doc-src/Nitpick/nitpick.tex
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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[french,english]{babel}
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\usepackage{color}
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\usepackage{graphicx}
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%\usepackage{mathpazo}
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\usepackage{multicol}
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\usepackage{stmaryrd}
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%\usepackage[scaled=.85]{beramono}
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\usepackage{../iman,../pdfsetup}
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%\oddsidemargin=4.6mm
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%\evensidemargin=4.6mm
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%\textwidth=150mm
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%\topmargin=4.6mm
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%\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\undef{\textit{undefined}}
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\def\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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\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 2010}
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\author{\hbox{} \\
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Jasmin Christian Blanchette \\
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{\normalsize Fakult\"at f\"ur Informatik, Technische Universit\"at M\"unchen} \\
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\hbox{}}
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\maketitle
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\tableofcontents
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\setlength{\parskip}{.7em plus .2em minus .1em}
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\setlength{\parindent}{0pt}
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\setlength{\abovedisplayskip}{\parskip}
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\setlength{\abovedisplayshortskip}{.9\parskip}
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\setlength{\belowdisplayskip}{\parskip}
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\setlength{\belowdisplayshortskip}{.9\parskip}
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% General-purpose enum environment with correct spacing
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\newenvironment{enum}%
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    {\begin{list}{}{%
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        \setlength{\topsep}{.1\parskip}%
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        \setlength{\partopsep}{.1\parskip}%
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        \setlength{\itemsep}{\parskip}%
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        \advance\itemsep by-\parsep}}
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    {\end{list}}
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\def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin
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\advance\rightskip by\leftmargin}
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\def\post{\vskip0pt plus1ex\endgroup}
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\def\prew{\pre\advance\rightskip by-\leftmargin}
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\def\postw{\post}
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\section{Introduction}
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\label{introduction}
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Nitpick \cite{blanchette-nipkow-2009} is a counterexample generator for
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Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
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combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
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quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
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first-order relational model finder developed by the Software Design Group at
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MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
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borrows many ideas and code fragments, but it benefits from Kodkod's
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optimizations and a new encoding scheme. The name Nitpick is shamelessly
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appropriated from a now retired Alloy precursor.
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Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
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theorem and wait a few seconds. Nonetheless, there are situations where knowing
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how it works under the hood and how it reacts to various options helps
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increase the test coverage. This manual also explains how to install the tool on
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your workstation. Should the motivation fail you, think of the many hours of
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hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.
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Another common use of Nitpick is to find out whether the axioms of a locale are
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satisfiable, while the locale is being developed. To check this, it suffices to
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write
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\prew
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\textbf{lemma}~``$\textit{False}$'' \\
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\textbf{nitpick}~[\textit{show\_all}]
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\postw
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after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
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must find a model for the axioms. If it finds no model, we have an indication
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that the axioms might be unsatisfiable.
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\newbox\boxA
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\setbox\boxA=\hbox{\texttt{nospam}}
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The known bugs and limitations at the time of writing are listed in
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\S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
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or this manual should be directed to
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\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
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in.\allowbreak tum.\allowbreak de}.
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\vskip2.5\smallskipamount
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\textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
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suggesting several textual improvements.
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% and Perry James for reporting a typo.
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\section{First Steps}
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\label{first-steps}
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This section introduces Nitpick by presenting small examples. If possible, you
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should try out the examples on your workstation. Your theory file should start
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the standard way:
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\prew
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\textbf{theory}~\textit{Scratch} \\
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\textbf{imports}~\textit{Main} \\
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\textbf{begin}
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\postw
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The results presented here were obtained using the JNI version of MiniSat and
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with multithreading disabled to reduce nondeterminism. This was done by adding
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the line
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\prew
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\textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSatJNI}, \,\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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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 by specifying
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\prew
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\textbf{nitpick\_params} [\textit{auto}]
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\postw
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at the beginning of the theory file. In this mode, Nitpick is run for up to 5
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seconds (by default) on every newly entered theorem, much like Auto Quickcheck.
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\subsection{Propositional Logic}
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\label{propositional-logic}
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Let's start with a trivial example from propositional logic:
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\prew
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\textbf{lemma}~``$P \longleftrightarrow Q$'' \\
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\textbf{nitpick}
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\postw
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You should get the following output:
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\prew
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\slshape
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Nitpick found a counterexample: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \textit{True}$ \\
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\hbox{}\qquad\qquad $Q = \textit{False}$
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\postw
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Nitpick can also be invoked on individual subgoals, as in the example below:
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\prew
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\textbf{apply}~\textit{auto} \\[2\smallskipamount]
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{\slshape goal (2 subgoals): \\
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\ 1. $P\,\Longrightarrow\, Q$ \\
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\ 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 $\mathit{The} = \undef{}
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    (\!\begin{aligned}[t]%
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    & \{\} := a_3,\> \{a_3\} := a_3,\> \{a_2\} := a_2, \\[-2pt] %% TYPESETTING
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    & \{a_2, a_3\} := a_1,\> \{a_1\} := a_1,\> \{a_1, a_3\} := a_3, \\[-2pt]
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    & \{a_1, a_2\} := a_3,\> \{a_1, a_2, a_3\} := a_3)\end{aligned}$
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\postw
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Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
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just like before.\footnote{The \undef{} symbol's presence is explained as
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follows: In higher-order logic, any function can be built from the undefined
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function using repeated applications of the function update operator $f(x :=
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y)$, just like any list can be built from the empty list using $x \mathbin{\#}
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xs$.}
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Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
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unique $x$ such that'') at the front of our putative lemma's assumption:
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\prew
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\textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
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\postw
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The fix appears to work:
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\prew
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\textbf{nitpick} \\[2\smallskipamount]
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\slshape Nitpick found no counterexample.
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\postw
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We can further increase our confidence in the formula by exhausting all
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cardinalities up to 50:
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\prew
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\textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
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can be entered as \texttt{-} (hyphen) or
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\texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
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\slshape Nitpick found no counterexample.
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\postw
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Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
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\prew
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\textbf{sledgehammer} \\[2\smallskipamount]
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blanchet
parents:
diff changeset
   351
{\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   352
$\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   353
Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   354
\textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   355
{\slshape No subgoals!}% \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   356
%\textbf{done}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   357
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   358
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   359
This must be our lucky day.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   360
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   361
\subsection{Skolemization}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   362
\label{skolemization}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   363
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   364
Are all invertible functions onto? Let's find out:
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{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   368
 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   369
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   370
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   371
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   372
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   373
\hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   374
\hbox{}\qquad Skolem constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   375
\hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   376
\hbox{}\qquad\qquad $y = a_2$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   377
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   378
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   379
Although $f$ is the only free variable occurring in the formula, Nitpick also
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   380
displays values for the bound variables $g$ and $y$. These values are available
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   381
to Nitpick because it performs skolemization as a preprocessing step.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   382
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   383
In the previous example, skolemization only affected the outermost quantifiers.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   384
This is not always the case, as illustrated below:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   385
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   386
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   387
\textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   388
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   389
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   390
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   391
\hbox{}\qquad Skolem constant: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   392
\hbox{}\qquad\qquad $\lambda x.\; f =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   393
    \undef{}(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   394
    & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   395
    & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   396
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   397
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   398
The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   399
$x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   400
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
   401
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
   402
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   403
The source of the Skolem constants is sometimes more obscure:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   404
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   405
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   406
\textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   407
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   408
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   409
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   410
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   411
\hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   412
\hbox{}\qquad Skolem constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   413
\hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   414
\hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   415
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   416
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   417
What happened here is that Nitpick expanded the \textit{sym} constant to its
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   418
definition:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   419
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   420
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   421
$\mathit{sym}~r \,\equiv\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   422
 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   423
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   424
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   425
As their names suggest, the Skolem constants $\mathit{sym}.x$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   426
$\mathit{sym}.y$ are simply the bound variables $x$ and $y$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   427
from \textit{sym}'s definition.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   428
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   429
Although skolemization is a useful optimization, you can disable it by invoking
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   430
Nitpick with \textit{dont\_skolemize}. See \S\ref{optimizations} for details.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   431
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   432
\subsection{Natural Numbers and Integers}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   433
\label{natural-numbers-and-integers}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   434
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   435
Because of the axiom of infinity, the type \textit{nat} does not admit any
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   436
finite models. To deal with this, Nitpick considers prefixes $\{0,\, 1,\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   437
\ldots,\, K - 1\}$ of \textit{nat} (where $K = \textit{card}~\textit{nat}$) and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   438
maps all other numbers to the undefined value ($\unk$). The type \textit{int} is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   439
handled in a similar way: If $K = \textit{card}~\textit{int}$, the subset of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   440
\textit{int} known to Nitpick is $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   441
K/2 \rfloor\}$. Undefined values lead to a three-valued logic.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   442
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   443
Here is an example involving \textit{int}:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   444
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   445
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   446
\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
   447
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   448
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   449
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   450
\hbox{}\qquad\qquad $i = 0$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   451
\hbox{}\qquad\qquad $j = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   452
\hbox{}\qquad\qquad $m = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   453
\hbox{}\qquad\qquad $n = 0$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   454
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   455
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   456
With infinite types, we don't always have the luxury of a genuine counterexample
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   457
and must often content ourselves with a potential one. The tedious task of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   458
finding out whether the potential counterexample is in fact genuine can be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   459
outsourced to \textit{auto} by passing the option \textit{check\_potential}. For
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   460
example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   461
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   462
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   463
\textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   464
\textbf{nitpick} [\textit{card~nat}~= 100,\, \textit{check\_potential}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   465
\slshape Nitpick found a potential counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   466
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   467
\hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   468
Confirmation by ``\textit{auto}'': The above counterexample is genuine.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   469
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   470
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   471
You might wonder why the counterexample is first reported as potential. The root
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   472
of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   473
\mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   474
that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   475
\textit{False}; but otherwise, it does not know anything about values of $n \ge
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   476
\textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   477
\textit{True}. Since the assumption can never be satisfied, the putative lemma
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   478
can never be falsified.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   479
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   480
Incidentally, if you distrust the so-called genuine counterexamples, you can
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   481
enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   482
aware that \textit{auto} will often fail to prove that the counterexample is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   483
genuine or spurious.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   484
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   485
Some conjectures involving elementary number theory make Nitpick look like a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   486
giant with feet of clay:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   487
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   488
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   489
\textbf{lemma} ``$P~\textit{Suc}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   490
\textbf{nitpick} [\textit{card} = 1--6] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   491
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   492
Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   493
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   494
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   495
For any cardinality $k$, \textit{Suc} is the partial function $\{0 \mapsto 1,\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   496
1 \mapsto 2,\, \ldots,\, k - 1 \mapsto \unk\}$, which evaluates to $\unk$ when
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   497
it is passed as argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   498
The next example is similar:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   499
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   500
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   501
\textbf{lemma} ``$P~(\textit{op}~{+}\Colon
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   502
\textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   503
\textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   504
{\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   505
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   506
\hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   507
\textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   508
{\slshape Nitpick found no counterexample.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   509
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   510
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   511
The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   512
$\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   513
1\}$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   514
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   515
Because numbers are infinite and are approximated using a three-valued logic,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   516
there is usually no need to systematically enumerate domain sizes. If Nitpick
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   517
cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   518
unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   519
example above is an exception to this principle.) Nitpick nonetheless enumerates
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   520
all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   521
cardinalities are fast to handle and give rise to simpler counterexamples. This
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   522
is explained in more detail in \S\ref{scope-monotonicity}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   523
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   524
\subsection{Inductive Datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   525
\label{inductive-datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   526
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   527
Like natural numbers and integers, inductive datatypes with recursive
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   528
constructors admit no finite models and must be approximated by a subterm-closed
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   529
subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   530
Nitpick looks for all counterexamples that can be built using at most 10
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   531
different lists.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   532
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   533
Let's see with an example involving \textit{hd} (which returns the first element
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   534
of a list) and $@$ (which concatenates two lists):
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   535
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   536
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   537
\textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   538
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   539
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   540
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   541
\hbox{}\qquad\qquad $\textit{xs} = []$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   542
\hbox{}\qquad\qquad $\textit{y} = a_3$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   543
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   544
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   545
To see why the counterexample is genuine, we enable \textit{show\_consts}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   546
and \textit{show\_\allowbreak datatypes}:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   547
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   548
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   549
{\slshape Datatype:} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   550
\hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_3, a_3],\, [a_3],\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   551
{\slshape Constants:} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   552
\hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_3, a_3],\> [a_3, a_3] := \unk,\> [a_3] := \unk)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   553
\hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_3, a_3] := a_3,\> [a_3] := a_3)$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   554
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   555
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   556
Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   557
including $a_2$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   558
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   559
The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   560
append operator whose second argument is fixed to be $[y, y]$. Appending $[a_3,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   561
a_3]$ to $[a_3]$ would normally give $[a_3, a_3, a_3]$, but this value is not
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   562
representable in the subset of $'a$~\textit{list} considered by Nitpick, which
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   563
is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   564
appending $[a_3, a_3]$ to itself gives $\unk$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   565
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   566
Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   567
considers the following subsets:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   568
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   569
\kern-.5\smallskipamount %% TYPESETTING
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
\begin{multicols}{3}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   573
$\{[],\, [a_1],\, [a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   574
$\{[],\, [a_1],\, [a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   575
$\{[],\, [a_2],\, [a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   576
$\{[],\, [a_1],\, [a_1, a_1]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   577
$\{[],\, [a_1],\, [a_2, a_1]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   578
$\{[],\, [a_1],\, [a_3, a_1]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   579
$\{[],\, [a_2],\, [a_1, a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   580
$\{[],\, [a_2],\, [a_2, a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   581
$\{[],\, [a_2],\, [a_3, a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   582
$\{[],\, [a_3],\, [a_1, a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   583
$\{[],\, [a_3],\, [a_2, a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   584
$\{[],\, [a_3],\, [a_3, a_3]\}$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   585
\end{multicols}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   586
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   587
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   588
\kern-2\smallskipamount %% TYPESETTING
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   589
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   590
All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   591
are listed and only those. As an example of a non-subterm-closed subset,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   592
consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_3]\}$, and observe
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   593
that $[a_1, a_3]$ (i.e., $a_1 \mathbin{\#} [a_3]$) has $[a_3] \notin
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   594
\mathcal{S}$ as a subterm.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   595
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   596
Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   597
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   598
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   599
\textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   600
\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   601
\\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   602
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   603
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   604
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   605
\hbox{}\qquad\qquad $\textit{xs} = [a_2]$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   606
\hbox{}\qquad\qquad $\textit{ys} = [a_3]$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   607
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   608
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   609
\hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_3],\, [a_2],\, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   610
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   611
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   612
Because datatypes are approximated using a three-valued logic, there is usually
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   613
no need to systematically enumerate cardinalities: If Nitpick cannot find a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   614
genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   615
unlikely that one could be found for smaller cardinalities.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   616
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   617
\subsection{Typedefs, Records, Rationals, and Reals}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   618
\label{typedefs-records-rationals-and-reals}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   619
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   620
Nitpick generally treats types declared using \textbf{typedef} as datatypes
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   621
whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   622
For example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   623
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   624
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   625
\textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   626
\textbf{by}~\textit{blast} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   627
\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
   628
\textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   629
\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
   630
\textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   631
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   632
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   633
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   634
\hbox{}\qquad\qquad $P = \{\Abs{1},\, \Abs{0}\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   635
\hbox{}\qquad\qquad $x = \Abs{2}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   636
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   637
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   638
\hbox{}\qquad\qquad $\textit{three} = \{\Abs{2},\, \Abs{1},\, \Abs{0},\, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   639
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   640
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   641
%% MARK
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   642
In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   643
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   644
%% MARK
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   645
Records, which are implemented as \textbf{typedef}s behind the scenes, are
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   646
handled in much the same way:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   647
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   648
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   649
\textbf{record} \textit{point} = \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   650
\hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   651
\hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   652
\textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   653
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   654
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   655
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   656
\hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   657
\hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   658
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   659
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   660
\hbox{}\qquad\qquad $\textit{point} = \{\lparr\textit{Xcoord} = 1,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   661
\textit{Ycoord} = 1\rparr,\> \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr,\, \unr\}$\kern-1pt %% QUIET
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   662
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   663
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   664
Finally, Nitpick provides rudimentary support for rationals and reals using a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   665
similar approach:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   666
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   667
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   668
\textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   669
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   670
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   671
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   672
\hbox{}\qquad\qquad $x = 1/2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   673
\hbox{}\qquad\qquad $y = -1/2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   674
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   675
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   676
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   677
\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
   678
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   679
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   680
\subsection{Inductive and Coinductive Predicates}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   681
\label{inductive-and-coinductive-predicates}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   682
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   683
Inductively defined predicates (and sets) are particularly problematic for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   684
counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   685
loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   686
the problem is that they are defined using a least fixed point construction.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   687
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   688
Nitpick's philosophy is that not all inductive predicates are equal. Consider
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   689
the \textit{even} predicate below:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   690
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   691
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   692
\textbf{inductive}~\textit{even}~\textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   693
``\textit{even}~0'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   694
``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   695
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   696
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   697
This predicate enjoys the desirable property of being well-founded, which means
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   698
that the introduction rules don't give rise to infinite chains of the form
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   699
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   700
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   701
$\cdots\,\Longrightarrow\, \textit{even}~k''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   702
       \,\Longrightarrow\, \textit{even}~k'
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   703
       \,\Longrightarrow\, \textit{even}~k.$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   704
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   705
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   706
For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   707
$k/2 + 1$:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   708
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   709
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   710
$\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   711
       \,\Longrightarrow\, \textit{even}~(k - 2)
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   712
       \,\Longrightarrow\, \textit{even}~k.$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   713
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   714
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   715
Wellfoundedness is desirable because it enables Nitpick to use a very efficient
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   716
fixed point computation.%
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   717
\footnote{If an inductive predicate is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   718
well-founded, then it has exactly one fixed point, which is simultaneously the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   719
least and the greatest fixed point. In these circumstances, the computation of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   720
the least fixed point amounts to the computation of an arbitrary fixed point,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   721
which can be performed using a straightforward recursive equation.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   722
Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   723
just as Isabelle's \textbf{function} package usually discharges termination
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   724
proof obligations automatically.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   725
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   726
Let's try an example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   727
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   728
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   729
\textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   730
\textbf{nitpick}~[\textit{card nat}~= 100,\, \textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   731
\slshape The inductive predicate ``\textit{even}'' was proved well-founded.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   732
Nitpick can compute it efficiently. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   733
Trying 1 scope: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   734
\hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   735
Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   736
\hbox{}\qquad Empty assignment \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   737
Nitpick could not find a better counterexample. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   738
Total time: 2274 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   739
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   740
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   741
No genuine counterexample is possible because Nitpick cannot rule out the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   742
existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   743
$\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   744
existential quantifier:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   745
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   746
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   747
\textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   748
\textbf{nitpick}~[\textit{card nat}~= 100] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   749
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   750
\hbox{}\qquad Empty assignment
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   751
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   752
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   753
So far we were blessed by the wellfoundedness of \textit{even}. What happens if
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   754
we use the following definition instead?
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   755
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   756
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   757
\textbf{inductive} $\textit{even}'$ \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   758
``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   759
``$\textit{even}'~2$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   760
``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   761
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   762
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   763
This definition is not well-founded: From $\textit{even}'~0$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   764
$\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   765
predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   766
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   767
Let's check a property involving $\textit{even}'$. To make up for the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   768
foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   769
\textit{nat}'s cardinality to a mere 10:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   770
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   771
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   772
\textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   773
\lnot\;\textit{even}'~n$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   774
\textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   775
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   776
The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   777
Nitpick might need to unroll it. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   778
Trying 6 scopes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   779
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   780
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   781
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   782
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   783
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   784
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   785
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
   786
\hbox{}\qquad Constant: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   787
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   788
& 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   789
& 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   790
& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   791
Total time: 1140 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   792
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   793
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   794
Nitpick's output is very instructive. First, it tells us that the predicate is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   795
unrolled, meaning that it is computed iteratively from the empty set. Then it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   796
lists six scopes specifying different bounds on the numbers of iterations:\ 0,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   797
1, 2, 4, 8, and~9.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   798
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   799
The output also shows how each iteration contributes to $\textit{even}'$. The
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   800
notation $\lambda i.\; \textit{even}'$ indicates that the value of the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   801
predicate depends on an iteration counter. Iteration 0 provides the basis
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   802
elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   803
throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   804
iterations would not contribute any new elements.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   805
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   806
Some values are marked with superscripted question
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   807
marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   808
predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   809
\textit{True} or $\unk$, never \textit{False}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   810
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   811
When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   812
iterations. However, these numbers are bounded by the cardinality of the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   813
predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   814
ever needed to compute the value of a \textit{nat} predicate. You can specify
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   815
the number of iterations using the \textit{iter} option, as explained in
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   816
\S\ref{scope-of-search}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   817
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   818
In the next formula, $\textit{even}'$ occurs both positively and negatively:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   819
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   820
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   821
\textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   822
\textbf{nitpick} [\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   823
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   824
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   825
\hbox{}\qquad\qquad $n = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   826
\hbox{}\qquad Constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   827
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   828
& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$  \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   829
\hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   830
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   831
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   832
Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   833
8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   834
fixed point (not necessarily the least one). It is used to falsify
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   835
$\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   836
$\textit{even}'~(n - 2)$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   837
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   838
Coinductive predicates are handled dually. For example:
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{coinductive} \textit{nats} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   842
``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   843
\textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   844
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   845
\slshape Nitpick found a counterexample:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   846
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   847
\hbox{}\qquad Constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   848
\hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   849
& 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   850
& \unr\})\end{aligned}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   851
\hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   852
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   853
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   854
As a special case, Nitpick uses Kodkod's transitive closure operator to encode
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   855
negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   856
inductive predicates for which each the predicate occurs in at most one
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   857
assumption of each introduction rule. For example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   858
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   859
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   860
\textbf{inductive} \textit{odd} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   861
``$\textit{odd}~1$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   862
``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   863
\textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   864
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   865
\slshape Nitpick found a counterexample:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   866
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   867
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   868
\hbox{}\qquad\qquad $n = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   869
\hbox{}\qquad Constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   870
\hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   871
\hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   872
\hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   873
\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   874
  & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   875
  & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   876
       (3, 5), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   877
  & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   878
  & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   879
\hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   880
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   881
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   882
\noindent
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   883
In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   884
$\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   885
elements from known ones. The set $\textit{odd}$ consists of all the values
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   886
reachable through the reflexive transitive closure of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   887
$\textit{odd}_{\textrm{step}}$ starting with any element from
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   888
$\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   889
transitive closure to encode linear predicates is normally either more thorough
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   890
or more efficient than unrolling (depending on the value of \textit{iter}), but
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   891
for those cases where it isn't you can disable it by passing the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   892
\textit{dont\_star\_linear\_preds} option.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   893
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   894
\subsection{Coinductive Datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   895
\label{coinductive-datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   896
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   897
While Isabelle regrettably lacks a high-level mechanism for defining coinductive
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   898
datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   899
list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   900
these lazy lists seamlessly and provides a hook, described in
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   901
\S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   902
datatypes.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   903
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   904
(Co)intuitively, a coinductive datatype is similar to an inductive datatype but
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   905
allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   906
\ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   907
1, 2, 3, \ldots]$ can be defined as lazy lists using the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   908
$\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   909
$\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   910
\mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   911
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   912
Although it is otherwise no friend of infinity, Nitpick can find counterexamples
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   913
involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   914
finite lists:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   915
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   916
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   917
\textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   918
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   919
\slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   920
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   921
\hbox{}\qquad\qquad $\textit{a} = a_1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   922
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   923
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   924
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   925
The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   926
for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   927
infinite list $[a_1, a_1, a_1, \ldots]$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   928
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   929
The next example is more interesting:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   930
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   931
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   932
\textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   933
\textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   934
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   935
\slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   936
some scopes. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   937
Trying 8 scopes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   938
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 1,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   939
and \textit{bisim\_depth}~= 0. \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   940
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   941
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 8,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   942
and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   943
Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   944
\textit{card}~``\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   945
depth}~= 1:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   946
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   947
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   948
\hbox{}\qquad\qquad $\textit{a} = a_2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   949
\hbox{}\qquad\qquad $\textit{b} = a_1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   950
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   951
\hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_1~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega)$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   952
Total time: 726 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   953
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   954
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   955
The lazy list $\textit{xs}$ is simply $[a_2, a_2, a_2, \ldots]$, whereas
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   956
$\textit{ys}$ is $[a_1, a_2, a_2, a_2, \ldots]$, i.e., a lasso-shaped list with
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   957
$[a_1]$ as its stem and $[a_2]$ as its cycle. In general, the list segment
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   958
within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   959
the segment leading to the binder is the stem.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   960
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   961
A salient property of coinductive datatypes is that two objects are considered
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   962
equal if and only if they lead to the same observations. For example, the lazy
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   963
lists $\textrm{THE}~\omega.\; \omega =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   964
\textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   965
$\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   966
\textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   967
to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   968
equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   969
concept of equality for coinductive datatypes is called bisimulation and is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   970
defined coinductively.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   971
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   972
Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   973
the Kodkod problem to ensure that distinct objects lead to different
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   974
observations. This precaution is somewhat expensive and often unnecessary, so it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   975
can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   976
bisimilarity check is then performed \textsl{after} the counterexample has been
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   977
found to ensure correctness. If this after-the-fact check fails, the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   978
counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   979
again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   980
check for the previous example saves approximately 150~milli\-seconds; the speed
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   981
gains can be more significant for larger scopes.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   982
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   983
The next formula illustrates the need for bisimilarity (either as a Kodkod
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   984
predicate or as an after-the-fact check) to prevent spurious counterexamples:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   985
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   986
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   987
\textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   988
\,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   989
\textbf{nitpick} [\textit{bisim\_depth} = $-1$,\, \textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   990
\slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   991
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   992
\hbox{}\qquad\qquad $a = a_2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   993
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   994
\textit{LCons}~a_2~\omega$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   995
\hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   996
\hbox{}\qquad Codatatype:\strut \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   997
\hbox{}\qquad\qquad $'a~\textit{llist} =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   998
\{\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   999
  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1000
  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega,\> \unr\}\end{aligned}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1001
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1002
Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1003
that the counterexample is genuine. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1004
{\upshape\textbf{nitpick}} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1005
\slshape Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1006
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1007
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1008
In the first \textbf{nitpick} invocation, the after-the-fact check discovered 
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1009
that the two known elements of type $'a~\textit{llist}$ are bisimilar.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1010
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1011
A compromise between leaving out the bisimilarity predicate from the Kodkod
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1012
problem and performing the after-the-fact check is to specify a lower
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1013
nonnegative \textit{bisim\_depth} value than the default one provided by
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1014
Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1015
be distinguished from each other by their prefixes of length $K$. Be aware that
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1016
setting $K$ to a too low value can overconstrain Nitpick, preventing it from
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1017
finding any counterexamples.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1018
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1019
\subsection{Boxing}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1020
\label{boxing}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1021
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1022
Nitpick normally maps function and product types directly to the corresponding
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1023
Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1024
cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1025
\Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1026
off to treat these types in the same way as plain datatypes, by approximating
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1027
them by a subset of a given cardinality. This technique is called ``boxing'' and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1028
is particularly useful for functions passed as arguments to other functions, for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1029
high-arity functions, and for large tuples. Under the hood, boxing involves
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1030
wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1031
isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1032
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1033
To illustrate boxing, we consider a formalization of $\lambda$-terms represented
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1034
using de Bruijn's notation:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1035
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1036
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1037
\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
  1038
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1039
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1040
The $\textit{lift}~t~k$ function increments all variables with indices greater
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1041
than or equal to $k$ by one:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1042
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1043
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1044
\textbf{primrec} \textit{lift} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1045
``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1046
``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1047
``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1048
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1049
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1050
The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1051
term $t$ has a loose variable with index $k$ or more:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1052
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1053
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1054
\textbf{primrec}~\textit{loose} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1055
``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1056
``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1057
``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1058
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1059
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1060
Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1061
on $t$:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1062
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1063
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1064
\textbf{primrec}~\textit{subst} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1065
``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1066
``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1067
\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
  1068
``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1069
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1070
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1071
A substitution is a function that maps variable indices to terms. Observe that
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1072
$\sigma$ is a function passed as argument and that Nitpick can't optimize it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1073
away, because the recursive call for the \textit{Lam} case involves an altered
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1074
version. Also notice the \textit{lift} call, which increments the variable
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1075
indices when moving under a \textit{Lam}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1076
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1077
A reasonable property to expect of substitution is that it should leave closed
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1078
terms unchanged. Alas, even this simple property does not hold:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1079
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1080
\pre
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1081
\textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1082
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1083
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1084
Trying 8 scopes: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1085
\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
  1086
\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
  1087
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1088
\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
  1089
Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1090
and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1091
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1092
\hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1093
& 0 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1094
  1 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1095
  2 := \textit{Var}~0, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1096
& 3 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1097
  4 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1098
  5 := \textit{Var}~0)\end{aligned}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1099
\hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1100
Total time: $4679$ ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1101
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1102
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1103
Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1104
\textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1105
$\lambda$-term notation, $t$~is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1106
$\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1107
The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1108
replaced with $\textit{lift}~(\sigma~m)~0$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1109
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1110
An interesting aspect of Nitpick's verbose output is that it assigned inceasing
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1111
cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1112
For the formula of interest, knowing 6 values of that type was enough to find
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1113
the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1114
considered, a hopeless undertaking:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1115
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1116
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1117
\textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1118
{\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1119
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1120
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1121
{\looseness=-1
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1122
Boxing can be enabled or disabled globally or on a per-type basis using the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1123
\textit{box} option. Moreover, setting the cardinality of a function or
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1124
product type implicitly enables boxing for that type. Nitpick usually performs
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1125
reasonable choices about which types should be boxed, but option tweaking
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1126
sometimes helps.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1127
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1128
}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1129
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1130
\subsection{Scope Monotonicity}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1131
\label{scope-monotonicity}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1132
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1133
The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1134
and \textit{max}) controls which scopes are actually tested. In general, to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1135
exhaust all models below a certain cardinality bound, the number of scopes that
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1136
Nitpick must consider increases exponentially with the number of type variables
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1137
(and \textbf{typedecl}'d types) occurring in the formula. Given the default
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1138
cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1139
considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1140
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1141
Fortunately, many formulas exhibit a property called \textsl{scope
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1142
monotonicity}, meaning that if the formula is falsifiable for a given scope,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1143
it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1144
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1145
Consider the formula
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1146
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1147
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1148
\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
  1149
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1150
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1151
where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1152
$'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1153
exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1154
that any counterexample found with a small scope would still be a counterexample
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1155
in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1156
by the larger scope. Nitpick comes to the same conclusion after a careful
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1157
inspection of the formula and the relevant definitions:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1158
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1159
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1160
\textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1161
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1162
The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1163
Nitpick might be able to skip some scopes.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1164
 \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1165
Trying 8 scopes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1166
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1167
\textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1168
\textit{list}''~= 1, \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1169
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1170
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1171
\hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1172
\textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1173
\textit{list}''~= 2, \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1174
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1175
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1176
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1177
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1178
\textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1179
\textit{list}''~= 8, \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1180
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1181
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1182
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1183
Nitpick found a counterexample for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1184
\textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1185
\textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1186
\textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1187
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1188
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1189
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1190
\hbox{}\qquad\qquad $\textit{xs} = [a_4, a_5]$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1191
\hbox{}\qquad\qquad $\textit{ys} = [b_3, b_3]$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1192
Total time: 1636 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1193
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1194
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1195
In theory, it should be sufficient to test a single scope:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1196
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1197
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1198
\textbf{nitpick}~[\textit{card}~= 8]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1199
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1200
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1201
However, this is often less efficient in practice and may lead to overly complex
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1202
counterexamples.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1203
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1204
If the monotonicity check fails but we believe that the formula is monotonic (or
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1205
we don't mind missing some counterexamples), we can pass the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1206
\textit{mono} option. To convince yourself that this option is risky,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1207
simply consider this example from \S\ref{skolemization}:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1208
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1209
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1210
\textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1211
 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1212
\textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1213
{\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1214
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1215
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1216
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1217
\hbox{}\qquad $\vdots$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1218
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1219
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1220
(It turns out the formula holds if and only if $\textit{card}~'a \le
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1221
\textit{card}~'b$.) Although this is rarely advisable, the automatic
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1222
monotonicity checks can be disabled by passing \textit{non\_mono}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1223
(\S\ref{optimizations}).
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1224
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1225
As insinuated in \S\ref{natural-numbers-and-integers} and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1226
\S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1227
are normally monotonic and treated as such. The same is true for record types,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1228
\textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1229
cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1230
\textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1231
consider only 8~scopes instead of $32\,768$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1232
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1233
\section{Case Studies}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1234
\label{case-studies}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1235
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1236
As a didactic device, the previous section focused mostly on toy formulas whose
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1237
validity can easily be assessed just by looking at the formula. We will now
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1238
review two somewhat more realistic case studies that are within Nitpick's
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1239
reach:\ a context-free grammar modeled by mutually inductive sets and a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1240
functional implementation of AA trees. The results presented in this
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1241
section were produced with the following settings:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1242
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1243
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1244
\textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1245
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1246
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1247
\subsection{A Context-Free Grammar}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1248
\label{a-context-free-grammar}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1249
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1250
Our first case study is taken from section 7.4 in the Isabelle tutorial
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1251
\cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1252
Ullman, produces all strings with an equal number of $a$'s and $b$'s:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1253
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1254
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1255
\begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1256
$S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1257
$A$ & $::=$ & $aS \mid bAA$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1258
$B$ & $::=$ & $bS \mid aBB$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1259
\end{tabular}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1260
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1261
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1262
The intuition behind the grammar is that $A$ generates all string with one more
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1263
$a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1264
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1265
The alphabet consists exclusively of $a$'s and $b$'s:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1266
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1267
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1268
\textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1269
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1270
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1271
Strings over the alphabet are represented by \textit{alphabet list}s.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1272
Nonterminals in the grammar become sets of strings. The production rules
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1273
presented above can be expressed as a mutually inductive definition:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1274
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1275
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1276
\textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1277
\textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1278
\textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1279
\textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1280
\textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1281
\textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1282
\textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1283
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1284
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1285
The conversion of the grammar into the inductive definition was done manually by
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1286
Joe Blow, an underpaid undergraduate student. As a result, some errors might
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1287
have sneaked in.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1288
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1289
Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1290
d'\^etre}. A good approach is to state desirable properties of the specification
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1291
(here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1292
as $b$'s) and check them with Nitpick. If the properties are correctly stated,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1293
counterexamples will point to bugs in the specification. For our grammar
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1294
example, we will proceed in two steps, separating the soundness and the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1295
completeness of the set $S$. First, soundness:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1296
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1297
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1298
\textbf{theorem}~\textit{S\_sound}: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1299
``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1300
  \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1301
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1302
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1303
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1304
\hbox{}\qquad\qquad $w = [b]$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1305
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1306
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1307
It would seem that $[b] \in S$. How could this be? An inspection of the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1308
introduction rules reveals that the only rule with a right-hand side of the form
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1309
$b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1310
\textit{R5}:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1311
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1312
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1313
``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1314
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1315
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1316
On closer inspection, we can see that this rule is wrong. To match the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1317
production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1318
again:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1319
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1320
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1321
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1322
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1323
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1324
\hbox{}\qquad\qquad $w = [a, a, b]$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1325
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1326
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1327
Some detective work is necessary to find out what went wrong here. To get $[a,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1328
a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1329
from \textit{R6}:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1330
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1331
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1332
``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1333
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1334
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1335
Now, this formula must be wrong: The same assumption occurs twice, and the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1336
variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1337
the assumptions should have been a $w$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1338
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1339
With the correction made, we don't get any counterexample from Nitpick. Let's
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1340
move on and check completeness:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1341
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1342
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1343
\textbf{theorem}~\textit{S\_complete}: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1344
``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1345
   \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1346
  \longrightarrow w \in S$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1347
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1348
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1349
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1350
\hbox{}\qquad\qquad $w = [b, b, a, a]$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1351
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1352
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1353
Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1354
$a$'s and $b$'s. But since our inductive definition passed the soundness check,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1355
the introduction rules we have are probably correct. Perhaps we simply lack an
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1356
introduction rule. Comparing the grammar with the inductive definition, our
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1357
suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1358
without which the grammar cannot generate two or more $b$'s in a row. So we add
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1359
the rule
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1360
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1361
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1362
``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1363
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1364
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1365
With this last change, we don't get any counterexamples from Nitpick for either
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1366
soundness or completeness. We can even generalize our result to cover $A$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1367
$B$ as well:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1368
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1369
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1370
\textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1371
``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1372
``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1373
``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1374
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1375
\slshape Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1376
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1377
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1378
\subsection{AA Trees}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1379
\label{aa-trees}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1380
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1381
AA trees are a kind of balanced trees discovered by Arne Andersson that provide
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1382
similar performance to red-black trees, but with a simpler implementation
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1383
\cite{andersson-1993}. They can be used to store sets of elements equipped with
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1384
a total order $<$. We start by defining the datatype and some basic extractor
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1385
functions:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1386
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1387
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1388
\textbf{datatype} $'a$~\textit{tree} = $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{tree}'' ``\kern1pt$'a$ \textit{tree}''  \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1389
\textbf{primrec} \textit{data} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1390
``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1391
``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1392
\textbf{primrec} \textit{dataset} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1393
``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1394
``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1395
\textbf{primrec} \textit{level} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1396
``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1397
``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1398
\textbf{primrec} \textit{left} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1399
``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1400
``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1401
\textbf{primrec} \textit{right} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1402
``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1403
``$\textit{right}~(N~\_~\_~\_~u) = u$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1404
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1405
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1406
The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1407
as follows \cite{wikipedia-2009-aa-trees}:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1408
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1409
\kern.2\parskip %% TYPESETTING
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1410
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1411
\pre
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1412
Each node has a level field, and the following invariants must remain true for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1413
the tree to be valid:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1414
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1415
\raggedright
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1416
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1417
\kern-.4\parskip %% TYPESETTING
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1418
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1419
\begin{enum}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1420
\item[]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1421
\begin{enum}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1422
\item[1.] The level of a leaf node is one.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1423
\item[2.] The level of a left child is strictly less than that of its parent.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1424
\item[3.] The level of a right child is less than or equal to that of its parent.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1425
\item[4.] The level of a right grandchild is strictly less than that of its grandparent.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1426
\item[5.] Every node of level greater than one must have two children.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1427
\end{enum}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1428
\end{enum}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1429
\post
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1430
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1431
\kern.4\parskip %% TYPESETTING
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1432
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1433
The \textit{wf} predicate formalizes this description:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1434
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1435
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1436
\textbf{primrec} \textit{wf} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1437
``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1438
``$\textit{wf}~(N~\_~k~t~u) =$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1439
\phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1440
\phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1441
\phantom{``$($}$\textrm{else}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1442
\hbox{\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1443
\mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1444
\mathrel{\land} \textit{level}~u \le k \mathrel{\land}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1445
\textit{level}~(\textit{right}~u) < k)$''}\kern-200mm
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1446
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1447
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1448
Rebalancing the tree upon insertion and removal of elements is performed by two
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1449
auxiliary functions called \textit{skew} and \textit{split}, defined below:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1450
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1451
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1452
\textbf{primrec} \textit{skew} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1453
``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1454
``$\textit{skew}~(N~x~k~t~u) = {}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1455
\phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1456
\textit{level}~t~\textrm{then}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1457
\phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1458
(\textit{right}~t)~u)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1459
\phantom{``(}$\textrm{else}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1460
\phantom{``(\quad}$N~x~k~t~u)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1461
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1462
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1463
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1464
\textbf{primrec} \textit{split} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1465
``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1466
``$\textit{split}~(N~x~k~t~u) = {}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1467
\phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1468
\textit{level}~(\textit{right}~u)~\textrm{then}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1469
\phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1470
(N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1471
\phantom{``(}$\textrm{else}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1472
\phantom{``(\quad}$N~x~k~t~u)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1473
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1474
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1475
Performing a \textit{skew} or a \textit{split} should have no impact on the set
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1476
of elements stored in the tree:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1477
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1478
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1479
\textbf{theorem}~\textit{dataset\_skew\_split}:\\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1480
``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1481
``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1482
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1483
{\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1484
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1485
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1486
Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1487
should not alter the tree:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1488
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1489
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1490
\textbf{theorem}~\textit{wf\_skew\_split}:\\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1491
``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1492
``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1493
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1494
{\slshape Nitpick found no counterexample.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1495