n$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   858
\textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   859
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   860
The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   861
Nitpick might need to unroll it. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   862
Trying 6 scopes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   863
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   864
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   865
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   866
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   867
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   868
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   869
Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   870
\hbox{}\qquad Constant: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   871
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   872
& 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   873
& 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   874
& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   875
Total time: 1140 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   876
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   877
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   878
Nitpick's output is very instructive. First, it tells us that the predicate is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   879
unrolled, meaning that it is computed iteratively from the empty set. Then it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   880
lists six scopes specifying different bounds on the numbers of iterations:\ 0,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   881
1, 2, 4, 8, and~9.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   882
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   883
The output also shows how each iteration contributes to $\textit{even}'$. The
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   884
notation $\lambda i.\; \textit{even}'$ indicates that the value of the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   885
predicate depends on an iteration counter. Iteration 0 provides the basis
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   886
elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   887
throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   888
iterations would not contribute any new elements.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   889
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   890
Some values are marked with superscripted question
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   891
marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   892
predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   893
\textit{True} or $\unk$, never \textit{False}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   894
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   895
When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   896
iterations. However, these numbers are bounded by the cardinality of the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   897
predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   898
ever needed to compute the value of a \textit{nat} predicate. You can specify
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   899
the number of iterations using the \textit{iter} option, as explained in
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   900
\S\ref{scope-of-search}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   901
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   902
In the next formula, $\textit{even}'$ occurs both positively and negatively:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   903
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   904
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   905
\textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   906
\textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   907
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   908
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   909
\hbox{}\qquad\qquad $n = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   910
\hbox{}\qquad Constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   911
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   912
& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$  \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   913
\hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   914
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   915
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   916
Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   917
8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   918
fixed point (not necessarily the least one). It is used to falsify
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   919
$\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   920
$\textit{even}'~(n - 2)$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   921
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   922
Coinductive predicates are handled dually. For example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   923
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   924
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   925
\textbf{coinductive} \textit{nats} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   926
``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   927
\textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   928
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   929
\slshape Nitpick found a counterexample:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   930
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   931
\hbox{}\qquad Constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   932
\hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   933
& 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   934
& \unr\})\end{aligned}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   935
\hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   936
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   937
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   938
As a special case, Nitpick uses Kodkod's transitive closure operator to encode
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   939
negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   940
inductive predicates for which each the predicate occurs in at most one
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   941
assumption of each introduction rule. For example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   942
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   943
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   944
\textbf{inductive} \textit{odd} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   945
``$\textit{odd}~1$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   946
``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   947
\textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   948
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   949
\slshape Nitpick found a counterexample:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   950
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   951
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   952
\hbox{}\qquad\qquad $n = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   953
\hbox{}\qquad Constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   954
\hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   955
\hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   956
\hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   957
\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   958
  & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   959
  & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   960
       (3, 5), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   961
  & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   962
  & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   963
\hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   964
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   965
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   966
\noindent
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   967
In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   968
$\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   969
elements from known ones. The set $\textit{odd}$ consists of all the values
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   970
reachable through the reflexive transitive closure of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   971
$\textit{odd}_{\textrm{step}}$ starting with any element from
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   972
$\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   973
transitive closure to encode linear predicates is normally either more thorough
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   974
or more efficient than unrolling (depending on the value of \textit{iter}), but
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   975
for those cases where it isn't you can disable it by passing the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   976
\textit{dont\_star\_linear\_preds} option.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   977
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   978
\subsection{Coinductive Datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   979
\label{coinductive-datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   980
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   981
While Isabelle regrettably lacks a high-level mechanism for defining coinductive
35665
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
   982
datatypes, the \textit{Coinductive\_List} theory from Andreas Lochbihler's
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
   983
\textit{Coinductive} AFP entry \cite{lochbihler-2010} provides a coinductive
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
   984
``lazy list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
   985
supports these lazy lists seamlessly and provides a hook, described in
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   986
\S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   987
datatypes.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   988
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   989
(Co)intuitively, a coinductive datatype is similar to an inductive datatype but
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   990
allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   991
\ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   992
1, 2, 3, \ldots]$ can be defined as lazy lists using the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   993
$\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   994
$\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   995
\mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   996
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   997
Although it is otherwise no friend of infinity, Nitpick can find counterexamples
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   998
involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   999
finite lists:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1000
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1001
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1002
\textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1003
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1004
\slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1005
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1006
\hbox{}\qquad\qquad $\textit{a} = a_1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1007
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1008
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1009
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1010
The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1011
for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1012
infinite list $[a_1, a_1, a_1, \ldots]$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1013
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1014
The next example is more interesting:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1015
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1016
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1017
\textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1018
\textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1019
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1020
\slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1021
some scopes. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1022
Trying 8 scopes: \\
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
  1023
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 1,
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1024
and \textit{bisim\_depth}~= 0. \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1025
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
  1026
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 8,
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1027
and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1028
Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
  1029
\textit{card}~``\kern1pt$'a~\textit{list\/}$''~= 2, and \textit{bisim\_\allowbreak
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1030
depth}~= 1:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1031
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1032
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1033
\hbox{}\qquad\qquad $\textit{a} = a_1$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1034
\hbox{}\qquad\qquad $\textit{b} = a_2$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1035
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1036
\hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1037
Total time: 726 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1038
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1039
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1040
The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1041
$\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1042
$[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1043
within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1044
the segment leading to the binder is the stem.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1045
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1046
A salient property of coinductive datatypes is that two objects are considered
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1047
equal if and only if they lead to the same observations. For example, the lazy
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1048
lists $\textrm{THE}~\omega.\; \omega =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1049
\textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1050
$\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1051
\textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1052
to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1053
equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1054
concept of equality for coinductive datatypes is called bisimulation and is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1055
defined coinductively.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1056
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1057
Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1058
the Kodkod problem to ensure that distinct objects lead to different
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1059
observations. This precaution is somewhat expensive and often unnecessary, so it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1060
can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1061
bisimilarity check is then performed \textsl{after} the counterexample has been
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1062
found to ensure correctness. If this after-the-fact check fails, the
35695
80b2c22f8f00 fixed soundness bug in Nitpick
blanchet
parents: 35665
diff changeset
  1063
counterexample is tagged as ``quasi genuine'' and Nitpick recommends to try
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1064
again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1065
check for the previous example saves approximately 150~milli\-seconds; the speed
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1066
gains can be more significant for larger scopes.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1067
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1068
The next formula illustrates the need for bisimilarity (either as a Kodkod
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1069
predicate or as an after-the-fact check) to prevent spurious counterexamples:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1070
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1071
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1072
\textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1073
\,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
  1074
\textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
35695
80b2c22f8f00 fixed soundness bug in Nitpick
blanchet
parents: 35665
diff changeset
  1075
\slshape Nitpick found a quasi genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1076
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1077
\hbox{}\qquad\qquad $a = a_1$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1078
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1079
\textit{LCons}~a_1~\omega$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1080
\hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1081
\hbox{}\qquad Codatatype:\strut \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1082
\hbox{}\qquad\qquad $'a~\textit{llist} =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1083
\{\!\begin{aligned}[t]
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1084
  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1085
  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1086
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1087
Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1088
that the counterexample is genuine. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1089
{\upshape\textbf{nitpick}} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1090
\slshape Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1091
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1092
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1093
In the first \textbf{nitpick} invocation, the after-the-fact check discovered 
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1094
that the two known elements of type $'a~\textit{llist}$ are bisimilar.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1095
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1096
A compromise between leaving out the bisimilarity predicate from the Kodkod
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1097
problem and performing the after-the-fact check is to specify a lower
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1098
nonnegative \textit{bisim\_depth} value than the default one provided by
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1099
Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1100
be distinguished from each other by their prefixes of length $K$. Be aware that
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1101
setting $K$ to a too low value can overconstrain Nitpick, preventing it from
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1102
finding any counterexamples.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1103
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1104
\subsection{Boxing}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1105
\label{boxing}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1106
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1107
Nitpick normally maps function and product types directly to the corresponding
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1108
Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1109
cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1110
\Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1111
off to treat these types in the same way as plain datatypes, by approximating
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1112
them by a subset of a given cardinality. This technique is called ``boxing'' and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1113
is particularly useful for functions passed as arguments to other functions, for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1114
high-arity functions, and for large tuples. Under the hood, boxing involves
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1115
wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1116
isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1117
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1118
To illustrate boxing, we consider a formalization of $\lambda$-terms represented
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1119
using de Bruijn's notation:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1120
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1121
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1122
\textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1123
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1124
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1125
The $\textit{lift}~t~k$ function increments all variables with indices greater
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1126
than or equal to $k$ by one:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1127
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1128
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1129
\textbf{primrec} \textit{lift} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1130
``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1131
``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1132
``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1133
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1134
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1135
The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1136
term $t$ has a loose variable with index $k$ or more:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1137
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1138
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1139
\textbf{primrec}~\textit{loose} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1140
``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1141
``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1142
``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1143
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1144
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1145
Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1146
on $t$:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1147
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1148
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1149
\textbf{primrec}~\textit{subst} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1150
``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1151
``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1152
\phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1153
``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1154
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1155
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1156
A substitution is a function that maps variable indices to terms. Observe that
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1157
$\sigma$ is a function passed as argument and that Nitpick can't optimize it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1158
away, because the recursive call for the \textit{Lam} case involves an altered
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1159
version. Also notice the \textit{lift} call, which increments the variable
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1160
indices when moving under a \textit{Lam}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1161
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1162
A reasonable property to expect of substitution is that it should leave closed
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1163
terms unchanged. Alas, even this simple property does not hold:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1164
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1165
\pre
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1166
\textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1167
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1168
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1169
Trying 8 scopes: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1170
\hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1171
\hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1172
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1173
\hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1174
Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1175
and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1176
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1177
\hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1178
& 0 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1179
  1 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1180
  2 := \textit{Var}~0, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1181
& 3 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1182
  4 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1183
  5 := \textit{Var}~0)\end{aligned}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1184
\hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1185
Total time: $4679$ ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1186
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1187
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1188
Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1189
\textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1190
$\lambda$-term notation, $t$~is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1191
$\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
  1192
The bug is in \textit{subst\/}: The $\textit{lift}~(\sigma~m)~1$ call should be
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1193
replaced with $\textit{lift}~(\sigma~m)~0$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1194
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1195
An interesting aspect of Nitpick's verbose output is that it assigned inceasing
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1196
cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1197
For the formula of interest, knowing 6 values of that type was enough to find
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1198
the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1199
considered, a hopeless undertaking:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1200
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1201
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1202
\textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1203
{\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1204
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1205
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1206
{\looseness=-1
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1207
Boxing can be enabled or disabled globally or on a per-type basis using the
35665
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
  1208
\textit{box} option. Nitpick usually performs reasonable choices about which
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
  1209
types should be boxed, but option tweaking sometimes helps. A related optimization,
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
  1210
``finalization,'' attempts to wrap functions that constant at all but finitely
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
  1211
many points (e.g., finite sets); see the documentation for the \textit{finalize}
ff2bf50505ab added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
blanchet
parents: 35386
diff changeset
  1212
option in \S\ref{scope-of-search} for details.
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1213
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1214
}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1215
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1216
\subsection{Scope Monotonicity}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1217
\label{scope-monotonicity}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1218
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1219
The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1220
and \textit{max}) controls which scopes are actually tested. In general, to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1221
exhaust all models below a certain cardinality bound, the number of scopes that
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1222
Nitpick must consider increases exponentially with the number of type variables
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1223
(and \textbf{typedecl}'d types) occurring in the formula. Given the default
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1224
cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1225
considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1226
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1227
Fortunately, many formulas exhibit a property called \textsl{scope
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1228
monotonicity}, meaning that if the formula is falsifiable for a given scope,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1229
it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1230
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1231
Consider the formula
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1232
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1233
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1234
\textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1235
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1236
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1237
where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1238
$'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1239
exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1240
that any counterexample found with a small scope would still be a counterexample
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1241
in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1242
by the larger scope. Nitpick comes to the same conclusion after a careful
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1243
inspection of the formula and the relevant definitions:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1244
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1245
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1246
\textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1247
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1248
The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1249
Nitpick might be able to skip some scopes.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1250
 \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1251
Trying 8 scopes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1252
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1253
\textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1254
\textit{list\/}''~= 1, \\
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1255
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 1, and
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1256
\textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 1. \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1257
\hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1258
\textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1259
\textit{list\/}''~= 2, \\
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1260
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 2, and
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1261
\textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 2. \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1262
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1263
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1264
\textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1265
\textit{list\/}''~= 8, \\
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1266
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 8, and
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1267
\textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 8.
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1268
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1269
Nitpick found a counterexample for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1270
\textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1271
\textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
35712
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1272
\textit{list\/}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 5, and
77aa29bf14ee added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents: 35710
diff changeset
  1273
\textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 5:
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1274
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1275
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1276
\hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1277
\hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1278
Total time: 1636 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1279
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1280
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1281
In theory, it should be sufficient to test a single scope:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1282
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1283
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1284
\textbf{nitpick}~[\textit{card}~= 8]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1285
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1286
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1287
However, this is often less efficient in practice and may lead to overly complex
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1288
counterexamples.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1289
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1290
If the monotonicity check fails but we believe that the formula is monotonic (or
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1291
we don't mind missing some counterexamples), we can pass the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1292
\textit{mono} option. To convince yourself that this option is risky,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1293
simply consider this example from \S\ref{skolemization}:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1294
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1295
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1296
\textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1297
 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1298
\textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1299
{\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1300
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1301
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1302
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1303
\hbox{}\qquad $\vdots$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1304
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1305
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1306
(It turns out the formula holds if and only if $\textit{card}~'a \le
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1307
\textit{card}~'b$.) Although this is rarely advisable, the automatic
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1308
monotonicity checks can be disabled by passing \textit{non\_mono}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1309
(\S\ref{optimizations}).
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1310
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1311
As insinuated in \S\ref{natural-numbers-and-integers} and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1312
\S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1313
are normally monotonic and treated as such. The same is true for record types,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1314
\textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1315
cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1316
\textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1317
consider only 8~scopes instead of $32\,768$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1318
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1319
\subsection{Inductive Properties}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1320
\label{inductive-properties}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1321
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1322
Inductive properties are a particular pain to prove, because the failure to
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1323
establish an induction step can mean several things:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1324
%
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1325
\begin{enumerate}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1326
\item The property is invalid.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1327
\item The property is valid but is too weak to support the induction step.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1328
\item The property is valid and strong enough; it's just that we haven't found
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1329
the proof yet.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1330
\end{enumerate}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1331
%
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1332
Depending on which scenario applies, we would take the appropriate course of
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1333
action:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1334
%
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1335
\begin{enumerate}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1336
\item Repair the statement of the property so that it becomes valid.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1337
\item Generalize the property and/or prove auxiliary properties.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1338
\item Work harder on a proof.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1339
\end{enumerate}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1340
%
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1341
How can we distinguish between the three scenarios? Nitpick's normal mode of
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1342
operation can often detect scenario 1, and Isabelle's automatic tactics help with
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1343
scenario 3. Using appropriate techniques, it is also often possible to use
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1344
Nitpick to identify scenario 2. Consider the following transition system,
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1345
in which natural numbers represent states:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1346
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1347
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1348
\textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1349
``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1350
``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1351
``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1352
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1353
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1354
We will try to prove that only even numbers are reachable:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1355
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1356
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1357
\textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1358
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1359
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1360
Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1361
so let's attempt a proof by induction:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1362
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1363
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1364
\textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1365
\textbf{apply}~\textit{auto}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1366
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1367
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1368
This leaves us in the following proof state:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1369
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1370
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1371
{\slshape goal (2 subgoals): \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1372
\phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1373
\phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1374
}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1375
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1376
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1377
If we run Nitpick on the first subgoal, it still won't find any
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1378
counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1379
is helpless. However, notice the $n \in \textit{reach}$ assumption, which
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1380
strengthens the induction hypothesis but is not immediately usable in the proof.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1381
If we remove it and invoke Nitpick, this time we get a counterexample:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1382
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1383
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1384
\textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1385
\textbf{nitpick} \\[2\smallskipamount]
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1386
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1387
\hbox{}\qquad Skolem constant: \nopagebreak \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1388
\hbox{}\qquad\qquad $n = 0$
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1389
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1390
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1391
Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1392
to strength the lemma:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1393
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1394
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1395
\textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1396
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1397
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1398
Unfortunately, the proof by induction still gets stuck, except that Nitpick now
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1399
finds the counterexample $n = 2$. We generalize the lemma further to
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1400
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1401
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1402
\textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1403
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1404
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1405
and this time \textit{arith} can finish off the subgoals.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1406
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1407
A similar technique can be employed for structural induction. The
35180
c57dba973391 more work on Nitpick's support for nonstandard models + fix in model reconstruction
blanchet
parents: 35178
diff changeset
  1408
following mini formalization of full binary trees will serve as illustration:
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1409
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1410
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1411
\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]
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1412
\textbf{primrec}~\textit{labels}~\textbf{where} \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1413
``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1414
``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1415
\textbf{primrec}~\textit{swap}~\textbf{where} \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1416
``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1417
\phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1418
``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1419
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1420
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1421
The \textit{labels} function returns the set of labels occurring on leaves of a
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: