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[english,french]{babel}
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\usepackage{color}
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\usepackage{graphicx}
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%\usepackage{mathpazo}
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\usepackage{multicol}
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\usepackage{stmaryrd}
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%\usepackage[scaled=.85]{beramono}
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\usepackage{../iman,../pdfsetup}
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%\oddsidemargin=4.6mm
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%\evensidemargin=4.6mm
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%\textwidth=150mm
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%\topmargin=4.6mm
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%\headheight=0mm
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%\headsep=0mm
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%\textheight=234mm
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\def\Colon{\mathord{:\mkern-1.5mu:}}
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%\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}}
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%\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}}
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\def\lparr{\mathopen{(\mkern-4mu\mid}}
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\def\rparr{\mathclose{\mid\mkern-4mu)}}
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\def\unk{{?}}
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\def\undef{(\lambda x.\; \unk)}
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%\def\unr{\textit{others}}
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\def\unr{\ldots}
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\def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
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\def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
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\hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
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counter-example counter-examples data-type data-types co-data-type 
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co-data-types in-duc-tive co-in-duc-tive}
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\urlstyle{tt}
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\begin{document}
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\selectlanguage{english}
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\title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
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Picking Nits \\[\smallskipamount]
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\Large A User's Guide to Nitpick for Isabelle/HOL}
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\author{\hbox{} \\
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Jasmin Christian Blanchette \\
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{\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\
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\hbox{}}
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\maketitle
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\tableofcontents
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\setlength{\parskip}{.7em plus .2em minus .1em}
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\setlength{\parindent}{0pt}
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\setlength{\belowdisplayskip}{\parskip}
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\setlength{\belowdisplayshortskip}{.9\parskip}
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% General-purpose enum environment with correct spacing
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\newenvironment{enum}%
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    {\begin{list}{}{%
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        \setlength{\topsep}{.1\parskip}%
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        \setlength{\partopsep}{.1\parskip}%
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        \setlength{\itemsep}{\parskip}%
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        \advance\itemsep by-\parsep}}
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    {\end{list}}
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\def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin
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\advance\rightskip by\leftmargin}
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\def\post{\vskip0pt plus1ex\endgroup}
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\def\prew{\pre\advance\rightskip by-\leftmargin}
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\def\postw{\post}
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\section{Introduction}
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\label{introduction}
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Nitpick \cite{blanchette-nipkow-2009} is a counterexample generator for
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Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
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combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
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quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
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first-order relational model finder developed by the Software Design Group at
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MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
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borrows many ideas and code fragments, but it benefits from Kodkod's
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optimizations and a new encoding scheme. The name Nitpick is shamelessly
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appropriated from a now retired Alloy precursor.
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Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
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theorem and wait a few seconds. Nonetheless, there are situations where knowing
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how it works under the hood and how it reacts to various options helps
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increase the test coverage. This manual also explains how to install the tool on
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your workstation. Should the motivation fail you, think of the many hours of
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hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.
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Another common use of Nitpick is to find out whether the axioms of a locale are
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satisfiable, while the locale is being developed. To check this, it suffices to
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write
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\prew
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\textbf{lemma}~``$\textit{False}$'' \\
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\textbf{nitpick}~[\textit{show\_all}]
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\postw
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after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
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must find a model for the axioms. If it finds no model, we have an indication
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that the axioms might be unsatisfiable.
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Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
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machine called \texttt{java}. The examples presented in this manual can be found
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in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.
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Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
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Nitpick also provides an automatic mode that can be enabled using the
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``Auto Nitpick'' option from the ``Isabelle'' menu in Proof General. In this
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mode, Nitpick is run on every newly entered theorem, much like Auto Quickcheck.
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The collective time limit for Auto Nitpick and Auto Quickcheck can be set using
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the ``Auto Counterexample Time Limit'' option.
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\newbox\boxA
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\setbox\boxA=\hbox{\texttt{nospam}}
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The known bugs and limitations at the time of writing are listed in
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\S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
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or this manual should be directed to
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\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
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in.\allowbreak tum.\allowbreak de}.
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\vskip2.5\smallskipamount
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\textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
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suggesting several textual improvements.
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% and Perry James for reporting a typo.
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\section{First Steps}
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\label{first-steps}
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This section introduces Nitpick by presenting small examples. If possible, you
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should try out the examples on your workstation. Your theory file should start
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as follows:
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\prew
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\textbf{theory}~\textit{Scratch} \\
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\textbf{imports}~\textit{Main~Coinductive\_List~Quotient\_Product~RealDef} \\
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\textbf{begin}
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\postw
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The results presented here were obtained using the JNI version of MiniSat and
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with multithreading disabled to reduce nondeterminism. 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{MiniSat\_JNI}, \,\textit{max\_threads}~= 1]
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\postw
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after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
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Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
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be installed, as explained in \S\ref{optimizations}. If you have already
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configured SAT solvers in Isabelle (e.g., for Refute), these will also be
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available to Nitpick.
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\subsection{Propositional Logic}
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\label{propositional-logic}
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Let's start with a trivial example from propositional logic:
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\prew
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\textbf{lemma}~``$P \longleftrightarrow Q$'' \\
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\textbf{nitpick}
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\postw
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You should get the following output:
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\prew
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\slshape
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Nitpick found a counterexample: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \textit{True}$ \\
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\hbox{}\qquad\qquad $Q = \textit{False}$
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\postw
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Nitpick can also be invoked on individual subgoals, as in the example below:
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\prew
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\textbf{apply}~\textit{auto} \\[2\smallskipamount]
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{\slshape goal (2 subgoals): \\
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\phantom{0}1. $P\,\Longrightarrow\, Q$ \\
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\phantom{0}2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
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\textbf{nitpick}~1 \\[2\smallskipamount]
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{\slshape Nitpick found a counterexample: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \textit{True}$ \\
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\hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
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\textbf{nitpick}~2 \\[2\smallskipamount]
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{\slshape Nitpick found a counterexample: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \textit{False}$ \\
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\hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
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\textbf{oops}
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\postw
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\subsection{Type Variables}
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\label{type-variables}
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If you are left unimpressed by the previous example, don't worry. The next
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one is more mind- and computer-boggling:
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\prew
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\textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
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\postw
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\pagebreak[2] %% TYPESETTING
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The putative lemma involves the definite description operator, {THE}, presented
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in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
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operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
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lemma is merely asserting the indefinite description operator axiom with {THE}
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substituted for {SOME}.
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The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
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containing type variables, Nitpick enumerates the possible domains for each type
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variable, up to a given cardinality (8 by default), looking for a finite
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countermodel:
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\prew
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\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
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\slshape
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Trying 8 scopes: \nopagebreak \\
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\hbox{}\qquad \textit{card}~$'a$~= 1; \\
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\hbox{}\qquad \textit{card}~$'a$~= 2; \\
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\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
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\hbox{}\qquad \textit{card}~$'a$~= 8. \\[2\smallskipamount]
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Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
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\hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
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Total time: 580 ms.
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\postw
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Nitpick found a counterexample in which $'a$ has cardinality 3. (For
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cardinalities 1 and 2, the formula holds.) In the counterexample, the three
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values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
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The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
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\textit{verbose} is enabled. You can specify \textit{verbose} each time you
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invoke \textbf{nitpick}, or you can set it globally using the command
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\prew
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\textbf{nitpick\_params} [\textit{verbose}]
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\postw
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This command also displays the current default values for all of the options
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supported by Nitpick. The options are listed in \S\ref{option-reference}.
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\subsection{Constants}
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\label{constants}
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By just looking at Nitpick's output, it might not be clear why the
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counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
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this time telling it to show the values of the constants that occur in the
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formula:
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\prew
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\textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
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\textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
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\slshape
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Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
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\hbox{}\qquad\qquad $x = a_3$ \\
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\hbox{}\qquad Constant: \nopagebreak \\
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\hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
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\postw
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We can see more clearly now. Since the predicate $P$ isn't true for a unique
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value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
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$a_1$. Since $P~a_1$ is false, the entire formula is falsified.
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As an optimization, Nitpick's preprocessor introduced the special constant
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``\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
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$\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
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satisfying $P~y$. We disable this optimization by passing the
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\textit{full\_descrs} option:
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\prew
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\textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
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\slshape
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Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
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\hbox{}\qquad Free variables: \nopagebreak \\
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\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
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\hbox{}\qquad\qquad $x = a_3$ \\
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\hbox{}\qquad Constant: \nopagebreak \\
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\hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
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\postw
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As the result of another optimization, Nitpick directly assigned a value to the
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subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
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disable this second optimization by using the command
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\prew
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\textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
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\textit{show\_consts}]
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\postw
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we finally get \textit{The}:
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\prew
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\slshape Constant: \nopagebreak \\
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\hbox{}\qquad $\mathit{The} = \undef{}
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    (\!\begin{aligned}[t]%
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    & \{a_1, a_2, a_3\} := a_3,\> \{a_1, a_2\} := a_3,\> \{a_1, a_3\} := a_3, \\[-2pt] %% TYPESETTING
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    & \{a_1\} := a_1,\> \{a_2, a_3\} := a_1,\> \{a_2\} := a_2, \\[-2pt]
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    & \{a_3\} := a_3,\> \{\} := a_3)\end{aligned}$
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\postw
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Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
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just like before.\footnote{The Isabelle/HOL notation $f(x :=
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y)$ denotes the function that maps $x$ to $y$ and that otherwise behaves like
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$f$.}
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Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
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unique $x$ such that'') at the front of our putative lemma's assumption:
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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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diff changeset
   339
cardinalities up to 50:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   340
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   341
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   342
\textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   343
can be entered as \texttt{-} (hyphen) or
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   344
\texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   345
\slshape Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   346
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   347
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   348
Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   349
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   350
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   351
\textbf{sledgehammer} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   352
{\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   353
$\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   354
Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   355
\textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   356
{\slshape No subgoals!}% \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   357
%\textbf{done}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   358
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   359
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   360
This must be our lucky day.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   361
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   362
\subsection{Skolemization}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   363
\label{skolemization}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   364
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   365
Are all invertible functions onto? Let's find out:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   366
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   367
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   368
\textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   369
 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   370
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   371
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   372
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   373
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   374
\hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   375
\hbox{}\qquad Skolem constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   376
\hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   377
\hbox{}\qquad\qquad $y = a_2$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   378
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   379
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   380
Although $f$ is the only free variable occurring in the formula, Nitpick also
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   381
displays values for the bound variables $g$ and $y$. These values are available
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   382
to Nitpick because it performs skolemization as a preprocessing step.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   383
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   384
In the previous example, skolemization only affected the outermost quantifiers.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   385
This is not always the case, as illustrated below:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   386
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   387
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   388
\textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   389
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   390
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   391
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   392
\hbox{}\qquad Skolem constant: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   393
\hbox{}\qquad\qquad $\lambda x.\; f =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   394
    \undef{}(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   395
    & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   396
    & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   397
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   398
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   399
The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   400
$x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   401
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
   402
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
   403
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   404
The source of the Skolem constants is sometimes more obscure:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   405
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   406
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   407
\textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   408
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   409
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   410
Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   411
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   412
\hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   413
\hbox{}\qquad Skolem constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   414
\hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   415
\hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   416
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   417
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   418
What happened here is that Nitpick expanded the \textit{sym} constant to its
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   419
definition:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   420
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   421
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   422
$\mathit{sym}~r \,\equiv\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   423
 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   424
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   425
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   426
As their names suggest, the Skolem constants $\mathit{sym}.x$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   427
$\mathit{sym}.y$ are simply the bound variables $x$ and $y$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   428
from \textit{sym}'s definition.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   429
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   430
Although skolemization is a useful optimization, you can disable it by invoking
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   431
Nitpick with \textit{dont\_skolemize}. See \S\ref{optimizations} for details.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   432
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   433
\subsection{Natural Numbers and Integers}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   434
\label{natural-numbers-and-integers}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   435
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   436
Because of the axiom of infinity, the type \textit{nat} does not admit any
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   437
finite models. To deal with this, Nitpick's approach is to consider finite
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   438
subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   439
value (displayed as `$\unk$'). The type \textit{int} is handled similarly.
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   440
Internally, undefined values lead to a three-valued logic.
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   441
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   442
Here is an example involving \textit{int\/}:
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   443
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   444
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   445
\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
   446
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   447
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   448
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   449
\hbox{}\qquad\qquad $i = 0$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   450
\hbox{}\qquad\qquad $j = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   451
\hbox{}\qquad\qquad $m = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   452
\hbox{}\qquad\qquad $n = 0$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   453
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   454
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   455
Internally, Nitpick uses either a unary or a binary representation of numbers.
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   456
The unary representation is more efficient but only suitable for numbers very
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   457
close to zero. By default, Nitpick attempts to choose the more appropriate
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   458
encoding by inspecting the formula at hand. This behavior can be overridden by
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   459
passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   460
binary notation, the number of bits to use can be specified using
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   461
the \textit{bits} option. For example:
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   462
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   463
\prew
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   464
\textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$]
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   465
\postw
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   466
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   467
With infinite types, we don't always have the luxury of a genuine counterexample
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   468
and must often content ourselves with a potential one. The tedious task of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   469
finding out whether the potential counterexample is in fact genuine can be
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   470
outsourced to \textit{auto} by passing \textit{check\_potential}. For example:
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   471
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   472
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   473
\textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   474
\textbf{nitpick} [\textit{card~nat}~= 100, \textit{check\_potential}] \\[2\smallskipamount]
35220
2bcdae5f4fdb added support for nonstandard "nat"s to Nitpick and fixed bugs in binary "nat"s and "int"s
blanchet
parents: 35189
diff changeset
   475
\slshape Warning: The conjecture either trivially holds for the given scopes or (more likely) lies outside Nitpick's supported
35185
9b8f351cced6 added yet another hint to Nitpick's output, this time warning about problems for which nothing was effectively tested
blanchet
parents: 35183
diff changeset
   476
fragment. Only potential counterexamples may be found. \\[2\smallskipamount]
9b8f351cced6 added yet another hint to Nitpick's output, this time warning about problems for which nothing was effectively tested
blanchet
parents: 35183
diff changeset
   477
Nitpick found a potential counterexample: \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   478
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   479
\hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   480
Confirmation by ``\textit{auto}'': The above counterexample is genuine.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   481
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   482
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   483
You might wonder why the counterexample is first reported as potential. The root
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   484
of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   485
\mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   486
that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   487
\textit{False}; but otherwise, it does not know anything about values of $n \ge
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   488
\textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   489
\textit{True}. Since the assumption can never be satisfied, the putative lemma
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   490
can never be falsified.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   491
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   492
Incidentally, if you distrust the so-called genuine counterexamples, you can
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   493
enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   494
aware that \textit{auto} will usually fail to prove that the counterexample is
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   495
genuine or spurious.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   496
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   497
Some conjectures involving elementary number theory make Nitpick look like a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   498
giant with feet of clay:
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} ``$P~\textit{Suc}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   502
\textbf{nitpick} [\textit{card} = 1--6] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   503
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   504
Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   505
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   506
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   507
On any finite set $N$, \textit{Suc} is a partial function; for example, if $N =
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   508
\{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\,
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   509
\ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when passed as
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   510
argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$. The next
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   511
example is similar:
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   512
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   513
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   514
\textbf{lemma} ``$P~(\textit{op}~{+}\Colon
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   515
\textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   516
\textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   517
{\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   518
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   519
\hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   520
\textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   521
{\slshape Nitpick found no counterexample.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   522
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   523
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   524
The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   525
$\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   526
1\}$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   527
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   528
Because numbers are infinite and are approximated using a three-valued logic,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   529
there is usually no need to systematically enumerate domain sizes. If Nitpick
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   530
cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   531
unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   532
example above is an exception to this principle.) Nitpick nonetheless enumerates
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   533
all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   534
cardinalities are fast to handle and give rise to simpler counterexamples. This
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   535
is explained in more detail in \S\ref{scope-monotonicity}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   536
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   537
\subsection{Inductive Datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   538
\label{inductive-datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   539
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   540
Like natural numbers and integers, inductive datatypes with recursive
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   541
constructors admit no finite models and must be approximated by a subterm-closed
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   542
subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   543
Nitpick looks for all counterexamples that can be built using at most 10
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   544
different lists.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   545
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   546
Let's see with an example involving \textit{hd} (which returns the first element
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   547
of a list) and $@$ (which concatenates two lists):
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   548
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   549
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   550
\textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   551
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   552
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   553
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   554
\hbox{}\qquad\qquad $\textit{xs} = []$ \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   555
\hbox{}\qquad\qquad $\textit{y} = a_1$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   556
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   557
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   558
To see why the counterexample is genuine, we enable \textit{show\_consts}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   559
and \textit{show\_\allowbreak datatypes}:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   560
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   561
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   562
{\slshape Datatype:} \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   563
\hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   564
{\slshape Constants:} \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   565
\hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_1, a_1])$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   566
\hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   567
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   568
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   569
Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   570
including $a_2$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   571
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   572
The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   573
append operator whose second argument is fixed to be $[y, y]$. Appending $[a_1,
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   574
a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   575
representable in the subset of $'a$~\textit{list} considered by Nitpick, which
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   576
is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   577
appending $[a_1, a_1]$ to itself gives $\unk$.
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   578
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   579
Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   580
considers the following subsets:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   581
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   582
\kern-.5\smallskipamount %% TYPESETTING
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   583
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   584
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   585
\begin{multicols}{3}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   586
$\{[],\, [a_1],\, [a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   587
$\{[],\, [a_1],\, [a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   588
$\{[],\, [a_2],\, [a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   589
$\{[],\, [a_1],\, [a_1, a_1]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   590
$\{[],\, [a_1],\, [a_2, a_1]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   591
$\{[],\, [a_1],\, [a_3, a_1]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   592
$\{[],\, [a_2],\, [a_1, a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   593
$\{[],\, [a_2],\, [a_2, a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   594
$\{[],\, [a_2],\, [a_3, a_2]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   595
$\{[],\, [a_3],\, [a_1, a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   596
$\{[],\, [a_3],\, [a_2, a_3]\}$; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   597
$\{[],\, [a_3],\, [a_3, a_3]\}$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   598
\end{multicols}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   599
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   600
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   601
\kern-2\smallskipamount %% TYPESETTING
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   602
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   603
All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   604
are listed and only those. As an example of a non-subterm-closed subset,
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   605
consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   606
that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   607
\mathcal{S}$ as a subterm.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   608
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   609
Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   610
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   611
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   612
\textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   613
\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   614
\\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   615
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   616
\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   617
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   618
\hbox{}\qquad\qquad $\textit{xs} = [a_1]$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   619
\hbox{}\qquad\qquad $\textit{ys} = [a_2]$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   620
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   621
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   622
\hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   623
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   624
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   625
Because datatypes are approximated using a three-valued logic, there is usually
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   626
no need to systematically enumerate cardinalities: If Nitpick cannot find a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   627
genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   628
unlikely that one could be found for smaller cardinalities.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   629
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   630
\subsection{Typedefs, Quotient Types, Records, Rationals, and Reals}
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   631
\label{typedefs-records-rationals-and-reals}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   632
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   633
Nitpick generally treats types declared using \textbf{typedef} as datatypes
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   634
whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   635
For example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   636
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   637
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   638
\textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   639
\textbf{by}~\textit{blast} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   640
\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
   641
\textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   642
\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
   643
\textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   644
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   645
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   646
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   647
\hbox{}\qquad\qquad $P = \{\Abs{0},\, \Abs{1}\}$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   648
\hbox{}\qquad\qquad $x = \Abs{2}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   649
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   650
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   651
\hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   652
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   653
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   654
In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   655
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   656
Quotient types are handled in much the same way. The following fragment defines
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   657
the integer type \textit{my\_int} by encoding the integer $x$ by a pair of
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   658
natural numbers $(m, n)$ such that $x + n = m$:
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   659
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   660
\prew
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   661
\textbf{fun} \textit{my\_int\_rel} \textbf{where} \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   662
``$\textit{my\_int\_rel}~(x,\, y)~(u,\, v) = (x + v = u + y)$'' \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   663
%
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   664
\textbf{quotient\_type}~\textit{my\_int} = ``$\textit{nat} \times \textit{nat\/}$''$\;{/}\;$\textit{my\_int\_rel} \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   665
\textbf{by}~(\textit{auto simp add\/}:\ \textit{equivp\_def expand\_fun\_eq}) \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   666
%
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   667
\textbf{definition}~\textit{add\_raw}~\textbf{where} \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   668
``$\textit{add\_raw} \,\equiv\, \lambda(x,\, y)~(u,\, v).\; (x + (u\Colon\textit{nat}), y + (v\Colon\textit{nat}))$'' \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   669
%
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   670
\textbf{quotient\_definition} ``$\textit{add\/}\Colon\textit{my\_int} \Rightarrow \textit{my\_int} \Rightarrow \textit{my\_int\/}$'' \textbf{is} \textit{add\_raw} \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   671
%
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   672
\textbf{lemma} ``$\textit{add}~x~y = \textit{add}~x~x$'' \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   673
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   674
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   675
\hbox{}\qquad Free variables: \nopagebreak \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   676
\hbox{}\qquad\qquad $x = \Abs{(0,\, 0)}$ \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   677
\hbox{}\qquad\qquad $y = \Abs{(1,\, 0)}$ \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   678
\hbox{}\qquad Datatypes: \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   679
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, \unr\}$ \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   680
\hbox{}\qquad\qquad $\textit{nat} \times \textit{nat} = \{(0,\, 0),\> (1,\, 0),\> \unr\}$ \\
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   681
\hbox{}\qquad\qquad $\textit{my\_int} = \{\Abs{(0,\, 0)},\> \Abs{(1,\, 0)},\> \unr\}$
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   682
\postw
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   683
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   684
In the counterexample, $\Abs{(0,\, 0)}$ and $\Abs{(1,\, 0)}$ represent the
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   685
integers $0$ and $1$, respectively. Other representants would have been
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   686
possible---e.g., $\Abs{(5,\, 5)}$ and $\Abs{(12,\, 11)}$.
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   687
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   688
Records are also handled as datatypes with a single constructor:
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   689
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   690
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   691
\textbf{record} \textit{point} = \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   692
\hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   693
\hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   694
\textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   695
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   696
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   697
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   698
\hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   699
\hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   700
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   701
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   702
\hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t]
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   703
& \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   704
& \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   705
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   706
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   707
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   708
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   709
Finally, Nitpick provides rudimentary support for rationals and reals using a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   710
similar approach:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   711
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   712
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   713
\textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   714
\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   715
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   716
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   717
\hbox{}\qquad\qquad $x = 1/2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   718
\hbox{}\qquad\qquad $y = -1/2$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   719
\hbox{}\qquad Datatypes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   720
\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   721
\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   722
\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
   723
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   724
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   725
\subsection{Inductive and Coinductive Predicates}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   726
\label{inductive-and-coinductive-predicates}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   727
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   728
Inductively defined predicates (and sets) are particularly problematic for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   729
counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   730
loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   731
the problem is that they are defined using a least fixed point construction.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   732
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   733
Nitpick's philosophy is that not all inductive predicates are equal. Consider
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   734
the \textit{even} predicate below:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   735
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   736
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   737
\textbf{inductive}~\textit{even}~\textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   738
``\textit{even}~0'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   739
``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   740
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   741
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   742
This predicate enjoys the desirable property of being well-founded, which means
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   743
that the introduction rules don't give rise to infinite chains of the form
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   744
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   745
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   746
$\cdots\,\Longrightarrow\, \textit{even}~k''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   747
       \,\Longrightarrow\, \textit{even}~k'
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   748
       \,\Longrightarrow\, \textit{even}~k.$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   749
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   750
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   751
For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   752
$k/2 + 1$:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   753
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   754
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   755
$\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   756
       \,\Longrightarrow\, \textit{even}~(k - 2)
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   757
       \,\Longrightarrow\, \textit{even}~k.$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   758
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   759
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   760
Wellfoundedness is desirable because it enables Nitpick to use a very efficient
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   761
fixed point computation.%
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   762
\footnote{If an inductive predicate is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   763
well-founded, then it has exactly one fixed point, which is simultaneously the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   764
least and the greatest fixed point. In these circumstances, the computation of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   765
the least fixed point amounts to the computation of an arbitrary fixed point,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   766
which can be performed using a straightforward recursive equation.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   767
Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   768
just as Isabelle's \textbf{function} package usually discharges termination
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   769
proof obligations automatically.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   770
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   771
Let's try an example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   772
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   773
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   774
\textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
34126
8a2c5d7aff51 polished Nitpick's binary integer support etc.;
blanchet
parents: 34124
diff changeset
   775
\textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   776
\slshape The inductive predicate ``\textit{even}'' was proved well-founded.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   777
Nitpick can compute it efficiently. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   778
Trying 1 scope: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   779
\hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   780
Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   781
\hbox{}\qquad Empty assignment \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   782
Nitpick could not find a better counterexample. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   783
Total time: 2274 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   784
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   785
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   786
No genuine counterexample is possible because Nitpick cannot rule out the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   787
existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   788
$\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   789
existential quantifier:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   790
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   791
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   792
\textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
34126
8a2c5d7aff51 polished Nitpick's binary integer support etc.;
blanchet
parents: 34124
diff changeset
   793
\textbf{nitpick}~[\textit{card nat}~= 100, \textit{unary\_ints}] \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   794
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   795
\hbox{}\qquad Empty assignment
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   796
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   797
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   798
So far we were blessed by the wellfoundedness of \textit{even}. What happens if
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   799
we use the following definition instead?
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   800
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   801
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   802
\textbf{inductive} $\textit{even}'$ \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   803
``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   804
``$\textit{even}'~2$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   805
``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   806
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   807
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   808
This definition is not well-founded: From $\textit{even}'~0$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   809
$\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   810
predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   811
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   812
Let's check a property involving $\textit{even}'$. To make up for the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   813
foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   814
\textit{nat}'s cardinality to a mere 10:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   815
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   816
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   817
\textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   818
\lnot\;\textit{even}'~n$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   819
\textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   820
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   821
The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   822
Nitpick might need to unroll it. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   823
Trying 6 scopes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   824
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   825
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   826
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   827
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   828
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   829
\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   830
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
   831
\hbox{}\qquad Constant: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   832
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   833
& 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   834
& 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   835
& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   836
Total time: 1140 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   837
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   838
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   839
Nitpick's output is very instructive. First, it tells us that the predicate is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   840
unrolled, meaning that it is computed iteratively from the empty set. Then it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   841
lists six scopes specifying different bounds on the numbers of iterations:\ 0,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   842
1, 2, 4, 8, and~9.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   843
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   844
The output also shows how each iteration contributes to $\textit{even}'$. The
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   845
notation $\lambda i.\; \textit{even}'$ indicates that the value of the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   846
predicate depends on an iteration counter. Iteration 0 provides the basis
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   847
elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   848
throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   849
iterations would not contribute any new elements.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   850
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   851
Some values are marked with superscripted question
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   852
marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   853
predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   854
\textit{True} or $\unk$, never \textit{False}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   855
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   856
When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   857
iterations. However, these numbers are bounded by the cardinality of the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   858
predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   859
ever needed to compute the value of a \textit{nat} predicate. You can specify
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   860
the number of iterations using the \textit{iter} option, as explained in
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   861
\S\ref{scope-of-search}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   862
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   863
In the next formula, $\textit{even}'$ occurs both positively and negatively:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   864
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   865
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   866
\textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
   867
\textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   868
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   869
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   870
\hbox{}\qquad\qquad $n = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   871
\hbox{}\qquad Constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   872
\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   873
& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$  \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   874
\hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   875
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   876
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   877
Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   878
8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   879
fixed point (not necessarily the least one). It is used to falsify
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   880
$\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   881
$\textit{even}'~(n - 2)$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   882
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   883
Coinductive predicates are handled dually. For example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   884
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   885
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   886
\textbf{coinductive} \textit{nats} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   887
``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   888
\textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   889
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   890
\slshape Nitpick found a counterexample:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   891
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   892
\hbox{}\qquad Constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   893
\hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   894
& 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   895
& \unr\})\end{aligned}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   896
\hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   897
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   898
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   899
As a special case, Nitpick uses Kodkod's transitive closure operator to encode
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   900
negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   901
inductive predicates for which each the predicate occurs in at most one
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   902
assumption of each introduction rule. For example:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   903
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   904
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   905
\textbf{inductive} \textit{odd} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   906
``$\textit{odd}~1$'' $\,\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   907
``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   908
\textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   909
\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   910
\slshape Nitpick found a counterexample:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   911
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   912
\hbox{}\qquad Free variable: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   913
\hbox{}\qquad\qquad $n = 1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   914
\hbox{}\qquad Constants: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   915
\hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   916
\hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   917
\hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   918
\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   919
  & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   920
  & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   921
       (3, 5), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   922
  & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   923
  & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   924
\hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   925
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   926
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   927
\noindent
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   928
In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   929
$\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   930
elements from known ones. The set $\textit{odd}$ consists of all the values
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   931
reachable through the reflexive transitive closure of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   932
$\textit{odd}_{\textrm{step}}$ starting with any element from
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   933
$\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   934
transitive closure to encode linear predicates is normally either more thorough
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   935
or more efficient than unrolling (depending on the value of \textit{iter}), but
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   936
for those cases where it isn't you can disable it by passing the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   937
\textit{dont\_star\_linear\_preds} option.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   938
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   939
\subsection{Coinductive Datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   940
\label{coinductive-datatypes}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   941
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   942
While Isabelle regrettably lacks a high-level mechanism for defining coinductive
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   943
datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   944
list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   945
these lazy lists seamlessly and provides a hook, described in
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   946
\S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   947
datatypes.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   948
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   949
(Co)intuitively, a coinductive datatype is similar to an inductive datatype but
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   950
allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   951
\ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   952
1, 2, 3, \ldots]$ can be defined as lazy lists using the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   953
$\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   954
$\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   955
\mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   956
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   957
Although it is otherwise no friend of infinity, Nitpick can find counterexamples
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   958
involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   959
finite lists:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   960
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   961
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   962
\textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   963
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   964
\slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   965
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   966
\hbox{}\qquad\qquad $\textit{a} = a_1$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   967
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   968
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   969
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   970
The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   971
for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   972
infinite list $[a_1, a_1, a_1, \ldots]$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   973
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   974
The next example is more interesting:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   975
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   976
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   977
\textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   978
\textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   979
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   980
\slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   981
some scopes. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   982
Trying 8 scopes: \\
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   983
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 1,
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   984
and \textit{bisim\_depth}~= 0. \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   985
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   986
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 8,
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   987
and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   988
Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
   989
\textit{card}~``\kern1pt$'a~\textit{list\/}$''~= 2, and \textit{bisim\_\allowbreak
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   990
depth}~= 1:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   991
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   992
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   993
\hbox{}\qquad\qquad $\textit{a} = a_1$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   994
\hbox{}\qquad\qquad $\textit{b} = a_2$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   995
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
   996
\hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   997
Total time: 726 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   998
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
   999
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1000
The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1001
$\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1002
$[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1003
within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1004
the segment leading to the binder is the stem.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1005
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1006
A salient property of coinductive datatypes is that two objects are considered
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1007
equal if and only if they lead to the same observations. For example, the lazy
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1008
lists $\textrm{THE}~\omega.\; \omega =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1009
\textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1010
$\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1011
\textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1012
to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1013
equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1014
concept of equality for coinductive datatypes is called bisimulation and is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1015
defined coinductively.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1016
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1017
Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1018
the Kodkod problem to ensure that distinct objects lead to different
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1019
observations. This precaution is somewhat expensive and often unnecessary, so it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1020
can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1021
bisimilarity check is then performed \textsl{after} the counterexample has been
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1022
found to ensure correctness. If this after-the-fact check fails, the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1023
counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1024
again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1025
check for the previous example saves approximately 150~milli\-seconds; the speed
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1026
gains can be more significant for larger scopes.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1027
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1028
The next formula illustrates the need for bisimilarity (either as a Kodkod
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1029
predicate or as an after-the-fact check) to prevent spurious counterexamples:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1030
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1031
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1032
\textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1033
\,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
34124
c4628a1dcf75 added support for binary nat/int representation to Nitpick
blanchet
parents: 34038
diff changeset
  1034
\textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1035
\slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1036
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1037
\hbox{}\qquad\qquad $a = a_1$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1038
\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1039
\textit{LCons}~a_1~\omega$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1040
\hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1041
\hbox{}\qquad Codatatype:\strut \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1042
\hbox{}\qquad\qquad $'a~\textit{llist} =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1043
\{\!\begin{aligned}[t]
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1044
  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt]
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1045
  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1046
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1047
Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1048
that the counterexample is genuine. \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1049
{\upshape\textbf{nitpick}} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1050
\slshape Nitpick found no counterexample.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1051
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1052
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1053
In the first \textbf{nitpick} invocation, the after-the-fact check discovered 
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1054
that the two known elements of type $'a~\textit{llist}$ are bisimilar.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1055
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1056
A compromise between leaving out the bisimilarity predicate from the Kodkod
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1057
problem and performing the after-the-fact check is to specify a lower
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1058
nonnegative \textit{bisim\_depth} value than the default one provided by
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1059
Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1060
be distinguished from each other by their prefixes of length $K$. Be aware that
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1061
setting $K$ to a too low value can overconstrain Nitpick, preventing it from
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1062
finding any counterexamples.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1063
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1064
\subsection{Boxing}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1065
\label{boxing}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1066
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1067
Nitpick normally maps function and product types directly to the corresponding
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1068
Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1069
cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1070
\Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1071
off to treat these types in the same way as plain datatypes, by approximating
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1072
them by a subset of a given cardinality. This technique is called ``boxing'' and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1073
is particularly useful for functions passed as arguments to other functions, for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1074
high-arity functions, and for large tuples. Under the hood, boxing involves
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1075
wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1076
isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1077
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1078
To illustrate boxing, we consider a formalization of $\lambda$-terms represented
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1079
using de Bruijn's notation:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1080
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1081
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1082
\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
  1083
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1084
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1085
The $\textit{lift}~t~k$ function increments all variables with indices greater
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1086
than or equal to $k$ by one:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1087
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1088
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1089
\textbf{primrec} \textit{lift} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1090
``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1091
``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1092
``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1093
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1094
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1095
The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1096
term $t$ has a loose variable with index $k$ or more:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1097
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1098
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1099
\textbf{primrec}~\textit{loose} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1100
``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1101
``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1102
``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1103
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1104
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1105
Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1106
on $t$:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1107
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1108
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1109
\textbf{primrec}~\textit{subst} \textbf{where} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1110
``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1111
``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1112
\phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1113
``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1114
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1115
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1116
A substitution is a function that maps variable indices to terms. Observe that
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1117
$\sigma$ is a function passed as argument and that Nitpick can't optimize it
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1118
away, because the recursive call for the \textit{Lam} case involves an altered
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1119
version. Also notice the \textit{lift} call, which increments the variable
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1120
indices when moving under a \textit{Lam}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1121
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1122
A reasonable property to expect of substitution is that it should leave closed
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1123
terms unchanged. Alas, even this simple property does not hold:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1124
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1125
\pre
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1126
\textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1127
\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1128
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1129
Trying 8 scopes: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1130
\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
  1131
\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
  1132
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1133
\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
  1134
Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1135
and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1136
\hbox{}\qquad Free variables: \nopagebreak \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1137
\hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1138
& 0 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1139
  1 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1140
  2 := \textit{Var}~0, \\[-2pt]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1141
& 3 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1142
  4 := \textit{Var}~0,\>
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1143
  5 := \textit{Var}~0)\end{aligned}$ \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1144
\hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1145
Total time: $4679$ ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1146
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1147
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1148
Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1149
\textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1150
$\lambda$-term notation, $t$~is
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1151
$\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
35284
9edc2bd6d2bd enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents: 35220
diff changeset
  1152
The bug is in \textit{subst\/}: The $\textit{lift}~(\sigma~m)~1$ call should be
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1153
replaced with $\textit{lift}~(\sigma~m)~0$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1154
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1155
An interesting aspect of Nitpick's verbose output is that it assigned inceasing
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1156
cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1157
For the formula of interest, knowing 6 values of that type was enough to find
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1158
the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1159
considered, a hopeless undertaking:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1160
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1161
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1162
\textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1163
{\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1164
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1165
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1166
{\looseness=-1
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1167
Boxing can be enabled or disabled globally or on a per-type basis using the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1168
\textit{box} option. Moreover, setting the cardinality of a function or
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1169
product type implicitly enables boxing for that type. Nitpick usually performs
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1170
reasonable choices about which types should be boxed, but option tweaking
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1171
sometimes helps.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1172
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1173
}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1174
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1175
\subsection{Scope Monotonicity}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1176
\label{scope-monotonicity}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1177
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1178
The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1179
and \textit{max}) controls which scopes are actually tested. In general, to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1180
exhaust all models below a certain cardinality bound, the number of scopes that
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1181
Nitpick must consider increases exponentially with the number of type variables
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1182
(and \textbf{typedecl}'d types) occurring in the formula. Given the default
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1183
cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1184
considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1185
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1186
Fortunately, many formulas exhibit a property called \textsl{scope
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1187
monotonicity}, meaning that if the formula is falsifiable for a given scope,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1188
it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1189
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1190
Consider the formula
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1192
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1193
\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
  1194
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1195
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1196
where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1197
$'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1198
exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1199
that any counterexample found with a small scope would still be a counterexample
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1200
in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1201
by the larger scope. Nitpick comes to the same conclusion after a careful
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1202
inspection of the formula and the relevant definitions:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1203
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1204
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1205
\textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1206
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1207
The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1208
Nitpick might be able to skip some scopes.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1209
 \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1210
Trying 8 scopes: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1211
\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1212
\textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1213
\textit{list}''~= 1, \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1214
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1215
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1216
\hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1217
\textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1218
\textit{list}''~= 2, \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1219
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1220
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1221
\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1222
\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1223
\textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1224
\textit{list}''~= 8, \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1225
\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1226
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1227
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1228
Nitpick found a counterexample for
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1229
\textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1230
\textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1231
\textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1232
\textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1233
\\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1234
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1235
\hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1236
\hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount]
33191
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1237
Total time: 1636 ms.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1238
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1239
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1240
In theory, it should be sufficient to test a single scope:
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1241
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1242
\prew
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1243
\textbf{nitpick}~[\textit{card}~= 8]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1244
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1245
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1246
However, this is often less efficient in practice and may lead to overly complex
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1247
counterexamples.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1248
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1249
If the monotonicity check fails but we believe that the formula is monotonic (or
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1250
we don't mind missing some counterexamples), we can pass the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1251
\textit{mono} option. To convince yourself that this option is risky,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1252
simply consider this example from \S\ref{skolemization}:
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
\textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1256
 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1257
\textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1258
{\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1259
\textbf{nitpick} \\[2\smallskipamount]
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1260
\slshape
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1261
Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1262
\hbox{}\qquad $\vdots$
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1263
\postw
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1264
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1265
(It turns out the formula holds if and only if $\textit{card}~'a \le
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1266
\textit{card}~'b$.) Although this is rarely advisable, the automatic
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1267
monotonicity checks can be disabled by passing \textit{non\_mono}
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1268
(\S\ref{optimizations}).
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1269
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1270
As insinuated in \S\ref{natural-numbers-and-integers} and
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1271
\S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1272
are normally monotonic and treated as such. The same is true for record types,
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1273
\textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1274
cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1275
\textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1276
consider only 8~scopes instead of $32\,768$.
fe3c65d9c577 Added Nitpick manual.
blanchet
parents:
diff changeset
  1277
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1278
\subsection{Inductive Properties}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1279
\label{inductive-properties}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1280
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1281
Inductive properties are a particular pain to prove, because the failure to
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1282
establish an induction step can mean several things:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1283
%
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1284
\begin{enumerate}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1285
\item The property is invalid.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1286
\item The property is valid but is too weak to support the induction step.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1287
\item The property is valid and strong enough; it's just that we haven't found
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1288
the proof yet.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1289
\end{enumerate}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1290
%
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1291
Depending on which scenario applies, we would take the appropriate course of
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1292
action:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1293
%
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1294
\begin{enumerate}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1295
\item Repair the statement of the property so that it becomes valid.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1296
\item Generalize the property and/or prove auxiliary properties.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1297
\item Work harder on a proof.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1298
\end{enumerate}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1299
%
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1300
How can we distinguish between the three scenarios? Nitpick's normal mode of
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1301
operation can often detect scenario 1, and Isabelle's automatic tactics help with
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1302
scenario 3. Using appropriate techniques, it is also often possible to use
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1303
Nitpick to identify scenario 2. Consider the following transition system,
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1304
in which natural numbers represent states:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1305
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1306
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1307
\textbf{inductive\_set}~\textit{reach}~\textbf{where} \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1308
``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1309
``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1310
``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1311
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1312
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1313
We will try to prove that only even numbers are reachable:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1314
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1315
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1316
\textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1317
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1318
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1319
Does this property hold? Nitpick cannot find a counterexample within 30 seconds,
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1320
so let's attempt a proof by induction:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1321
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1322
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1323
\textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1324
\textbf{apply}~\textit{auto}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1325
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1326
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1327
This leaves us in the following proof state:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1328
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1329
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1330
{\slshape goal (2 subgoals): \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1331
\phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1332
\phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1333
}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1334
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1335
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1336
If we run Nitpick on the first subgoal, it still won't find any
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1337
counterexample; and yet, \textit{auto} fails to go further, and \textit{arith}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1338
is helpless. However, notice the $n \in \textit{reach}$ assumption, which
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1339
strengthens the induction hypothesis but is not immediately usable in the proof.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1340
If we remove it and invoke Nitpick, this time we get a counterexample:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1341
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1342
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1343
\textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1344
\textbf{nitpick} \\[2\smallskipamount]
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1345
\slshape Nitpick found a counterexample: \\[2\smallskipamount]
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1346
\hbox{}\qquad Skolem constant: \nopagebreak \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1347
\hbox{}\qquad\qquad $n = 0$
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1348
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1349
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1350
Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1351
to strength the lemma:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1352
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1353
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1354
\textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1355
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1356
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1357
Unfortunately, the proof by induction still gets stuck, except that Nitpick now
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1358
finds the counterexample $n = 2$. We generalize the lemma further to
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1359
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1360
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1361
\textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1362
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1363
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1364
and this time \textit{arith} can finish off the subgoals.
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1365
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1366
A similar technique can be employed for structural induction. The
35180
c57dba973391 more work on Nitpick's support for nonstandard models + fix in model reconstruction
blanchet
parents: 35178
diff changeset
  1367
following mini formalization of full binary trees will serve as illustration:
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1368
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1369
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1370
\textbf{datatype} $\kern1pt'a$~\textit{bin\_tree} = $\textit{Leaf}~{\kern1pt'a}$ $\mid$ $\textit{Branch}$ ``\kern1pt$'a$ \textit{bin\_tree}'' ``\kern1pt$'a$ \textit{bin\_tree}'' \\[2\smallskipamount]
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1371
\textbf{primrec}~\textit{labels}~\textbf{where} \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1372
``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1373
``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount]
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1374
\textbf{primrec}~\textit{swap}~\textbf{where} \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1375
``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1376
\phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1377
``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$''
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1378
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1379
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1380
The \textit{labels} function returns the set of labels occurring on leaves of a
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1381
tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1382
labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1383
obtained by swapping $a$ and $b$:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1384
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1385
\prew
35180
c57dba973391 more work on Nitpick's support for nonstandard models + fix in model reconstruction
blanchet
parents: 35178
diff changeset
  1386
\textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$''
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1387
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1388
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1389
Nitpick can't find any counterexample, so we proceed with induction
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1390
(this time favoring a more structured style):
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1391
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1392
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1393
\textbf{proof}~(\textit{induct}~$t$) \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1394
\hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1395
\textbf{next} \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1396
\hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case}
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1397
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1398
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1399
Nitpick can't find any counterexample at this point either, but it makes the
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1400
following suggestion:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1401
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1402
\prew
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1403
\slshape
35178
29a0e3be0be1 minor fixes to Nitpick
blanchet
parents: 35078
diff changeset
  1404
Hint: To check that the induction hypothesis is general enough, try this command:
35183
8580ba651489 reintroduce structural induction hint in Nitpick
blanchet
parents: 35180
diff changeset
  1405
\textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}].
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1406
\postw
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1407
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1408
If we follow the hint, we get a ``nonstandard'' counterexample for the step:
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1409
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1410
\prew
35180
c57dba973391 more work on Nitpick's support for nonstandard models + fix in model reconstruction
blanchet
parents: 35178
diff changeset
  1411
\slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount]
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1412
\hbox{}\qquad Free variables: \nopagebreak \\
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1413
\hbox{}\qquad\qquad $a = a_1$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1414
\hbox{}\qquad\qquad $b = a_2$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1415
\hbox{}\qquad\qquad $t = \xi_1$ \\
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1416
\hbox{}\qquad\qquad $u = \xi_2$ \\
35180
c57dba973391 more work on Nitpick's support for nonstandard models + fix in model reconstruction
blanchet
parents: 35178
diff changeset
  1417
\hbox{}\qquad Datatype: \nopagebreak \\
c57dba973391 more work on Nitpick's support for nonstandard models + fix in model reconstruction
blanchet
parents: 35178
diff changeset
  1418
\hbox{}\qquad\qquad $\alpha~\textit{btree} = \{\xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \textit{Branch}~\xi_1~\xi_2\}$ \\
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1419
\hbox{}\qquad {\slshape Constants:} \nopagebreak \\
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1420
\hbox{}\qquad\qquad $\textit{labels} = \undef
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1421
    (\!\begin{aligned}[t]%
35180
c57dba973391 more work on Nitpick's support for nonstandard models + fix in model reconstruction
blanchet
parents: 35178
diff changeset
  1422
    & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt]
c57dba973391 more work on Nitpick's support for nonstandard models + fix in model reconstruction
blanchet
parents: 35178
diff changeset
  1423
    & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1424
\hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \undef
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1425
    (\!\begin{aligned}[t]%
35078
6fd1052fe463 optimization to quantifiers in Nitpick's handling of simp rules + renamed some SAT solvers
blanchet
parents: 35072
diff changeset
  1426
    & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt]
35180
c57dba973391 more work on Nitpick's support for nonstandard models + fix in model reconstruction
blanchet
parents: 35178
diff changeset
  1427
    & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount]
34982
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents: 34126
diff changeset
  1428
The existence of a nonstandard model suggests that the induction hypothesis is not general enough or perhaps
7b8c366e34a2 added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanch&#