1 \documentclass[a4paper,12pt]{article} |
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2 \usepackage[T1]{fontenc} |
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3 \usepackage{amsmath} |
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4 \usepackage{amssymb} |
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5 \usepackage[english,french]{babel} |
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6 \usepackage{color} |
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7 \usepackage{footmisc} |
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8 \usepackage{graphicx} |
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9 %\usepackage{mathpazo} |
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10 \usepackage{multicol} |
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11 \usepackage{stmaryrd} |
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12 %\usepackage[scaled=.85]{beramono} |
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13 \usepackage{isabelle,iman,pdfsetup} |
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14 |
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15 %\oddsidemargin=4.6mm |
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16 %\evensidemargin=4.6mm |
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17 %\textwidth=150mm |
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18 %\topmargin=4.6mm |
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19 %\headheight=0mm |
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20 %\headsep=0mm |
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21 %\textheight=234mm |
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22 |
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23 \def\Colon{\mathord{:\mkern-1.5mu:}} |
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24 %\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}} |
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25 %\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}} |
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26 \def\lparr{\mathopen{(\mkern-4mu\mid}} |
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27 \def\rparr{\mathclose{\mid\mkern-4mu)}} |
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28 |
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29 \def\unk{{?}} |
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30 \def\unkef{(\lambda x.\; \unk)} |
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31 \def\undef{(\lambda x.\; \_)} |
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32 %\def\unr{\textit{others}} |
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33 \def\unr{\ldots} |
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34 \def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}} |
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35 \def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}} |
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36 |
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37 \hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick |
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38 counter-example counter-examples data-type data-types co-data-type |
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39 co-data-types in-duc-tive co-in-duc-tive} |
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40 |
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41 \urlstyle{tt} |
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42 |
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43 \begin{document} |
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44 |
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45 %%% TYPESETTING |
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46 %\renewcommand\labelitemi{$\bullet$} |
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47 \renewcommand\labelitemi{\raise.065ex\hbox{\small\textbullet}} |
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48 |
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49 \selectlanguage{english} |
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50 |
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51 \title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex] |
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52 Picking Nits \\[\smallskipamount] |
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53 \Large A User's Guide to Nitpick for Isabelle/HOL} |
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54 \author{\hbox{} \\ |
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55 Jasmin Christian Blanchette \\ |
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56 {\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\ |
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57 \hbox{}} |
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58 |
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59 \maketitle |
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60 |
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61 \tableofcontents |
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62 |
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63 \setlength{\parskip}{.7em plus .2em minus .1em} |
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64 \setlength{\parindent}{0pt} |
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65 \setlength{\abovedisplayskip}{\parskip} |
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66 \setlength{\abovedisplayshortskip}{.9\parskip} |
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67 \setlength{\belowdisplayskip}{\parskip} |
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68 \setlength{\belowdisplayshortskip}{.9\parskip} |
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69 |
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70 % General-purpose enum environment with correct spacing |
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71 \newenvironment{enum}% |
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72 {\begin{list}{}{% |
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73 \setlength{\topsep}{.1\parskip}% |
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74 \setlength{\partopsep}{.1\parskip}% |
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75 \setlength{\itemsep}{\parskip}% |
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76 \advance\itemsep by-\parsep}} |
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77 {\end{list}} |
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78 |
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79 \def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin |
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80 \advance\rightskip by\leftmargin} |
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81 \def\post{\vskip0pt plus1ex\endgroup} |
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82 |
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83 \def\prew{\pre\advance\rightskip by-\leftmargin} |
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84 \def\postw{\post} |
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85 |
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86 \section{Introduction} |
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87 \label{introduction} |
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88 |
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89 Nitpick \cite{blanchette-nipkow-2010} is a counterexample generator for |
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90 Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas |
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91 combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and |
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92 quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized |
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93 first-order relational model finder developed by the Software Design Group at |
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94 MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it |
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95 borrows many ideas and code fragments, but it benefits from Kodkod's |
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96 optimizations and a new encoding scheme. The name Nitpick is shamelessly |
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97 appropriated from a now retired Alloy precursor. |
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98 |
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99 Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative |
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100 theorem and wait a few seconds. Nonetheless, there are situations where knowing |
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101 how it works under the hood and how it reacts to various options helps |
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102 increase the test coverage. This manual also explains how to install the tool on |
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103 your workstation. Should the motivation fail you, think of the many hours of |
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104 hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}. |
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105 |
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106 Another common use of Nitpick is to find out whether the axioms of a locale are |
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107 satisfiable, while the locale is being developed. To check this, it suffices to |
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108 write |
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109 |
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110 \prew |
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111 \textbf{lemma}~``$\textit{False\/}$'' \\ |
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112 \textbf{nitpick}~[\textit{show\_all}] |
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113 \postw |
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114 |
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115 after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick |
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116 must find a model for the axioms. If it finds no model, we have an indication |
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117 that the axioms might be unsatisfiable. |
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118 |
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119 You can also invoke Nitpick from the ``Commands'' submenu of the |
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120 ``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c C-a |
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121 C-n. This is equivalent to entering the \textbf{nitpick} command with no |
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122 arguments in the theory text. |
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123 |
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124 Throughout this manual, we will explicitly invoke the \textbf{nitpick} command. |
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125 Nitpick also provides an automatic mode that can be enabled via the ``Auto |
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126 Nitpick'' option from the ``Isabelle'' menu in Proof General. In this mode, |
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127 Nitpick is run on every newly entered theorem. The time limit for Auto Nitpick |
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128 and other automatic tools can be set using the ``Auto Tools Time Limit'' option. |
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129 |
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130 \newbox\boxA |
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131 \setbox\boxA=\hbox{\texttt{nospam}} |
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132 |
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133 \newcommand\authoremail{\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak |
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134 in.\allowbreak tum.\allowbreak de}} |
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135 |
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136 To run Nitpick, you must also make sure that the theory \textit{Nitpick} is |
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137 imported---this is rarely a problem in practice since it is part of |
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138 \textit{Main}. The examples presented in this manual can be found |
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139 in Isabelle's \texttt{src/HOL/\allowbreak Nitpick\_Examples/Manual\_Nits.thy} theory. |
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140 The known bugs and limitations at the time of writing are listed in |
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141 \S\ref{known-bugs-and-limitations}. Comments and bug reports concerning either |
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142 the tool or the manual should be directed to the author at \authoremail. |
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143 |
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144 \vskip2.5\smallskipamount |
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145 |
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146 \textbf{Acknowledgment.} The author would like to thank Mark Summerfield for |
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147 suggesting several textual improvements. |
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148 % and Perry James for reporting a typo. |
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149 |
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150 \section{Installation} |
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151 \label{installation} |
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152 |
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153 Sledgehammer is part of Isabelle, so you don't need to install it. However, it |
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154 relies on a third-party Kodkod front-end called Kodkodi as well as a Java |
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155 virtual machine called \texttt{java} (version 1.5 or above). |
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156 |
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157 There are two main ways of installing Kodkodi: |
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158 |
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159 \begin{enum} |
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160 \item[\labelitemi] If you installed an official Isabelle package, |
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161 it should already include a properly setup version of Kodkodi. |
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162 |
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163 \item[\labelitemi] If you use a repository or snapshot version of Isabelle, you |
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164 an official Isabelle package, you can download the Isabelle-aware Kodkodi package |
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165 from \url{http://www21.in.tum.de/~blanchet/\#software}. Extract the archive, then add a |
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166 line to your \texttt{\$ISABELLE\_HOME\_USER\slash etc\slash components}% |
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167 \footnote{The variable \texttt{\$ISABELLE\_HOME\_USER} is set by Isabelle at |
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168 startup. Its value can be retrieved by executing \texttt{isabelle} |
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169 \texttt{getenv} \texttt{ISABELLE\_HOME\_USER} on the command line.} |
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170 file with the absolute path to Kodkodi. For example, if the |
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171 \texttt{components} file does not exist yet and you extracted Kodkodi to |
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172 \texttt{/usr/local/kodkodi-1.5.1}, create it with the single line |
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173 |
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174 \prew |
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175 \texttt{/usr/local/kodkodi-1.5.1} |
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176 \postw |
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177 |
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178 (including an invisible newline character) in it. |
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179 \end{enum} |
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180 |
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181 To check whether Kodkodi is successfully installed, you can try out the example |
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182 in \S\ref{propositional-logic}. |
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183 |
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184 \section{First Steps} |
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185 \label{first-steps} |
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186 |
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187 This section introduces Nitpick by presenting small examples. If possible, you |
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188 should try out the examples on your workstation. Your theory file should start |
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189 as follows: |
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190 |
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191 \prew |
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192 \textbf{theory}~\textit{Scratch} \\ |
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193 \textbf{imports}~\textit{Main~Quotient\_Product~RealDef} \\ |
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194 \textbf{begin} |
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195 \postw |
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196 |
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197 The results presented here were obtained using the JNI (Java Native Interface) |
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198 version of MiniSat and with multithreading disabled to reduce nondeterminism. |
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199 This was done by adding the line |
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200 |
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201 \prew |
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202 \textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSat\_JNI}, \,\textit{max\_threads}~= 1] |
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203 \postw |
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204 |
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205 after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with |
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206 Kodkodi and is precompiled for Linux, Mac~OS~X, and Windows (Cygwin). Other SAT |
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207 solvers can also be installed, as explained in \S\ref{optimizations}. If you |
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208 have already configured SAT solvers in Isabelle (e.g., for Refute), these will |
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209 also be available to Nitpick. |
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210 |
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211 \subsection{Propositional Logic} |
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212 \label{propositional-logic} |
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213 |
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214 Let's start with a trivial example from propositional logic: |
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215 |
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216 \prew |
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217 \textbf{lemma}~``$P \longleftrightarrow Q$'' \\ |
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218 \textbf{nitpick} |
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219 \postw |
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220 |
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221 You should get the following output: |
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222 |
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223 \prew |
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224 \slshape |
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225 Nitpick found a counterexample: \\[2\smallskipamount] |
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226 \hbox{}\qquad Free variables: \nopagebreak \\ |
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227 \hbox{}\qquad\qquad $P = \textit{True}$ \\ |
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228 \hbox{}\qquad\qquad $Q = \textit{False}$ |
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229 \postw |
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230 |
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231 Nitpick can also be invoked on individual subgoals, as in the example below: |
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232 |
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233 \prew |
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234 \textbf{apply}~\textit{auto} \\[2\smallskipamount] |
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235 {\slshape goal (2 subgoals): \\ |
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236 \phantom{0}1. $P\,\Longrightarrow\, Q$ \\ |
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237 \phantom{0}2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount] |
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238 \textbf{nitpick}~1 \\[2\smallskipamount] |
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239 {\slshape Nitpick found a counterexample: \\[2\smallskipamount] |
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240 \hbox{}\qquad Free variables: \nopagebreak \\ |
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241 \hbox{}\qquad\qquad $P = \textit{True}$ \\ |
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242 \hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount] |
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243 \textbf{nitpick}~2 \\[2\smallskipamount] |
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244 {\slshape Nitpick found a counterexample: \\[2\smallskipamount] |
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245 \hbox{}\qquad Free variables: \nopagebreak \\ |
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246 \hbox{}\qquad\qquad $P = \textit{False}$ \\ |
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247 \hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount] |
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248 \textbf{oops} |
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249 \postw |
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250 |
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251 \subsection{Type Variables} |
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252 \label{type-variables} |
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253 |
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254 If you are left unimpressed by the previous example, don't worry. The next |
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255 one is more mind- and computer-boggling: |
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256 |
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257 \prew |
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258 \textbf{lemma} ``$x \in A\,\Longrightarrow\, (\textrm{THE}~y.\;y \in A) \in A$'' |
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259 \postw |
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260 \pagebreak[2] %% TYPESETTING |
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261 |
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262 The putative lemma involves the definite description operator, {THE}, presented |
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263 in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The |
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264 operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative |
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265 lemma is merely asserting the indefinite description operator axiom with {THE} |
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266 substituted for {SOME}. |
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267 |
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268 The free variable $x$ and the bound variable $y$ have type $'a$. For formulas |
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269 containing type variables, Nitpick enumerates the possible domains for each type |
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270 variable, up to a given cardinality (10 by default), looking for a finite |
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271 countermodel: |
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272 |
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273 \prew |
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274 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount] |
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275 \slshape |
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276 Trying 10 scopes: \nopagebreak \\ |
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277 \hbox{}\qquad \textit{card}~$'a$~= 1; \\ |
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278 \hbox{}\qquad \textit{card}~$'a$~= 2; \\ |
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279 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount] |
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280 \hbox{}\qquad \textit{card}~$'a$~= 10. \\[2\smallskipamount] |
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281 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount] |
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282 \hbox{}\qquad Free variables: \nopagebreak \\ |
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283 \hbox{}\qquad\qquad $A = \{a_2,\, a_3\}$ \\ |
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284 \hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount] |
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285 Total time: 963 ms. |
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286 \postw |
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287 |
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288 Nitpick found a counterexample in which $'a$ has cardinality 3. (For |
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289 cardinalities 1 and 2, the formula holds.) In the counterexample, the three |
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290 values of type $'a$ are written $a_1$, $a_2$, and $a_3$. |
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291 |
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292 The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option |
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293 \textit{verbose} is enabled. You can specify \textit{verbose} each time you |
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294 invoke \textbf{nitpick}, or you can set it globally using the command |
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295 |
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296 \prew |
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297 \textbf{nitpick\_params} [\textit{verbose}] |
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298 \postw |
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299 |
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300 This command also displays the current default values for all of the options |
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301 supported by Nitpick. The options are listed in \S\ref{option-reference}. |
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302 |
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303 \subsection{Constants} |
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304 \label{constants} |
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305 |
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306 By just looking at Nitpick's output, it might not be clear why the |
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307 counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again, |
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308 this time telling it to show the values of the constants that occur in the |
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309 formula: |
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310 |
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311 \prew |
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312 \textbf{lemma} ``$x \in A\,\Longrightarrow\, (\textrm{THE}~y.\;y \in A) \in A$'' \\ |
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313 \textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount] |
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314 \slshape |
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315 Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount] |
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316 \hbox{}\qquad Free variables: \nopagebreak \\ |
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317 \hbox{}\qquad\qquad $A = \{a_2,\, a_3\}$ \\ |
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318 \hbox{}\qquad\qquad $x = a_3$ \\ |
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319 \hbox{}\qquad Constant: \nopagebreak \\ |
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320 \hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;y \in A = a_1$ |
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321 \postw |
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322 |
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323 As the result of an optimization, Nitpick directly assigned a value to the |
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324 subterm $\textrm{THE}~y.\;y \in A$, rather than to the \textit{The} constant. We |
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325 can disable this optimization by using the command |
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326 |
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327 \prew |
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328 \textbf{nitpick}~[\textit{dont\_specialize},\, \textit{show\_consts}] |
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329 \postw |
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330 |
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331 Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a |
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332 unique $x$ such that'') at the front of our putative lemma's assumption: |
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333 |
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334 \prew |
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335 \textbf{lemma} ``$\exists {!}x.\; x \in A\,\Longrightarrow\, (\textrm{THE}~y.\;y \in A) \in A$'' |
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336 \postw |
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337 |
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338 The fix appears to work: |
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339 |
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340 \prew |
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341 \textbf{nitpick} \\[2\smallskipamount] |
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342 \slshape Nitpick found no counterexample. |
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343 \postw |
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344 |
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345 We can further increase our confidence in the formula by exhausting all |
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346 cardinalities up to 50: |
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347 |
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348 \prew |
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349 \textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--' |
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350 can be entered as \texttt{-} (hyphen) or |
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351 \texttt{\char`\\\char`\<emdash\char`\>}.} \\[2\smallskipamount] |
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352 \slshape Nitpick found no counterexample. |
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353 \postw |
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354 |
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355 Let's see if Sledgehammer can find a proof: |
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356 |
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357 \prew |
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358 \textbf{sledgehammer} \\[2\smallskipamount] |
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359 {\slshape Sledgehammer: ``$e$'' on goal \\ |
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360 Try this: \textbf{by}~(\textit{metis~theI}) (42 ms).} \\ |
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361 \hbox{}\qquad\vdots \\[2\smallskipamount] |
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362 \textbf{by}~(\textit{metis~theI\/}) |
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363 \postw |
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364 |
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365 This must be our lucky day. |
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366 |
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367 \subsection{Skolemization} |
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368 \label{skolemization} |
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369 |
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370 Are all invertible functions onto? Let's find out: |
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371 |
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372 \prew |
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373 \textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x |
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374 \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\ |
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375 \textbf{nitpick} \\[2\smallskipamount] |
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376 \slshape |
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377 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount] |
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378 \hbox{}\qquad Free variable: \nopagebreak \\ |
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379 \hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\ |
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380 \hbox{}\qquad Skolem constants: \nopagebreak \\ |
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381 \hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\ |
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382 \hbox{}\qquad\qquad $y = a_2$ |
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383 \postw |
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384 |
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385 (The Isabelle/HOL notation $f(x := y)$ denotes the function that maps $x$ to $y$ |
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386 and that otherwise behaves like $f$.) |
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387 Although $f$ is the only free variable occurring in the formula, Nitpick also |
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388 displays values for the bound variables $g$ and $y$. These values are available |
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389 to Nitpick because it performs skolemization as a preprocessing step. |
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390 |
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391 In the previous example, skolemization only affected the outermost quantifiers. |
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392 This is not always the case, as illustrated below: |
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393 |
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394 \prew |
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395 \textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\ |
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396 \textbf{nitpick} \\[2\smallskipamount] |
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397 \slshape |
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398 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount] |
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399 \hbox{}\qquad Skolem constant: \nopagebreak \\ |
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400 \hbox{}\qquad\qquad $\lambda x.\; f = |
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401 \undef{}(\!\begin{aligned}[t] |
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402 & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt] |
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403 & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$ |
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404 \postw |
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405 |
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406 The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on |
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407 $x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the |
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408 function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$ |
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409 maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$. |
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410 |
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411 The source of the Skolem constants is sometimes more obscure: |
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412 |
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413 \prew |
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414 \textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\ |
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415 \textbf{nitpick} \\[2\smallskipamount] |
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416 \slshape |
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417 Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount] |
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418 \hbox{}\qquad Free variable: \nopagebreak \\ |
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419 \hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\ |
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420 \hbox{}\qquad Skolem constants: \nopagebreak \\ |
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421 \hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\ |
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422 \hbox{}\qquad\qquad $\mathit{sym}.y = a_1$ |
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423 \postw |
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424 |
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425 What happened here is that Nitpick expanded \textit{sym} to its definition: |
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426 |
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427 \prew |
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428 $\mathit{sym}~r \,\equiv\, |
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429 \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$ |
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430 \postw |
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431 |
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432 As their names suggest, the Skolem constants $\mathit{sym}.x$ and |
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433 $\mathit{sym}.y$ are simply the bound variables $x$ and $y$ |
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434 from \textit{sym}'s definition. |
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435 |
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436 \subsection{Natural Numbers and Integers} |
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437 \label{natural-numbers-and-integers} |
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438 |
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439 Because of the axiom of infinity, the type \textit{nat} does not admit any |
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440 finite models. To deal with this, Nitpick's approach is to consider finite |
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441 subsets $N$ of \textit{nat} and maps all numbers $\notin N$ to the undefined |
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442 value (displayed as `$\unk$'). The type \textit{int} is handled similarly. |
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443 Internally, undefined values lead to a three-valued logic. |
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444 |
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445 Here is an example involving \textit{int\/}: |
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446 |
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447 \prew |
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448 \textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\ |
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449 \textbf{nitpick} \\[2\smallskipamount] |
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450 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
451 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
452 \hbox{}\qquad\qquad $i = 0$ \\ |
|
453 \hbox{}\qquad\qquad $j = 1$ \\ |
|
454 \hbox{}\qquad\qquad $m = 1$ \\ |
|
455 \hbox{}\qquad\qquad $n = 0$ |
|
456 \postw |
|
457 |
|
458 Internally, Nitpick uses either a unary or a binary representation of numbers. |
|
459 The unary representation is more efficient but only suitable for numbers very |
|
460 close to zero. By default, Nitpick attempts to choose the more appropriate |
|
461 encoding by inspecting the formula at hand. This behavior can be overridden by |
|
462 passing either \textit{unary\_ints} or \textit{binary\_ints} as option. For |
|
463 binary notation, the number of bits to use can be specified using |
|
464 the \textit{bits} option. For example: |
|
465 |
|
466 \prew |
|
467 \textbf{nitpick} [\textit{binary\_ints}, \textit{bits}${} = 16$] |
|
468 \postw |
|
469 |
|
470 With infinite types, we don't always have the luxury of a genuine counterexample |
|
471 and must often content ourselves with a potentially spurious one. The tedious |
|
472 task of finding out whether the potentially spurious counterexample is in fact |
|
473 genuine can be delegated to \textit{auto} by passing \textit{check\_potential}. |
|
474 For example: |
|
475 |
|
476 \prew |
|
477 \textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\ |
|
478 \textbf{nitpick} [\textit{card~nat}~= 50, \textit{check\_potential}] \\[2\smallskipamount] |
|
479 \slshape Warning: The conjecture either trivially holds for the given scopes or lies outside Nitpick's supported |
|
480 fragment. Only potentially spurious counterexamples may be found. \\[2\smallskipamount] |
|
481 Nitpick found a potentially spurious counterexample: \\[2\smallskipamount] |
|
482 \hbox{}\qquad Free variable: \nopagebreak \\ |
|
483 \hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount] |
|
484 Confirmation by ``\textit{auto}'': The above counterexample is genuine. |
|
485 \postw |
|
486 |
|
487 You might wonder why the counterexample is first reported as potentially |
|
488 spurious. The root of the problem is that the bound variable in $\forall n.\; |
|
489 \textit{Suc}~n \mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds |
|
490 an $n$ such that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to |
|
491 \textit{False}; but otherwise, it does not know anything about values of $n \ge |
|
492 \textit{card~nat}$ and must therefore evaluate the assumption to~$\unk$, not |
|
493 \textit{True}. Since the assumption can never be satisfied, the putative lemma |
|
494 can never be falsified. |
|
495 |
|
496 Incidentally, if you distrust the so-called genuine counterexamples, you can |
|
497 enable \textit{check\_\allowbreak genuine} to verify them as well. However, be |
|
498 aware that \textit{auto} will usually fail to prove that the counterexample is |
|
499 genuine or spurious. |
|
500 |
|
501 Some conjectures involving elementary number theory make Nitpick look like a |
|
502 giant with feet of clay: |
|
503 |
|
504 \prew |
|
505 \textbf{lemma} ``$P~\textit{Suc\/}$'' \\ |
|
506 \textbf{nitpick} \\[2\smallskipamount] |
|
507 \slshape |
|
508 Nitpick found no counterexample. |
|
509 \postw |
|
510 |
|
511 On any finite set $N$, \textit{Suc} is a partial function; for example, if $N = |
|
512 \{0, 1, \ldots, k\}$, then \textit{Suc} is $\{0 \mapsto 1,\, 1 \mapsto 2,\, |
|
513 \ldots,\, k \mapsto \unk\}$, which evaluates to $\unk$ when passed as |
|
514 argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$. The next |
|
515 example is similar: |
|
516 |
|
517 \prew |
|
518 \textbf{lemma} ``$P~(\textit{op}~{+}\Colon |
|
519 \textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\ |
|
520 \textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount] |
|
521 {\slshape Nitpick found a counterexample:} \\[2\smallskipamount] |
|
522 \hbox{}\qquad Free variable: \nopagebreak \\ |
|
523 \hbox{}\qquad\qquad $P = \unkef(\unkef(0 := \unkef(0 := 0)) := \mathit{False})$ \\[2\smallskipamount] |
|
524 \textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount] |
|
525 {\slshape Nitpick found no counterexample.} |
|
526 \postw |
|
527 |
|
528 The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be |
|
529 $\{0\}$ but becomes partial as soon as we add $1$, because |
|
530 $1 + 1 \notin \{0, 1\}$. |
|
531 |
|
532 Because numbers are infinite and are approximated using a three-valued logic, |
|
533 there is usually no need to systematically enumerate domain sizes. If Nitpick |
|
534 cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very |
|
535 unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$ |
|
536 example above is an exception to this principle.) Nitpick nonetheless enumerates |
|
537 all cardinalities from 1 to 10 for \textit{nat}, mainly because smaller |
|
538 cardinalities are fast to handle and give rise to simpler counterexamples. This |
|
539 is explained in more detail in \S\ref{scope-monotonicity}. |
|
540 |
|
541 \subsection{Inductive Datatypes} |
|
542 \label{inductive-datatypes} |
|
543 |
|
544 Like natural numbers and integers, inductive datatypes with recursive |
|
545 constructors admit no finite models and must be approximated by a subterm-closed |
|
546 subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$, |
|
547 Nitpick looks for all counterexamples that can be built using at most 10 |
|
548 different lists. |
|
549 |
|
550 Let's see with an example involving \textit{hd} (which returns the first element |
|
551 of a list) and $@$ (which concatenates two lists): |
|
552 |
|
553 \prew |
|
554 \textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs\/}$'' \\ |
|
555 \textbf{nitpick} \\[2\smallskipamount] |
|
556 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount] |
|
557 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
558 \hbox{}\qquad\qquad $\textit{xs} = []$ \\ |
|
559 \hbox{}\qquad\qquad $\textit{y} = a_1$ |
|
560 \postw |
|
561 |
|
562 To see why the counterexample is genuine, we enable \textit{show\_consts} |
|
563 and \textit{show\_\allowbreak datatypes}: |
|
564 |
|
565 \prew |
|
566 {\slshape Datatype:} \\ |
|
567 \hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_1],\, [a_1, a_1],\, \unr\}$ \\ |
|
568 {\slshape Constants:} \\ |
|
569 \hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \unkef([] := [a_1, a_1])$ \\ |
|
570 \hbox{}\qquad $\textit{hd} = \unkef([] := a_2,\> [a_1] := a_1,\> [a_1, a_1] := a_1)$ |
|
571 \postw |
|
572 |
|
573 Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value, |
|
574 including $a_2$. |
|
575 |
|
576 The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the |
|
577 append operator whose second argument is fixed to be $[y, y]$. Appending $[a_1, |
|
578 a_1]$ to $[a_1]$ would normally give $[a_1, a_1, a_1]$, but this value is not |
|
579 representable in the subset of $'a$~\textit{list} considered by Nitpick, which |
|
580 is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly, |
|
581 appending $[a_1, a_1]$ to itself gives $\unk$. |
|
582 |
|
583 Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick |
|
584 considers the following subsets: |
|
585 |
|
586 \kern-.5\smallskipamount %% TYPESETTING |
|
587 |
|
588 \prew |
|
589 \begin{multicols}{3} |
|
590 $\{[],\, [a_1],\, [a_2]\}$; \\ |
|
591 $\{[],\, [a_1],\, [a_3]\}$; \\ |
|
592 $\{[],\, [a_2],\, [a_3]\}$; \\ |
|
593 $\{[],\, [a_1],\, [a_1, a_1]\}$; \\ |
|
594 $\{[],\, [a_1],\, [a_2, a_1]\}$; \\ |
|
595 $\{[],\, [a_1],\, [a_3, a_1]\}$; \\ |
|
596 $\{[],\, [a_2],\, [a_1, a_2]\}$; \\ |
|
597 $\{[],\, [a_2],\, [a_2, a_2]\}$; \\ |
|
598 $\{[],\, [a_2],\, [a_3, a_2]\}$; \\ |
|
599 $\{[],\, [a_3],\, [a_1, a_3]\}$; \\ |
|
600 $\{[],\, [a_3],\, [a_2, a_3]\}$; \\ |
|
601 $\{[],\, [a_3],\, [a_3, a_3]\}$. |
|
602 \end{multicols} |
|
603 \postw |
|
604 |
|
605 \kern-2\smallskipamount %% TYPESETTING |
|
606 |
|
607 All subterm-closed subsets of $'a~\textit{list}$ consisting of three values |
|
608 are listed and only those. As an example of a non-subterm-closed subset, |
|
609 consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_2]\}$, and observe |
|
610 that $[a_1, a_2]$ (i.e., $a_1 \mathbin{\#} [a_2]$) has $[a_2] \notin |
|
611 \mathcal{S}$ as a subterm. |
|
612 |
|
613 Here's another m\"ochtegern-lemma that Nitpick can refute without a blink: |
|
614 |
|
615 \prew |
|
616 \textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1 |
|
617 \rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys\/}$'' |
|
618 \\ |
|
619 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount] |
|
620 \slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount] |
|
621 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
622 \hbox{}\qquad\qquad $\textit{xs} = [a_2]$ \\ |
|
623 \hbox{}\qquad\qquad $\textit{ys} = [a_1]$ \\ |
|
624 \hbox{}\qquad Datatypes: \\ |
|
625 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\ |
|
626 \hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_1],\, [a_2],\, \unr\}$ |
|
627 \postw |
|
628 |
|
629 Because datatypes are approximated using a three-valued logic, there is usually |
|
630 no need to systematically enumerate cardinalities: If Nitpick cannot find a |
|
631 genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very |
|
632 unlikely that one could be found for smaller cardinalities. |
|
633 |
|
634 \subsection{Typedefs, Quotient Types, Records, Rationals, and Reals} |
|
635 \label{typedefs-quotient-types-records-rationals-and-reals} |
|
636 |
|
637 Nitpick generally treats types declared using \textbf{typedef} as datatypes |
|
638 whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function. |
|
639 For example: |
|
640 |
|
641 \prew |
|
642 \textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\ |
|
643 \textbf{by}~\textit{blast} \\[2\smallskipamount] |
|
644 \textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\ |
|
645 \textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\ |
|
646 \textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount] |
|
647 \textbf{lemma} ``$\lbrakk A \in X;\> B \in X\rbrakk \,\Longrightarrow\, c \in X$'' \\ |
|
648 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount] |
|
649 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
650 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
651 \hbox{}\qquad\qquad $X = \{\Abs{0},\, \Abs{1}\}$ \\ |
|
652 \hbox{}\qquad\qquad $c = \Abs{2}$ \\ |
|
653 \hbox{}\qquad Datatypes: \\ |
|
654 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\ |
|
655 \hbox{}\qquad\qquad $\textit{three} = \{\Abs{0},\, \Abs{1},\, \Abs{2},\, \unr\}$ |
|
656 \postw |
|
657 |
|
658 In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$. |
|
659 |
|
660 Quotient types are handled in much the same way. The following fragment defines |
|
661 the integer type \textit{my\_int} by encoding the integer $x$ by a pair of |
|
662 natural numbers $(m, n)$ such that $x + n = m$: |
|
663 |
|
664 \prew |
|
665 \textbf{fun} \textit{my\_int\_rel} \textbf{where} \\ |
|
666 ``$\textit{my\_int\_rel}~(x,\, y)~(u,\, v) = (x + v = u + y)$'' \\[2\smallskipamount] |
|
667 % |
|
668 \textbf{quotient\_type}~\textit{my\_int} = ``$\textit{nat} \times \textit{nat\/}$''$\;{/}\;$\textit{my\_int\_rel} \\ |
|
669 \textbf{by}~(\textit{auto simp add\/}:\ \textit{equivp\_def fun\_eq\_iff}) \\[2\smallskipamount] |
|
670 % |
|
671 \textbf{definition}~\textit{add\_raw}~\textbf{where} \\ |
|
672 ``$\textit{add\_raw} \,\equiv\, \lambda(x,\, y)~(u,\, v).\; (x + (u\Colon\textit{nat}), y + (v\Colon\textit{nat}))$'' \\[2\smallskipamount] |
|
673 % |
|
674 \textbf{quotient\_definition} ``$\textit{add\/}\Colon\textit{my\_int} \Rightarrow \textit{my\_int} \Rightarrow \textit{my\_int\/}$'' \textbf{is} \textit{add\_raw} \\[2\smallskipamount] |
|
675 % |
|
676 \textbf{lemma} ``$\textit{add}~x~y = \textit{add}~x~x$'' \\ |
|
677 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount] |
|
678 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
679 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
680 \hbox{}\qquad\qquad $x = \Abs{(0,\, 0)}$ \\ |
|
681 \hbox{}\qquad\qquad $y = \Abs{(0,\, 1)}$ \\ |
|
682 \hbox{}\qquad Datatypes: \\ |
|
683 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, \unr\}$ \\ |
|
684 \hbox{}\qquad\qquad $\textit{nat} \times \textit{nat}~[\textsl{boxed\/}] = \{(0,\, 0),\> (1,\, 0),\> \unr\}$ \\ |
|
685 \hbox{}\qquad\qquad $\textit{my\_int} = \{\Abs{(0,\, 0)},\> \Abs{(0,\, 1)},\> \unr\}$ |
|
686 \postw |
|
687 |
|
688 The values $\Abs{(0,\, 0)}$ and $\Abs{(0,\, 1)}$ represent the |
|
689 integers $0$ and $-1$, respectively. Other representants would have been |
|
690 possible---e.g., $\Abs{(5,\, 5)}$ and $\Abs{(11,\, 12)}$. If we are going to |
|
691 use \textit{my\_int} extensively, it pays off to install a term postprocessor |
|
692 that converts the pair notation to the standard mathematical notation: |
|
693 |
|
694 \prew |
|
695 $\textbf{ML}~\,\{{*} \\ |
|
696 \!\begin{aligned}[t] |
|
697 %& ({*}~\,\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \\[-2pt] |
|
698 %& \phantom{(*}~\,{\rightarrow}\;\textit{term}~\,{*}) \\[-2pt] |
|
699 & \textbf{fun}\,~\textit{my\_int\_postproc}~\_~\_~\_~T~(\textit{Const}~\_~\$~(\textit{Const}~\_~\$~\textit{t1}~\$~\textit{t2\/})) = {} \\[-2pt] |
|
700 & \phantom{fun}\,~\textit{HOLogic.mk\_number}~T~(\textit{snd}~(\textit{HOLogic.dest\_number~t1}) \\[-2pt] |
|
701 & \phantom{fun\,~\textit{HOLogic.mk\_number}~T~(}{-}~\textit{snd}~(\textit{HOLogic.dest\_number~t2\/})) \\[-2pt] |
|
702 & \phantom{fun}\!{\mid}\,~\textit{my\_int\_postproc}~\_~\_~\_~\_~t = t \\[-2pt] |
|
703 {*}\}\end{aligned}$ \\[2\smallskipamount] |
|
704 $\textbf{declaration}~\,\{{*} \\ |
|
705 \!\begin{aligned}[t] |
|
706 & \textit{Nitpick\_Model.register\_term\_postprocessor}~\!\begin{aligned}[t] |
|
707 & @\{\textrm{typ}~\textit{my\_int}\} \\[-2pt] |
|
708 & \textit{my\_int\_postproc}\end{aligned} \\[-2pt] |
|
709 {*}\}\end{aligned}$ |
|
710 \postw |
|
711 |
|
712 Records are handled as datatypes with a single constructor: |
|
713 |
|
714 \prew |
|
715 \textbf{record} \textit{point} = \\ |
|
716 \hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\ |
|
717 \hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount] |
|
718 \textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\ |
|
719 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount] |
|
720 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
721 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
722 \hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\ |
|
723 \hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\ |
|
724 \hbox{}\qquad Datatypes: \\ |
|
725 \hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\ |
|
726 \hbox{}\qquad\qquad $\textit{point} = \{\!\begin{aligned}[t] |
|
727 & \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr, \\[-2pt] %% TYPESETTING |
|
728 & \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr,\, \unr\}\end{aligned}$ |
|
729 \postw |
|
730 |
|
731 Finally, Nitpick provides rudimentary support for rationals and reals using a |
|
732 similar approach: |
|
733 |
|
734 \prew |
|
735 \textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\ |
|
736 \textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount] |
|
737 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
738 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
739 \hbox{}\qquad\qquad $x = 1/2$ \\ |
|
740 \hbox{}\qquad\qquad $y = -1/2$ \\ |
|
741 \hbox{}\qquad Datatypes: \\ |
|
742 \hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\ |
|
743 \hbox{}\qquad\qquad $\textit{int} = \{-3,\, -2,\, -1,\, 0,\, 1,\, 2,\, 3,\, 4,\, \unr\}$ \\ |
|
744 \hbox{}\qquad\qquad $\textit{real} = \{-3/2,\, -1/2,\, 0,\, 1/2,\, 1,\, 2,\, 3,\, 4,\, \unr\}$ |
|
745 \postw |
|
746 |
|
747 \subsection{Inductive and Coinductive Predicates} |
|
748 \label{inductive-and-coinductive-predicates} |
|
749 |
|
750 Inductively defined predicates (and sets) are particularly problematic for |
|
751 counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004} |
|
752 loop forever and Refute~\cite{weber-2008} run out of resources. The crux of |
|
753 the problem is that they are defined using a least fixed-point construction. |
|
754 |
|
755 Nitpick's philosophy is that not all inductive predicates are equal. Consider |
|
756 the \textit{even} predicate below: |
|
757 |
|
758 \prew |
|
759 \textbf{inductive}~\textit{even}~\textbf{where} \\ |
|
760 ``\textit{even}~0'' $\,\mid$ \\ |
|
761 ``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$'' |
|
762 \postw |
|
763 |
|
764 This predicate enjoys the desirable property of being well-founded, which means |
|
765 that the introduction rules don't give rise to infinite chains of the form |
|
766 |
|
767 \prew |
|
768 $\cdots\,\Longrightarrow\, \textit{even}~k'' |
|
769 \,\Longrightarrow\, \textit{even}~k' |
|
770 \,\Longrightarrow\, \textit{even}~k.$ |
|
771 \postw |
|
772 |
|
773 For \textit{even}, this is obvious: Any chain ending at $k$ will be of length |
|
774 $k/2 + 1$: |
|
775 |
|
776 \prew |
|
777 $\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots |
|
778 \,\Longrightarrow\, \textit{even}~(k - 2) |
|
779 \,\Longrightarrow\, \textit{even}~k.$ |
|
780 \postw |
|
781 |
|
782 Wellfoundedness is desirable because it enables Nitpick to use a very efficient |
|
783 fixed-point computation.% |
|
784 \footnote{If an inductive predicate is |
|
785 well-founded, then it has exactly one fixed point, which is simultaneously the |
|
786 least and the greatest fixed point. In these circumstances, the computation of |
|
787 the least fixed point amounts to the computation of an arbitrary fixed point, |
|
788 which can be performed using a straightforward recursive equation.} |
|
789 Moreover, Nitpick can prove wellfoundedness of most well-founded predicates, |
|
790 just as Isabelle's \textbf{function} package usually discharges termination |
|
791 proof obligations automatically. |
|
792 |
|
793 Let's try an example: |
|
794 |
|
795 \prew |
|
796 \textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\ |
|
797 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}, \textit{verbose}] \\[2\smallskipamount] |
|
798 \slshape The inductive predicate ``\textit{even}'' was proved well-founded. |
|
799 Nitpick can compute it efficiently. \\[2\smallskipamount] |
|
800 Trying 1 scope: \\ |
|
801 \hbox{}\qquad \textit{card nat}~= 50. \\[2\smallskipamount] |
|
802 Warning: The conjecture either trivially holds for the given scopes or lies outside Nitpick's supported fragment. Only |
|
803 potentially spurious counterexamples may be found. \\[2\smallskipamount] |
|
804 Nitpick found a potentially spurious counterexample for \textit{card nat}~= 50: \\[2\smallskipamount] |
|
805 \hbox{}\qquad Empty assignment \\[2\smallskipamount] |
|
806 Nitpick could not find a better counterexample. It checked 1 of 1 scope. \\[2\smallskipamount] |
|
807 Total time: 1.62 s. |
|
808 \postw |
|
809 |
|
810 No genuine counterexample is possible because Nitpick cannot rule out the |
|
811 existence of a natural number $n \ge 50$ such that both $\textit{even}~n$ and |
|
812 $\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the |
|
813 existential quantifier: |
|
814 |
|
815 \prew |
|
816 \textbf{lemma} ``$\exists n \mathbin{\le} 49.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\ |
|
817 \textbf{nitpick}~[\textit{card nat}~= 50, \textit{unary\_ints}] \\[2\smallskipamount] |
|
818 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
819 \hbox{}\qquad Empty assignment |
|
820 \postw |
|
821 |
|
822 So far we were blessed by the wellfoundedness of \textit{even}. What happens if |
|
823 we use the following definition instead? |
|
824 |
|
825 \prew |
|
826 \textbf{inductive} $\textit{even}'$ \textbf{where} \\ |
|
827 ``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\ |
|
828 ``$\textit{even}'~2$'' $\,\mid$ \\ |
|
829 ``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$'' |
|
830 \postw |
|
831 |
|
832 This definition is not well-founded: From $\textit{even}'~0$ and |
|
833 $\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the |
|
834 predicates $\textit{even}$ and $\textit{even}'$ are equivalent. |
|
835 |
|
836 Let's check a property involving $\textit{even}'$. To make up for the |
|
837 foreseeable computational hurdles entailed by non-wellfoundedness, we decrease |
|
838 \textit{nat}'s cardinality to a mere 10: |
|
839 |
|
840 \prew |
|
841 \textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\; |
|
842 \lnot\;\textit{even}'~n$'' \\ |
|
843 \textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount] |
|
844 \slshape |
|
845 The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded. |
|
846 Nitpick might need to unroll it. \\[2\smallskipamount] |
|
847 Trying 6 scopes: \\ |
|
848 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\ |
|
849 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\ |
|
850 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\ |
|
851 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\ |
|
852 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\ |
|
853 \hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount] |
|
854 Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount] |
|
855 \hbox{}\qquad Constant: \nopagebreak \\ |
|
856 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\unkef(\!\begin{aligned}[t] |
|
857 & 0 := \unkef(0 := \textit{True},\, 2 := \textit{True}),\, \\[-2pt] |
|
858 & 1 := \unkef(0 := \textit{True},\, 2 := \textit{True},\, 4 := \textit{True}),\, \\[-2pt] |
|
859 & 2 := \unkef(0 := \textit{True},\, 2 := \textit{True},\, 4 := \textit{True},\, \\[-2pt] |
|
860 & \phantom{2 := \unkef(}6 := \textit{True},\, 8 := \textit{True}))\end{aligned}$ \\[2\smallskipamount] |
|
861 Total time: 1.87 s. |
|
862 \postw |
|
863 |
|
864 Nitpick's output is very instructive. First, it tells us that the predicate is |
|
865 unrolled, meaning that it is computed iteratively from the empty set. Then it |
|
866 lists six scopes specifying different bounds on the numbers of iterations:\ 0, |
|
867 1, 2, 4, 8, and~9. |
|
868 |
|
869 The output also shows how each iteration contributes to $\textit{even}'$. The |
|
870 notation $\lambda i.\; \textit{even}'$ indicates that the value of the |
|
871 predicate depends on an iteration counter. Iteration 0 provides the basis |
|
872 elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2 |
|
873 throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further |
|
874 iterations would not contribute any new elements. |
|
875 The predicate $\textit{even}'$ evaluates to either \textit{True} or $\unk$, |
|
876 never \textit{False}. |
|
877 |
|
878 %Some values are marked with superscripted question |
|
879 %marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the |
|
880 %predicate evaluates to $\unk$. |
|
881 |
|
882 When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, 20, 24, and 28 |
|
883 iterations. However, these numbers are bounded by the cardinality of the |
|
884 predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are |
|
885 ever needed to compute the value of a \textit{nat} predicate. You can specify |
|
886 the number of iterations using the \textit{iter} option, as explained in |
|
887 \S\ref{scope-of-search}. |
|
888 |
|
889 In the next formula, $\textit{even}'$ occurs both positively and negatively: |
|
890 |
|
891 \prew |
|
892 \textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\ |
|
893 \textbf{nitpick} [\textit{card nat} = 10, \textit{show\_consts}] \\[2\smallskipamount] |
|
894 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
895 \hbox{}\qquad Free variable: \nopagebreak \\ |
|
896 \hbox{}\qquad\qquad $n = 1$ \\ |
|
897 \hbox{}\qquad Constants: \nopagebreak \\ |
|
898 \hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\unkef(\!\begin{aligned}[t] |
|
899 & 0 := \unkef(0 := \mathit{True},\, 2 := \mathit{True}))\end{aligned}$ \\ |
|
900 \hbox{}\qquad\qquad $\textit{even}' \leq \unkef(\!\begin{aligned}[t] |
|
901 & 0 := \mathit{True},\, 1 := \mathit{False},\, 2 := \mathit{True},\, \\[-2pt] |
|
902 & 4 := \mathit{True},\, 6 := \mathit{True},\, 8 := \mathit{True})\end{aligned}$ |
|
903 \postw |
|
904 |
|
905 Notice the special constraint $\textit{even}' \leq \ldots$ in the output, whose |
|
906 right-hand side represents an arbitrary fixed point (not necessarily the least |
|
907 one). It is used to falsify $\textit{even}'~n$. In contrast, the unrolled |
|
908 predicate is used to satisfy $\textit{even}'~(n - 2)$. |
|
909 |
|
910 Coinductive predicates are handled dually. For example: |
|
911 |
|
912 \prew |
|
913 \textbf{coinductive} \textit{nats} \textbf{where} \\ |
|
914 ``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount] |
|
915 \textbf{lemma} ``$\textit{nats} = (\lambda n.\; n \mathbin\in \{0, 1, 2, 3, 4\})$'' \\ |
|
916 \textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount] |
|
917 \slshape Nitpick found a counterexample: |
|
918 \\[2\smallskipamount] |
|
919 \hbox{}\qquad Constants: \nopagebreak \\ |
|
920 \hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \unkef(0 := \unkef,\, 1 := \unkef,\, 2 := \unkef)$ \\ |
|
921 \hbox{}\qquad\qquad $\textit{nats} \geq \unkef(3 := \textit{True},\, 4 := \textit{False},\, 5 := \textit{True})$ |
|
922 \postw |
|
923 |
|
924 As a special case, Nitpick uses Kodkod's transitive closure operator to encode |
|
925 negative occurrences of non-well-founded ``linear inductive predicates,'' i.e., |
|
926 inductive predicates for which each the predicate occurs in at most one |
|
927 assumption of each introduction rule. For example: |
|
928 |
|
929 \prew |
|
930 \textbf{inductive} \textit{odd} \textbf{where} \\ |
|
931 ``$\textit{odd}~1$'' $\,\mid$ \\ |
|
932 ``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount] |
|
933 \textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\ |
|
934 \textbf{nitpick}~[\textit{card nat} = 4,\, \textit{show\_consts}] \\[2\smallskipamount] |
|
935 \slshape Nitpick found a counterexample: |
|
936 \\[2\smallskipamount] |
|
937 \hbox{}\qquad Free variable: \nopagebreak \\ |
|
938 \hbox{}\qquad\qquad $n = 1$ \\ |
|
939 \hbox{}\qquad Constants: \nopagebreak \\ |
|
940 \hbox{}\qquad\qquad $\textit{even} = (λx. ?)(0 := True, 1 := False, 2 := True, 3 := False)$ \\ |
|
941 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = {}$ \\ |
|
942 \hbox{}\qquad\qquad\quad $\unkef(0 := \textit{False},\, 1 := \textit{True},\, 2 := \textit{False},\, 3 := \textit{False})$ \\ |
|
943 \hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \unkef$\\ |
|
944 \hbox{}\qquad\qquad\quad $( |
|
945 \!\begin{aligned}[t] |
|
946 & 0 := \unkef(0 := \textit{True},\, 1 := \textit{False},\, 2 := \textit{True},\, 3 := \textit{False}), \\[-2pt] |
|
947 & 1 := \unkef(0 := \textit{False},\, 1 := \textit{True},\, 2 := \textit{False},\, 3 := \textit{True}), \\[-2pt] |
|
948 & 2 := \unkef(0 := \textit{False},\, 1 := \textit{False},\, 2 := \textit{True},\, 3 := \textit{False}), \\[-2pt] |
|
949 & 3 := \unkef(0 := \textit{False},\, 1 := \textit{False},\, 2 := \textit{False},\, 3 := \textit{True})) |
|
950 \end{aligned}$ \\ |
|
951 \hbox{}\qquad\qquad $\textit{odd} \leq \unkef(0 := \textit{False},\, 1 := \textit{True},\, 2 := \textit{False},\, 3 := \textit{True})$ |
|
952 \postw |
|
953 |
|
954 \noindent |
|
955 In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and |
|
956 $\textit{odd}_{\textrm{step}}$ is a transition relation that computes new |
|
957 elements from known ones. The set $\textit{odd}$ consists of all the values |
|
958 reachable through the reflexive transitive closure of |
|
959 $\textit{odd}_{\textrm{step}}$ starting with any element from |
|
960 $\textit{odd}_{\textrm{base}}$, namely 1 and 3. Using Kodkod's |
|
961 transitive closure to encode linear predicates is normally either more thorough |
|
962 or more efficient than unrolling (depending on the value of \textit{iter}), but |
|
963 you can disable it by passing the \textit{dont\_star\_linear\_preds} option. |
|
964 |
|
965 \subsection{Coinductive Datatypes} |
|
966 \label{coinductive-datatypes} |
|
967 |
|
968 While Isabelle regrettably lacks a high-level mechanism for defining coinductive |
|
969 datatypes, the \textit{Coinductive\_List} theory from Andreas Lochbihler's |
|
970 \textit{Coinductive} AFP entry \cite{lochbihler-2010} provides a coinductive |
|
971 ``lazy list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick |
|
972 supports these lazy lists seamlessly and provides a hook, described in |
|
973 \S\ref{registration-of-coinductive-datatypes}, to register custom coinductive |
|
974 datatypes. |
|
975 |
|
976 (Co)intuitively, a coinductive datatype is similar to an inductive datatype but |
|
977 allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a, |
|
978 \ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0, |
|
979 1, 2, 3, \ldots]$ can be defined as lazy lists using the |
|
980 $\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and |
|
981 $\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist} |
|
982 \mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors. |
|
983 |
|
984 Although it is otherwise no friend of infinity, Nitpick can find counterexamples |
|
985 involving cyclic lists such as \textit{ps} and \textit{qs} above as well as |
|
986 finite lists: |
|
987 |
|
988 \prew |
|
989 \textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs\/}$'' \\ |
|
990 \textbf{nitpick} \\[2\smallskipamount] |
|
991 \slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount] |
|
992 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
993 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\ |
|
994 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ |
|
995 \postw |
|
996 |
|
997 The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands |
|
998 for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the |
|
999 infinite list $[a_1, a_1, a_1, \ldots]$. |
|
1000 |
|
1001 The next example is more interesting: |
|
1002 |
|
1003 \prew |
|
1004 \textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\, |
|
1005 \textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys\/}$'' \\ |
|
1006 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount] |
|
1007 \slshape The type $'a$ passed the monotonicity test. Nitpick might be able to skip |
|
1008 some scopes. \\[2\smallskipamount] |
|
1009 Trying 10 scopes: \\ |
|
1010 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 1, |
|
1011 and \textit{bisim\_depth}~= 0. \\ |
|
1012 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount] |
|
1013 \hbox{}\qquad \textit{card} $'a$~= 10, \textit{card} ``\kern1pt$'a~\textit{list\/}$''~= 10, |
|
1014 and \textit{bisim\_depth}~= 9. \\[2\smallskipamount] |
|
1015 Nitpick found a counterexample for {\itshape card}~$'a$ = 2, |
|
1016 \textit{card}~``\kern1pt$'a~\textit{llist\/}$''~= 2, and \textit{bisim\_\allowbreak |
|
1017 depth}~= 1: |
|
1018 \\[2\smallskipamount] |
|
1019 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
1020 \hbox{}\qquad\qquad $\textit{a} = a_1$ \\ |
|
1021 \hbox{}\qquad\qquad $\textit{b} = a_2$ \\ |
|
1022 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\ |
|
1023 \hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_2~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega)$ \\[2\smallskipamount] |
|
1024 Total time: 1.11 s. |
|
1025 \postw |
|
1026 |
|
1027 The lazy list $\textit{xs}$ is simply $[a_1, a_1, a_1, \ldots]$, whereas |
|
1028 $\textit{ys}$ is $[a_2, a_1, a_1, a_1, \ldots]$, i.e., a lasso-shaped list with |
|
1029 $[a_2]$ as its stem and $[a_1]$ as its cycle. In general, the list segment |
|
1030 within the scope of the {THE} binder corresponds to the lasso's cycle, whereas |
|
1031 the segment leading to the binder is the stem. |
|
1032 |
|
1033 A salient property of coinductive datatypes is that two objects are considered |
|
1034 equal if and only if they lead to the same observations. For example, the two |
|
1035 lazy lists |
|
1036 % |
|
1037 \begin{gather*} |
|
1038 \textrm{THE}~\omega.\; \omega = \textit{LCons}~a~(\textit{LCons}~b~\omega) \\ |
|
1039 \textit{LCons}~a~(\textrm{THE}~\omega.\; \omega = \textit{LCons}~b~(\textit{LCons}~a~\omega)) |
|
1040 \end{gather*} |
|
1041 % |
|
1042 are identical, because both lead |
|
1043 to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or, |
|
1044 equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This |
|
1045 concept of equality for coinductive datatypes is called bisimulation and is |
|
1046 defined coinductively. |
|
1047 |
|
1048 Internally, Nitpick encodes the coinductive bisimilarity predicate as part of |
|
1049 the Kodkod problem to ensure that distinct objects lead to different |
|
1050 observations. This precaution is somewhat expensive and often unnecessary, so it |
|
1051 can be disabled by setting the \textit{bisim\_depth} option to $-1$. The |
|
1052 bisimilarity check is then performed \textsl{after} the counterexample has been |
|
1053 found to ensure correctness. If this after-the-fact check fails, the |
|
1054 counterexample is tagged as ``quasi genuine'' and Nitpick recommends to try |
|
1055 again with \textit{bisim\_depth} set to a nonnegative integer. |
|
1056 |
|
1057 The next formula illustrates the need for bisimilarity (either as a Kodkod |
|
1058 predicate or as an after-the-fact check) to prevent spurious counterexamples: |
|
1059 |
|
1060 \prew |
|
1061 \textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk |
|
1062 \,\Longrightarrow\, \textit{xs} = \textit{ys\/}$'' \\ |
|
1063 \textbf{nitpick} [\textit{bisim\_depth} = $-1$, \textit{show\_datatypes}] \\[2\smallskipamount] |
|
1064 \slshape Nitpick found a quasi genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount] |
|
1065 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
1066 \hbox{}\qquad\qquad $a = a_1$ \\ |
|
1067 \hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = |
|
1068 \textit{LCons}~a_1~\omega$ \\ |
|
1069 \hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$ \\ |
|
1070 \hbox{}\qquad Codatatype:\strut \nopagebreak \\ |
|
1071 \hbox{}\qquad\qquad $'a~\textit{llist} = |
|
1072 \{\!\begin{aligned}[t] |
|
1073 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega, \\[-2pt] |
|
1074 & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega,\> \unr\}\end{aligned}$ |
|
1075 \\[2\smallskipamount] |
|
1076 Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm |
|
1077 that the counterexample is genuine. \\[2\smallskipamount] |
|
1078 {\upshape\textbf{nitpick}} \\[2\smallskipamount] |
|
1079 \slshape Nitpick found no counterexample. |
|
1080 \postw |
|
1081 |
|
1082 In the first \textbf{nitpick} invocation, the after-the-fact check discovered |
|
1083 that the two known elements of type $'a~\textit{llist}$ are bisimilar, prompting |
|
1084 Nitpick to label the example ``quasi genuine.'' |
|
1085 |
|
1086 A compromise between leaving out the bisimilarity predicate from the Kodkod |
|
1087 problem and performing the after-the-fact check is to specify a lower |
|
1088 nonnegative \textit{bisim\_depth} value than the default one provided by |
|
1089 Nitpick. In general, a value of $K$ means that Nitpick will require all lists to |
|
1090 be distinguished from each other by their prefixes of length $K$. Be aware that |
|
1091 setting $K$ to a too low value can overconstrain Nitpick, preventing it from |
|
1092 finding any counterexamples. |
|
1093 |
|
1094 \subsection{Boxing} |
|
1095 \label{boxing} |
|
1096 |
|
1097 Nitpick normally maps function and product types directly to the corresponding |
|
1098 Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has |
|
1099 cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a |
|
1100 \Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays |
|
1101 off to treat these types in the same way as plain datatypes, by approximating |
|
1102 them by a subset of a given cardinality. This technique is called ``boxing'' and |
|
1103 is particularly useful for functions passed as arguments to other functions, for |
|
1104 high-arity functions, and for large tuples. Under the hood, boxing involves |
|
1105 wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in |
|
1106 isomorphic datatypes, as can be seen by enabling the \textit{debug} option. |
|
1107 |
|
1108 To illustrate boxing, we consider a formalization of $\lambda$-terms represented |
|
1109 using de Bruijn's notation: |
|
1110 |
|
1111 \prew |
|
1112 \textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm} |
|
1113 \postw |
|
1114 |
|
1115 The $\textit{lift}~t~k$ function increments all variables with indices greater |
|
1116 than or equal to $k$ by one: |
|
1117 |
|
1118 \prew |
|
1119 \textbf{primrec} \textit{lift} \textbf{where} \\ |
|
1120 ``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\ |
|
1121 ``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\ |
|
1122 ``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$'' |
|
1123 \postw |
|
1124 |
|
1125 The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if |
|
1126 term $t$ has a loose variable with index $k$ or more: |
|
1127 |
|
1128 \prew |
|
1129 \textbf{primrec}~\textit{loose} \textbf{where} \\ |
|
1130 ``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\ |
|
1131 ``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\ |
|
1132 ``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$'' |
|
1133 \postw |
|
1134 |
|
1135 Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$ |
|
1136 on $t$: |
|
1137 |
|
1138 \prew |
|
1139 \textbf{primrec}~\textit{subst} \textbf{where} \\ |
|
1140 ``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\ |
|
1141 ``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\ |
|
1142 \phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\ |
|
1143 ``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$'' |
|
1144 \postw |
|
1145 |
|
1146 A substitution is a function that maps variable indices to terms. Observe that |
|
1147 $\sigma$ is a function passed as argument and that Nitpick can't optimize it |
|
1148 away, because the recursive call for the \textit{Lam} case involves an altered |
|
1149 version. Also notice the \textit{lift} call, which increments the variable |
|
1150 indices when moving under a \textit{Lam}. |
|
1151 |
|
1152 A reasonable property to expect of substitution is that it should leave closed |
|
1153 terms unchanged. Alas, even this simple property does not hold: |
|
1154 |
|
1155 \pre |
|
1156 \textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\ |
|
1157 \textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount] |
|
1158 \slshape |
|
1159 Trying 10 scopes: \nopagebreak \\ |
|
1160 \hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm\/}$'' = 1; \\ |
|
1161 \hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm\/}$'' = 2; \\ |
|
1162 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount] |
|
1163 \hbox{}\qquad \textit{card~nat}~= 10, \textit{card tm}~= 10, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm\/}$'' = 10. \\[2\smallskipamount] |
|
1164 Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6, |
|
1165 and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm\/}$''~= 6: \\[2\smallskipamount] |
|
1166 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
1167 \hbox{}\qquad\qquad $\sigma = \unkef(\!\begin{aligned}[t] |
|
1168 & 0 := \textit{Var}~0,\> |
|
1169 1 := \textit{Var}~0,\> |
|
1170 2 := \textit{Var}~0, \\[-2pt] |
|
1171 & 3 := \textit{Var}~0,\> |
|
1172 4 := \textit{Var}~0,\> |
|
1173 5 := \textit{Lam}~(\textit{Lam}~(\textit{Var}~0)))\end{aligned}$ \\ |
|
1174 \hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount] |
|
1175 Total time: 3.08 s. |
|
1176 \postw |
|
1177 |
|
1178 Using \textit{eval}, we find out that $\textit{subst}~\sigma~t = |
|
1179 \textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional |
|
1180 $\lambda$-calculus notation, $t$ is |
|
1181 $\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is (wrongly) $\lambda x\, y.\> y$. |
|
1182 The bug is in \textit{subst\/}: The $\textit{lift}~(\sigma~m)~1$ call should be |
|
1183 replaced with $\textit{lift}~(\sigma~m)~0$. |
|
1184 |
|
1185 An interesting aspect of Nitpick's verbose output is that it assigned inceasing |
|
1186 cardinalities from 1 to 10 to the type $\textit{nat} \Rightarrow \textit{tm}$ |
|
1187 of the higher-order argument $\sigma$ of \textit{subst}. |
|
1188 For the formula of interest, knowing 6 values of that type was enough to find |
|
1189 the counterexample. Without boxing, $6^6 = 46\,656$ values must be |
|
1190 considered, a hopeless undertaking: |
|
1191 |
|
1192 \prew |
|
1193 \textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount] |
|
1194 {\slshape Nitpick ran out of time after checking 3 of 10 scopes.} |
|
1195 \postw |
|
1196 |
|
1197 Boxing can be enabled or disabled globally or on a per-type basis using the |
|
1198 \textit{box} option. Nitpick usually performs reasonable choices about which |
|
1199 types should be boxed, but option tweaking sometimes helps. |
|
1200 |
|
1201 %A related optimization, |
|
1202 %``finitization,'' attempts to wrap functions that are constant at all but finitely |
|
1203 %many points (e.g., finite sets); see the documentation for the \textit{finitize} |
|
1204 %option in \S\ref{scope-of-search} for details. |
|
1205 |
|
1206 \subsection{Scope Monotonicity} |
|
1207 \label{scope-monotonicity} |
|
1208 |
|
1209 The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth}, |
|
1210 and \textit{max}) controls which scopes are actually tested. In general, to |
|
1211 exhaust all models below a certain cardinality bound, the number of scopes that |
|
1212 Nitpick must consider increases exponentially with the number of type variables |
|
1213 (and \textbf{typedecl}'d types) occurring in the formula. Given the default |
|
1214 cardinality specification of 1--10, no fewer than $10^4 = 10\,000$ scopes must be |
|
1215 considered for a formula involving $'a$, $'b$, $'c$, and $'d$. |
|
1216 |
|
1217 Fortunately, many formulas exhibit a property called \textsl{scope |
|
1218 monotonicity}, meaning that if the formula is falsifiable for a given scope, |
|
1219 it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}. |
|
1220 |
|
1221 Consider the formula |
|
1222 |
|
1223 \prew |
|
1224 \textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$'' |
|
1225 \postw |
|
1226 |
|
1227 where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type |
|
1228 $'b~\textit{list}$. A priori, Nitpick would need to consider $1\,000$ scopes to |
|
1229 exhaust the specification \textit{card}~= 1--10 (10 cardinalies for $'a$ |
|
1230 $\times$ 10 cardinalities for $'b$ $\times$ 10 cardinalities for the datatypes). |
|
1231 However, our intuition tells us that any counterexample found with a small scope |
|
1232 would still be a counterexample in a larger scope---by simply ignoring the fresh |
|
1233 $'a$ and $'b$ values provided by the larger scope. Nitpick comes to the same |
|
1234 conclusion after a careful inspection of the formula and the relevant |
|
1235 definitions: |
|
1236 |
|
1237 \prew |
|
1238 \textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount] |
|
1239 \slshape |
|
1240 The types $'a$ and $'b$ passed the monotonicity test. |
|
1241 Nitpick might be able to skip some scopes. |
|
1242 \\[2\smallskipamount] |
|
1243 Trying 10 scopes: \\ |
|
1244 \hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1, |
|
1245 \textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$ |
|
1246 \textit{list\/}''~= 1, \\ |
|
1247 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 1, and |
|
1248 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 1. \\ |
|
1249 \hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2, |
|
1250 \textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$ |
|
1251 \textit{list\/}''~= 2, \\ |
|
1252 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 2, and |
|
1253 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 2. \\ |
|
1254 \hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount] |
|
1255 \hbox{}\qquad \textit{card} $'a$~= 10, \textit{card} $'b$~= 10, |
|
1256 \textit{card} \textit{nat}~= 10, \textit{card} ``$('a \times {'}b)$ |
|
1257 \textit{list\/}''~= 10, \\ |
|
1258 \hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 10, and |
|
1259 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 10. |
|
1260 \\[2\smallskipamount] |
|
1261 Nitpick found a counterexample for |
|
1262 \textit{card} $'a$~= 5, \textit{card} $'b$~= 5, |
|
1263 \textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$ |
|
1264 \textit{list\/}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list\/}''~= 5, and |
|
1265 \textit{card} ``\kern1pt$'b$ \textit{list\/}''~= 5: |
|
1266 \\[2\smallskipamount] |
|
1267 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
1268 \hbox{}\qquad\qquad $\textit{xs} = [a_1, a_2]$ \\ |
|
1269 \hbox{}\qquad\qquad $\textit{ys} = [b_1, b_1]$ \\[2\smallskipamount] |
|
1270 Total time: 1.63 s. |
|
1271 \postw |
|
1272 |
|
1273 In theory, it should be sufficient to test a single scope: |
|
1274 |
|
1275 \prew |
|
1276 \textbf{nitpick}~[\textit{card}~= 10] |
|
1277 \postw |
|
1278 |
|
1279 However, this is often less efficient in practice and may lead to overly complex |
|
1280 counterexamples. |
|
1281 |
|
1282 If the monotonicity check fails but we believe that the formula is monotonic (or |
|
1283 we don't mind missing some counterexamples), we can pass the |
|
1284 \textit{mono} option. To convince yourself that this option is risky, |
|
1285 simply consider this example from \S\ref{skolemization}: |
|
1286 |
|
1287 \prew |
|
1288 \textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x |
|
1289 \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\ |
|
1290 \textbf{nitpick} [\textit{mono}] \\[2\smallskipamount] |
|
1291 {\slshape Nitpick found no counterexample.} \\[2\smallskipamount] |
|
1292 \textbf{nitpick} \\[2\smallskipamount] |
|
1293 \slshape |
|
1294 Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\ |
|
1295 \hbox{}\qquad $\vdots$ |
|
1296 \postw |
|
1297 |
|
1298 (It turns out the formula holds if and only if $\textit{card}~'a \le |
|
1299 \textit{card}~'b$.) Although this is rarely advisable, the automatic |
|
1300 monotonicity checks can be disabled by passing \textit{non\_mono} |
|
1301 (\S\ref{optimizations}). |
|
1302 |
|
1303 As insinuated in \S\ref{natural-numbers-and-integers} and |
|
1304 \S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes |
|
1305 are normally monotonic and treated as such. The same is true for record types, |
|
1306 \textit{rat}, and \textit{real}. Thus, given the |
|
1307 cardinality specification 1--10, a formula involving \textit{nat}, \textit{int}, |
|
1308 \textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to |
|
1309 consider only 10~scopes instead of $10^4 = 10\,000$. On the other hand, |
|
1310 \textbf{typedef}s and quotient types are generally nonmonotonic. |
|
1311 |
|
1312 \subsection{Inductive Properties} |
|
1313 \label{inductive-properties} |
|
1314 |
|
1315 Inductive properties are a particular pain to prove, because the failure to |
|
1316 establish an induction step can mean several things: |
|
1317 % |
|
1318 \begin{enumerate} |
|
1319 \item The property is invalid. |
|
1320 \item The property is valid but is too weak to support the induction step. |
|
1321 \item The property is valid and strong enough; it's just that we haven't found |
|
1322 the proof yet. |
|
1323 \end{enumerate} |
|
1324 % |
|
1325 Depending on which scenario applies, we would take the appropriate course of |
|
1326 action: |
|
1327 % |
|
1328 \begin{enumerate} |
|
1329 \item Repair the statement of the property so that it becomes valid. |
|
1330 \item Generalize the property and/or prove auxiliary properties. |
|
1331 \item Work harder on a proof. |
|
1332 \end{enumerate} |
|
1333 % |
|
1334 How can we distinguish between the three scenarios? Nitpick's normal mode of |
|
1335 operation can often detect scenario 1, and Isabelle's automatic tactics help with |
|
1336 scenario 3. Using appropriate techniques, it is also often possible to use |
|
1337 Nitpick to identify scenario 2. Consider the following transition system, |
|
1338 in which natural numbers represent states: |
|
1339 |
|
1340 \prew |
|
1341 \textbf{inductive\_set}~\textit{reach}~\textbf{where} \\ |
|
1342 ``$(4\Colon\textit{nat}) \in \textit{reach\/}$'' $\mid$ \\ |
|
1343 ``$\lbrakk n < 4;\> n \in \textit{reach\/}\rbrakk \,\Longrightarrow\, 3 * n + 1 \in \textit{reach\/}$'' $\mid$ \\ |
|
1344 ``$n \in \textit{reach} \,\Longrightarrow n + 2 \in \textit{reach\/}$'' |
|
1345 \postw |
|
1346 |
|
1347 We will try to prove that only even numbers are reachable: |
|
1348 |
|
1349 \prew |
|
1350 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n$'' |
|
1351 \postw |
|
1352 |
|
1353 Does this property hold? Nitpick cannot find a counterexample within 30 seconds, |
|
1354 so let's attempt a proof by induction: |
|
1355 |
|
1356 \prew |
|
1357 \textbf{apply}~(\textit{induct~set}{:}~\textit{reach\/}) \\ |
|
1358 \textbf{apply}~\textit{auto} |
|
1359 \postw |
|
1360 |
|
1361 This leaves us in the following proof state: |
|
1362 |
|
1363 \prew |
|
1364 {\slshape goal (2 subgoals): \\ |
|
1365 \phantom{0}1. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, n < 4;\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(3 * n)$ \\ |
|
1366 \phantom{0}2. ${\bigwedge}n.\;\, \lbrakk n \in \textit{reach\/};\, 2~\textsl{dvd}~n\rbrakk \,\Longrightarrow\, 2~\textsl{dvd}~\textit{Suc}~(\textit{Suc}~n)$ |
|
1367 } |
|
1368 \postw |
|
1369 |
|
1370 If we run Nitpick on the first subgoal, it still won't find any |
|
1371 counterexample; and yet, \textit{auto} fails to go further, and \textit{arith} |
|
1372 is helpless. However, notice the $n \in \textit{reach}$ assumption, which |
|
1373 strengthens the induction hypothesis but is not immediately usable in the proof. |
|
1374 If we remove it and invoke Nitpick, this time we get a counterexample: |
|
1375 |
|
1376 \prew |
|
1377 \textbf{apply}~(\textit{thin\_tac}~``$n \in \textit{reach\/}$'') \\ |
|
1378 \textbf{nitpick} \\[2\smallskipamount] |
|
1379 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
1380 \hbox{}\qquad Skolem constant: \nopagebreak \\ |
|
1381 \hbox{}\qquad\qquad $n = 0$ |
|
1382 \postw |
|
1383 |
|
1384 Indeed, 0 < 4, 2 divides 0, but 2 does not divide 1. We can use this information |
|
1385 to strength the lemma: |
|
1386 |
|
1387 \prew |
|
1388 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \not= 0$'' |
|
1389 \postw |
|
1390 |
|
1391 Unfortunately, the proof by induction still gets stuck, except that Nitpick now |
|
1392 finds the counterexample $n = 2$. We generalize the lemma further to |
|
1393 |
|
1394 \prew |
|
1395 \textbf{lemma}~``$n \in \textit{reach} \,\Longrightarrow\, 2~\textrm{dvd}~n \mathrel{\lor} n \ge 4$'' |
|
1396 \postw |
|
1397 |
|
1398 and this time \textit{arith} can finish off the subgoals. |
|
1399 |
|
1400 A similar technique can be employed for structural induction. The |
|
1401 following mini formalization of full binary trees will serve as illustration: |
|
1402 |
|
1403 \prew |
|
1404 \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] |
|
1405 \textbf{primrec}~\textit{labels}~\textbf{where} \\ |
|
1406 ``$\textit{labels}~(\textit{Leaf}~a) = \{a\}$'' $\mid$ \\ |
|
1407 ``$\textit{labels}~(\textit{Branch}~t~u) = \textit{labels}~t \mathrel{\cup} \textit{labels}~u$'' \\[2\smallskipamount] |
|
1408 \textbf{primrec}~\textit{swap}~\textbf{where} \\ |
|
1409 ``$\textit{swap}~(\textit{Leaf}~c)~a~b =$ \\ |
|
1410 \phantom{``}$(\textrm{if}~c = a~\textrm{then}~\textit{Leaf}~b~\textrm{else~if}~c = b~\textrm{then}~\textit{Leaf}~a~\textrm{else}~\textit{Leaf}~c)$'' $\mid$ \\ |
|
1411 ``$\textit{swap}~(\textit{Branch}~t~u)~a~b = \textit{Branch}~(\textit{swap}~t~a~b)~(\textit{swap}~u~a~b)$'' |
|
1412 \postw |
|
1413 |
|
1414 The \textit{labels} function returns the set of labels occurring on leaves of a |
|
1415 tree, and \textit{swap} exchanges two labels. Intuitively, if two distinct |
|
1416 labels $a$ and $b$ occur in a tree $t$, they should also occur in the tree |
|
1417 obtained by swapping $a$ and $b$: |
|
1418 |
|
1419 \prew |
|
1420 \textbf{lemma} $``\{a, b\} \subseteq \textit{labels}~t \,\Longrightarrow\, \textit{labels}~(\textit{swap}~t~a~b) = \textit{labels}~t$'' |
|
1421 \postw |
|
1422 |
|
1423 Nitpick can't find any counterexample, so we proceed with induction |
|
1424 (this time favoring a more structured style): |
|
1425 |
|
1426 \prew |
|
1427 \textbf{proof}~(\textit{induct}~$t$) \\ |
|
1428 \hbox{}\quad \textbf{case}~\textit{Leaf}~\textbf{thus}~\textit{?case}~\textbf{by}~\textit{simp} \\ |
|
1429 \textbf{next} \\ |
|
1430 \hbox{}\quad \textbf{case}~$(\textit{Branch}~t~u)$~\textbf{thus} \textit{?case} |
|
1431 \postw |
|
1432 |
|
1433 Nitpick can't find any counterexample at this point either, but it makes the |
|
1434 following suggestion: |
|
1435 |
|
1436 \prew |
|
1437 \slshape |
|
1438 Hint: To check that the induction hypothesis is general enough, try this command: |
|
1439 \textbf{nitpick}~[\textit{non\_std}, \textit{show\_all}]. |
|
1440 \postw |
|
1441 |
|
1442 If we follow the hint, we get a ``nonstandard'' counterexample for the step: |
|
1443 |
|
1444 \prew |
|
1445 \slshape Nitpick found a nonstandard counterexample for \textit{card} $'a$ = 3: \\[2\smallskipamount] |
|
1446 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
1447 \hbox{}\qquad\qquad $a = a_1$ \\ |
|
1448 \hbox{}\qquad\qquad $b = a_2$ \\ |
|
1449 \hbox{}\qquad\qquad $t = \xi_1$ \\ |
|
1450 \hbox{}\qquad\qquad $u = \xi_2$ \\ |
|
1451 \hbox{}\qquad Datatype: \nopagebreak \\ |
|
1452 \hbox{}\qquad\qquad $'a~\textit{bin\_tree} = |
|
1453 \{\!\begin{aligned}[t] |
|
1454 & \xi_1 \mathbin{=} \textit{Branch}~\xi_1~\xi_1,\> \xi_2 \mathbin{=} \textit{Branch}~\xi_2~\xi_2,\> \\[-2pt] |
|
1455 & \textit{Branch}~\xi_1~\xi_2,\> \unr\}\end{aligned}$ \\ |
|
1456 \hbox{}\qquad {\slshape Constants:} \nopagebreak \\ |
|
1457 \hbox{}\qquad\qquad $\textit{labels} = \unkef |
|
1458 (\!\begin{aligned}[t]% |
|
1459 & \xi_1 := \{a_2, a_3\},\> \xi_2 := \{a_1\},\> \\[-2pt] |
|
1460 & \textit{Branch}~\xi_1~\xi_2 := \{a_1, a_2, a_3\})\end{aligned}$ \\ |
|
1461 \hbox{}\qquad\qquad $\lambda x_1.\> \textit{swap}~x_1~a~b = \unkef |
|
1462 (\!\begin{aligned}[t]% |
|
1463 & \xi_1 := \xi_2,\> \xi_2 := \xi_2, \\[-2pt] |
|
1464 & \textit{Branch}~\xi_1~\xi_2 := \xi_2)\end{aligned}$ \\[2\smallskipamount] |
|
1465 The existence of a nonstandard model suggests that the induction hypothesis is not general enough or may even |
|
1466 be wrong. See the Nitpick manual's ``Inductive Properties'' section for details (``\textit{isabelle doc nitpick}''). |
|
1467 \postw |
|
1468 |
|
1469 Reading the Nitpick manual is a most excellent idea. |
|
1470 But what's going on? The \textit{non\_std} option told the tool to look for |
|
1471 nonstandard models of binary trees, which means that new ``nonstandard'' trees |
|
1472 $\xi_1, \xi_2, \ldots$, are now allowed in addition to the standard trees |
|
1473 generated by the \textit{Leaf} and \textit{Branch} constructors.% |
|
1474 \footnote{Notice the similarity between allowing nonstandard trees here and |
|
1475 allowing unreachable states in the preceding example (by removing the ``$n \in |
|
1476 \textit{reach\/}$'' assumption). In both cases, we effectively enlarge the |
|
1477 set of objects over which the induction is performed while doing the step |
|
1478 in order to test the induction hypothesis's strength.} |
|
1479 Unlike standard trees, these new trees contain cycles. We will see later that |
|
1480 every property of acyclic trees that can be proved without using induction also |
|
1481 holds for cyclic trees. Hence, |
|
1482 % |
|
1483 \begin{quote} |
|
1484 \textsl{If the induction |
|
1485 hypothesis is strong enough, the induction step will hold even for nonstandard |
|
1486 objects, and Nitpick won't find any nonstandard counterexample.} |
|
1487 \end{quote} |
|
1488 % |
|
1489 But here the tool find some nonstandard trees $t = \xi_1$ |
|
1490 and $u = \xi_2$ such that $a \notin \textit{labels}~t$, $b \in |
|
1491 \textit{labels}~t$, $a \in \textit{labels}~u$, and $b \notin \textit{labels}~u$. |
|
1492 Because neither tree contains both $a$ and $b$, the induction hypothesis tells |
|
1493 us nothing about the labels of $\textit{swap}~t~a~b$ and $\textit{swap}~u~a~b$, |
|
1494 and as a result we know nothing about the labels of the tree |
|
1495 $\textit{swap}~(\textit{Branch}~t~u)~a~b$, which by definition equals |
|
1496 $\textit{Branch}$ $(\textit{swap}~t~a~b)$ $(\textit{swap}~u~a~b)$, whose |
|
1497 labels are $\textit{labels}$ $(\textit{swap}~t~a~b) \mathrel{\cup} |
|
1498 \textit{labels}$ $(\textit{swap}~u~a~b)$. |
|
1499 |
|
1500 The solution is to ensure that we always know what the labels of the subtrees |
|
1501 are in the inductive step, by covering the cases where $a$ and/or~$b$ is not in |
|
1502 $t$ in the statement of the lemma: |
|
1503 |
|
1504 \prew |
|
1505 \textbf{lemma} ``$\textit{labels}~(\textit{swap}~t~a~b) = {}$ \\ |
|
1506 \phantom{\textbf{lemma} ``}$(\textrm{if}~a \in \textit{labels}~t~\textrm{then}$ \nopagebreak \\ |
|
1507 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~\textit{labels}~t~\textrm{else}~(\textit{labels}~t - \{a\}) \mathrel{\cup} \{b\}$ \\ |
|
1508 \phantom{\textbf{lemma} ``(}$\textrm{else}$ \\ |
|
1509 \phantom{\textbf{lemma} ``(\quad}$\textrm{if}~b \in \textit{labels}~t~\textrm{then}~(\textit{labels}~t - \{b\}) \mathrel{\cup} \{a\}~\textrm{else}~\textit{labels}~t)$'' |
|
1510 \postw |
|
1511 |
|
1512 This time, Nitpick won't find any nonstandard counterexample, and we can perform |
|
1513 the induction step using \textit{auto}. |
|
1514 |
|
1515 \section{Case Studies} |
|
1516 \label{case-studies} |
|
1517 |
|
1518 As a didactic device, the previous section focused mostly on toy formulas whose |
|
1519 validity can easily be assessed just by looking at the formula. We will now |
|
1520 review two somewhat more realistic case studies that are within Nitpick's |
|
1521 reach:\ a context-free grammar modeled by mutually inductive sets and a |
|
1522 functional implementation of AA trees. The results presented in this |
|
1523 section were produced with the following settings: |
|
1524 |
|
1525 \prew |
|
1526 \textbf{nitpick\_params} [\textit{max\_potential}~= 0] |
|
1527 \postw |
|
1528 |
|
1529 \subsection{A Context-Free Grammar} |
|
1530 \label{a-context-free-grammar} |
|
1531 |
|
1532 Our first case study is taken from section 7.4 in the Isabelle tutorial |
|
1533 \cite{isa-tutorial}. The following grammar, originally due to Hopcroft and |
|
1534 Ullman, produces all strings with an equal number of $a$'s and $b$'s: |
|
1535 |
|
1536 \prew |
|
1537 \begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}} |
|
1538 $S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\ |
|
1539 $A$ & $::=$ & $aS \mid bAA$ \\ |
|
1540 $B$ & $::=$ & $bS \mid aBB$ |
|
1541 \end{tabular} |
|
1542 \postw |
|
1543 |
|
1544 The intuition behind the grammar is that $A$ generates all strings with one more |
|
1545 $a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s. |
|
1546 |
|
1547 The alphabet consists exclusively of $a$'s and $b$'s: |
|
1548 |
|
1549 \prew |
|
1550 \textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$ |
|
1551 \postw |
|
1552 |
|
1553 Strings over the alphabet are represented by \textit{alphabet list}s. |
|
1554 Nonterminals in the grammar become sets of strings. The production rules |
|
1555 presented above can be expressed as a mutually inductive definition: |
|
1556 |
|
1557 \prew |
|
1558 \textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\ |
|
1559 \textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\ |
|
1560 \textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\ |
|
1561 \textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\ |
|
1562 \textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\ |
|
1563 \textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\ |
|
1564 \textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$'' |
|
1565 \postw |
|
1566 |
|
1567 The conversion of the grammar into the inductive definition was done manually by |
|
1568 Joe Blow, an underpaid undergraduate student. As a result, some errors might |
|
1569 have sneaked in. |
|
1570 |
|
1571 Debugging faulty specifications is at the heart of Nitpick's \textsl{raison |
|
1572 d'\^etre}. A good approach is to state desirable properties of the specification |
|
1573 (here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s |
|
1574 as $b$'s) and check them with Nitpick. If the properties are correctly stated, |
|
1575 counterexamples will point to bugs in the specification. For our grammar |
|
1576 example, we will proceed in two steps, separating the soundness and the |
|
1577 completeness of the set $S$. First, soundness: |
|
1578 |
|
1579 \prew |
|
1580 \textbf{theorem}~\textit{S\_sound\/}: \\ |
|
1581 ``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] = |
|
1582 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\ |
|
1583 \textbf{nitpick} \\[2\smallskipamount] |
|
1584 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
1585 \hbox{}\qquad Free variable: \nopagebreak \\ |
|
1586 \hbox{}\qquad\qquad $w = [b]$ |
|
1587 \postw |
|
1588 |
|
1589 It would seem that $[b] \in S$. How could this be? An inspection of the |
|
1590 introduction rules reveals that the only rule with a right-hand side of the form |
|
1591 $b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is |
|
1592 \textit{R5}: |
|
1593 |
|
1594 \prew |
|
1595 ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' |
|
1596 \postw |
|
1597 |
|
1598 On closer inspection, we can see that this rule is wrong. To match the |
|
1599 production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try |
|
1600 again: |
|
1601 |
|
1602 \prew |
|
1603 \textbf{nitpick} \\[2\smallskipamount] |
|
1604 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
1605 \hbox{}\qquad Free variable: \nopagebreak \\ |
|
1606 \hbox{}\qquad\qquad $w = [a, a, b]$ |
|
1607 \postw |
|
1608 |
|
1609 Some detective work is necessary to find out what went wrong here. To get $[a, |
|
1610 a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come |
|
1611 from \textit{R6}: |
|
1612 |
|
1613 \prew |
|
1614 ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$'' |
|
1615 \postw |
|
1616 |
|
1617 Now, this formula must be wrong: The same assumption occurs twice, and the |
|
1618 variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in |
|
1619 the assumptions should have been a $w$. |
|
1620 |
|
1621 With the correction made, we don't get any counterexample from Nitpick. Let's |
|
1622 move on and check completeness: |
|
1623 |
|
1624 \prew |
|
1625 \textbf{theorem}~\textit{S\_complete}: \\ |
|
1626 ``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] = |
|
1627 \textit{length}~[x\mathbin{\leftarrow} w.\; x = b] |
|
1628 \longrightarrow w \in S$'' \\ |
|
1629 \textbf{nitpick} \\[2\smallskipamount] |
|
1630 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
1631 \hbox{}\qquad Free variable: \nopagebreak \\ |
|
1632 \hbox{}\qquad\qquad $w = [b, b, a, a]$ |
|
1633 \postw |
|
1634 |
|
1635 Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of |
|
1636 $a$'s and $b$'s. But since our inductive definition passed the soundness check, |
|
1637 the introduction rules we have are probably correct. Perhaps we simply lack an |
|
1638 introduction rule. Comparing the grammar with the inductive definition, our |
|
1639 suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$, |
|
1640 without which the grammar cannot generate two or more $b$'s in a row. So we add |
|
1641 the rule |
|
1642 |
|
1643 \prew |
|
1644 ``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$'' |
|
1645 \postw |
|
1646 |
|
1647 With this last change, we don't get any counterexamples from Nitpick for either |
|
1648 soundness or completeness. We can even generalize our result to cover $A$ and |
|
1649 $B$ as well: |
|
1650 |
|
1651 \prew |
|
1652 \textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\ |
|
1653 ``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\ |
|
1654 ``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\ |
|
1655 ``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\ |
|
1656 \textbf{nitpick} \\[2\smallskipamount] |
|
1657 \slshape Nitpick found no counterexample. |
|
1658 \postw |
|
1659 |
|
1660 \subsection{AA Trees} |
|
1661 \label{aa-trees} |
|
1662 |
|
1663 AA trees are a kind of balanced trees discovered by Arne Andersson that provide |
|
1664 similar performance to red-black trees, but with a simpler implementation |
|
1665 \cite{andersson-1993}. They can be used to store sets of elements equipped with |
|
1666 a total order $<$. We start by defining the datatype and some basic extractor |
|
1667 functions: |
|
1668 |
|
1669 \prew |
|
1670 \textbf{datatype} $'a$~\textit{aa\_tree} = \\ |
|
1671 \hbox{}\quad $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder\/}$'' \textit{nat} ``\kern1pt$'a$ \textit{aa\_tree}'' ``\kern1pt$'a$ \textit{aa\_tree}'' \\[2\smallskipamount] |
|
1672 \textbf{primrec} \textit{data} \textbf{where} \\ |
|
1673 ``$\textit{data}~\Lambda = \unkef$'' $\,\mid$ \\ |
|
1674 ``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount] |
|
1675 \textbf{primrec} \textit{dataset} \textbf{where} \\ |
|
1676 ``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\ |
|
1677 ``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount] |
|
1678 \textbf{primrec} \textit{level} \textbf{where} \\ |
|
1679 ``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\ |
|
1680 ``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount] |
|
1681 \textbf{primrec} \textit{left} \textbf{where} \\ |
|
1682 ``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\ |
|
1683 ``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount] |
|
1684 \textbf{primrec} \textit{right} \textbf{where} \\ |
|
1685 ``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\ |
|
1686 ``$\textit{right}~(N~\_~\_~\_~u) = u$'' |
|
1687 \postw |
|
1688 |
|
1689 The wellformedness criterion for AA trees is fairly complex. Wikipedia states it |
|
1690 as follows \cite{wikipedia-2009-aa-trees}: |
|
1691 |
|
1692 \kern.2\parskip %% TYPESETTING |
|
1693 |
|
1694 \pre |
|
1695 Each node has a level field, and the following invariants must remain true for |
|
1696 the tree to be valid: |
|
1697 |
|
1698 \raggedright |
|
1699 |
|
1700 \kern-.4\parskip %% TYPESETTING |
|
1701 |
|
1702 \begin{enum} |
|
1703 \item[] |
|
1704 \begin{enum} |
|
1705 \item[1.] The level of a leaf node is one. |
|
1706 \item[2.] The level of a left child is strictly less than that of its parent. |
|
1707 \item[3.] The level of a right child is less than or equal to that of its parent. |
|
1708 \item[4.] The level of a right grandchild is strictly less than that of its grandparent. |
|
1709 \item[5.] Every node of level greater than one must have two children. |
|
1710 \end{enum} |
|
1711 \end{enum} |
|
1712 \post |
|
1713 |
|
1714 \kern.4\parskip %% TYPESETTING |
|
1715 |
|
1716 The \textit{wf} predicate formalizes this description: |
|
1717 |
|
1718 \prew |
|
1719 \textbf{primrec} \textit{wf} \textbf{where} \\ |
|
1720 ``$\textit{wf}~\Lambda = \textit{True\/}$'' $\,\mid$ \\ |
|
1721 ``$\textit{wf}~(N~\_~k~t~u) =$ \\ |
|
1722 \phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\ |
|
1723 \phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\ |
|
1724 \phantom{``$($}$\textrm{else}$ \\ |
|
1725 \hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u |
|
1726 \mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k |
|
1727 \mathrel{\land} \textit{level}~u \le k$ \\ |
|
1728 \hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$'' |
|
1729 \postw |
|
1730 |
|
1731 Rebalancing the tree upon insertion and removal of elements is performed by two |
|
1732 auxiliary functions called \textit{skew} and \textit{split}, defined below: |
|
1733 |
|
1734 \prew |
|
1735 \textbf{primrec} \textit{skew} \textbf{where} \\ |
|
1736 ``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\ |
|
1737 ``$\textit{skew}~(N~x~k~t~u) = {}$ \\ |
|
1738 \phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k = |
|
1739 \textit{level}~t~\textrm{then}$ \\ |
|
1740 \phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~ |
|
1741 (\textit{right}~t)~u)$ \\ |
|
1742 \phantom{``(}$\textrm{else}$ \\ |
|
1743 \phantom{``(\quad}$N~x~k~t~u)$'' |
|
1744 \postw |
|
1745 |
|
1746 \prew |
|
1747 \textbf{primrec} \textit{split} \textbf{where} \\ |
|
1748 ``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\ |
|
1749 ``$\textit{split}~(N~x~k~t~u) = {}$ \\ |
|
1750 \phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k = |
|
1751 \textit{level}~(\textit{right}~u)~\textrm{then}$ \\ |
|
1752 \phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~ |
|
1753 (N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\ |
|
1754 \phantom{``(}$\textrm{else}$ \\ |
|
1755 \phantom{``(\quad}$N~x~k~t~u)$'' |
|
1756 \postw |
|
1757 |
|
1758 Performing a \textit{skew} or a \textit{split} should have no impact on the set |
|
1759 of elements stored in the tree: |
|
1760 |
|
1761 \prew |
|
1762 \textbf{theorem}~\textit{dataset\_skew\_split\/}:\\ |
|
1763 ``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\ |
|
1764 ``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\ |
|
1765 \textbf{nitpick} \\[2\smallskipamount] |
|
1766 {\slshape Nitpick ran out of time after checking 9 of 10 scopes.} |
|
1767 \postw |
|
1768 |
|
1769 Furthermore, applying \textit{skew} or \textit{split} on a well-formed tree |
|
1770 should not alter the tree: |
|
1771 |
|
1772 \prew |
|
1773 \textbf{theorem}~\textit{wf\_skew\_split\/}:\\ |
|
1774 ``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\ |
|
1775 ``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\ |
|
1776 \textbf{nitpick} \\[2\smallskipamount] |
|
1777 {\slshape Nitpick found no counterexample.} |
|
1778 \postw |
|
1779 |
|
1780 Insertion is implemented recursively. It preserves the sort order: |
|
1781 |
|
1782 \prew |
|
1783 \textbf{primrec}~\textit{insort} \textbf{where} \\ |
|
1784 ``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\ |
|
1785 ``$\textit{insort}~(N~y~k~t~u)~x =$ \\ |
|
1786 \phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\ |
|
1787 \phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$'' |
|
1788 \postw |
|
1789 |
|
1790 Notice that we deliberately commented out the application of \textit{skew} and |
|
1791 \textit{split}. Let's see if this causes any problems: |
|
1792 |
|
1793 \prew |
|
1794 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\ |
|
1795 \textbf{nitpick} \\[2\smallskipamount] |
|
1796 \slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount] |
|
1797 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
1798 \hbox{}\qquad\qquad $t = N~a_1~1~\Lambda~\Lambda$ \\ |
|
1799 \hbox{}\qquad\qquad $x = a_2$ |
|
1800 \postw |
|
1801 |
|
1802 It's hard to see why this is a counterexample. To improve readability, we will |
|
1803 restrict the theorem to \textit{nat}, so that we don't need to look up the value |
|
1804 of the $\textit{op}~{<}$ constant to find out which element is smaller than the |
|
1805 other. In addition, we will tell Nitpick to display the value of |
|
1806 $\textit{insort}~t~x$ using the \textit{eval} option. This gives |
|
1807 |
|
1808 \prew |
|
1809 \textbf{theorem} \textit{wf\_insort\_nat\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\ |
|
1810 \textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount] |
|
1811 \slshape Nitpick found a counterexample: \\[2\smallskipamount] |
|
1812 \hbox{}\qquad Free variables: \nopagebreak \\ |
|
1813 \hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\ |
|
1814 \hbox{}\qquad\qquad $x = 0$ \\ |
|
1815 \hbox{}\qquad Evaluated term: \\ |
|
1816 \hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$ |
|
1817 \postw |
|
1818 |
|
1819 Nitpick's output reveals that the element $0$ was added as a left child of $1$, |
|
1820 where both nodes have a level of 1. This violates the second AA tree invariant, |
|
1821 which states that a left child's level must be less than its parent's. This |
|
1822 shouldn't come as a surprise, considering that we commented out the tree |
|
1823 rebalancing code. Reintroducing the code seems to solve the problem: |
|
1824 |
|
1825 \prew |
|
1826 \textbf{theorem}~\textit{wf\_insort\/}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\ |
|
1827 \textbf{nitpick} \\[2\smallskipamount] |
|
1828 {\slshape Nitpick ran out of time after checking 8 of 10 scopes.} |
|
1829 \postw |
|
1830 |
|
1831 Insertion should transform the set of elements represented by the tree in the |
|
1832 obvious way: |
|
1833 |
|
1834 \prew |
|
1835 \textbf{theorem} \textit{dataset\_insort\/}:\kern.4em |
|
1836 ``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\ |
|
1837 \textbf{nitpick} \\[2\smallskipamount] |
|
1838 {\slshape Nitpick ran out of time after checking 7 of 10 scopes.} |
|
1839 \postw |
|
1840 |
|
1841 We could continue like this and sketch a full-blown theory of AA trees. Once the |
|
1842 definitions and main theorems are in place and have been thoroughly tested using |
|
1843 Nitpick, we could start working on the proofs. Developing theories this way |
|
1844 usually saves time, because faulty theorems and definitions are discovered much |
|
1845 earlier in the process. |
|
1846 |
|
1847 \section{Option Reference} |
|
1848 \label{option-reference} |
|
1849 |
|
1850 \def\defl{\{} |
|
1851 \def\defr{\}} |
|
1852 |
|
1853 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}} |
|
1854 \def\qty#1{$\left<\textit{#1}\right>$} |
|
1855 \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$} |
|
1856 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\enskip \defl\textit{true}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]} |
|
1857 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\enskip \defl\textit{false}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]} |
|
1858 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\enskip \defl\textit{smart}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]} |
|
1859 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]} |
|
1860 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\enskip \defl\textit{#3}\defr} \nopagebreak\\[\parskip]} |
|
1861 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]} |
|
1862 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]} |
|
1863 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]} |
|
1864 |
|
1865 Nitpick's behavior can be influenced by various options, which can be specified |
|
1866 in brackets after the \textbf{nitpick} command. Default values can be set |
|
1867 using \textbf{nitpick\_\allowbreak params}. For example: |
|
1868 |
|
1869 \prew |
|
1870 \textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60] |
|
1871 \postw |
|
1872 |
|
1873 The options are categorized as follows:\ mode of operation |
|
1874 (\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output |
|
1875 format (\S\ref{output-format}), automatic counterexample checks |
|
1876 (\S\ref{authentication}), optimizations |
|
1877 (\S\ref{optimizations}), and timeouts (\S\ref{timeouts}). |
|
1878 |
|
1879 You can instruct Nitpick to run automatically on newly entered theorems by |
|
1880 enabling the ``Auto Nitpick'' option from the ``Isabelle'' menu in Proof |
|
1881 General. For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation}), |
|
1882 \textit{assms} (\S\ref{mode-of-operation}), and \textit{mono} |
|
1883 (\S\ref{scope-of-search}) are implicitly enabled, \textit{blocking} |
|
1884 (\S\ref{mode-of-operation}), \textit{verbose} (\S\ref{output-format}), and |
|
1885 \textit{debug} (\S\ref{output-format}) are disabled, \textit{max\_threads} |
|
1886 (\S\ref{optimizations}) is taken to be 1, \textit{max\_potential} |
|
1887 (\S\ref{output-format}) is taken to be 0, and \textit{timeout} |
|
1888 (\S\ref{timeouts}) is superseded by the ``Auto Tools Time Limit'' in |
|
1889 Proof General's ``Isabelle'' menu. Nitpick's output is also more concise. |
|
1890 |
|
1891 The number of options can be overwhelming at first glance. Do not let that worry |
|
1892 you: Nitpick's defaults have been chosen so that it almost always does the right |
|
1893 thing, and the most important options have been covered in context in |
|
1894 \S\ref{first-steps}. |
|
1895 |
|
1896 The descriptions below refer to the following syntactic quantities: |
|
1897 |
|
1898 \begin{enum} |
|
1899 \item[\labelitemi] \qtybf{string}: A string. |
|
1900 \item[\labelitemi] \qtybf{string\_list\/}: A space-separated list of strings |
|
1901 (e.g., ``\textit{ichi ni san}''). |
|
1902 \item[\labelitemi] \qtybf{bool\/}: \textit{true} or \textit{false}. |
|
1903 \item[\labelitemi] \qtybf{smart\_bool\/}: \textit{true}, \textit{false}, or \textit{smart}. |
|
1904 \item[\labelitemi] \qtybf{int\/}: An integer. Negative integers are prefixed with a hyphen. |
|
1905 \item[\labelitemi] \qtybf{smart\_int\/}: An integer or \textit{smart}. |
|
1906 \item[\labelitemi] \qtybf{int\_range}: An integer (e.g., 3) or a range |
|
1907 of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<emdash\char`\>}. |
|
1908 \item[\labelitemi] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8). |
|
1909 \item[\labelitemi] \qtybf{float\_or\_none}: An integer (e.g., 60) or floating-point number |
|
1910 (e.g., 0.5) expressing a number of seconds, or the keyword \textit{none} |
|
1911 ($\infty$ seconds). |
|
1912 \item[\labelitemi] \qtybf{const\/}: The name of a HOL constant. |
|
1913 \item[\labelitemi] \qtybf{term}: A HOL term (e.g., ``$f~x$''). |
|
1914 \item[\labelitemi] \qtybf{term\_list\/}: A space-separated list of HOL terms (e.g., |
|
1915 ``$f~x$''~``$g~y$''). |
|
1916 \item[\labelitemi] \qtybf{type}: A HOL type. |
|
1917 \end{enum} |
|
1918 |
|
1919 Default values are indicated in curly brackets (\textrm{\{\}}). Boolean options |
|
1920 have a negated counterpart (e.g., \textit{blocking} vs.\ |
|
1921 \textit{non\_blocking}). When setting them, ``= \textit{true}'' may be omitted. |
|
1922 |
|
1923 \subsection{Mode of Operation} |
|
1924 \label{mode-of-operation} |
|
1925 |
|
1926 \begin{enum} |
|
1927 \optrue{blocking}{non\_blocking} |
|
1928 Specifies whether the \textbf{nitpick} command should operate synchronously. |
|
1929 The asynchronous (non-blocking) mode lets the user start proving the putative |
|
1930 theorem while Nitpick looks for a counterexample, but it can also be more |
|
1931 confusing. For technical reasons, automatic runs currently always block. |
|
1932 |
|
1933 \optrue{falsify}{satisfy} |
|
1934 Specifies whether Nitpick should look for falsifying examples (countermodels) or |
|
1935 satisfying examples (models). This manual assumes throughout that |
|
1936 \textit{falsify} is enabled. |
|
1937 |
|
1938 \opsmart{user\_axioms}{no\_user\_axioms} |
|
1939 Specifies whether the user-defined axioms (specified using |
|
1940 \textbf{axiomatization} and \textbf{axioms}) should be considered. If the option |
|
1941 is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on |
|
1942 the constants that occur in the formula to falsify. The option is implicitly set |
|
1943 to \textit{true} for automatic runs. |
|
1944 |
|
1945 \textbf{Warning:} If the option is set to \textit{true}, Nitpick might |
|
1946 nonetheless ignore some polymorphic axioms. Counterexamples generated under |
|
1947 these conditions are tagged as ``quasi genuine.'' The \textit{debug} |
|
1948 (\S\ref{output-format}) option can be used to find out which axioms were |
|
1949 considered. |
|
1950 |
|
1951 \nopagebreak |
|
1952 {\small See also \textit{assms} (\S\ref{mode-of-operation}) and \textit{debug} |
|
1953 (\S\ref{output-format}).} |
|
1954 |
|
1955 \optrue{assms}{no\_assms} |
|
1956 Specifies whether the relevant assumptions in structured proofs should be |
|
1957 considered. The option is implicitly enabled for automatic runs. |
|
1958 |
|
1959 \nopagebreak |
|
1960 {\small See also \textit{user\_axioms} (\S\ref{mode-of-operation}).} |
|
1961 |
|
1962 \opfalse{overlord}{no\_overlord} |
|
1963 Specifies whether Nitpick should put its temporary files in |
|
1964 \texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for |
|
1965 debugging Nitpick but also unsafe if several instances of the tool are run |
|
1966 simultaneously. The files are identified by the extensions |
|
1967 \texttt{.kki}, \texttt{.cnf}, \texttt{.out}, and |
|
1968 \texttt{.err}; you may safely remove them after Nitpick has run. |
|
1969 |
|
1970 \nopagebreak |
|
1971 {\small See also \textit{debug} (\S\ref{output-format}).} |
|
1972 \end{enum} |
|
1973 |
|
1974 \subsection{Scope of Search} |
|
1975 \label{scope-of-search} |
|
1976 |
|
1977 \begin{enum} |
|
1978 \oparg{card}{type}{int\_seq} |
|
1979 Specifies the sequence of cardinalities to use for a given type. |
|
1980 For free types, and often also for \textbf{typedecl}'d types, it usually makes |
|
1981 sense to specify cardinalities as a range of the form \textit{$1$--$n$}. |
|
1982 |
|
1983 \nopagebreak |
|
1984 {\small See also \textit{box} (\S\ref{scope-of-search}) and \textit{mono} |
|
1985 (\S\ref{scope-of-search}).} |
|
1986 |
|
1987 \opdefault{card}{int\_seq}{\upshape 1--10} |
|
1988 Specifies the default sequence of cardinalities to use. This can be overridden |
|
1989 on a per-type basis using the \textit{card}~\qty{type} option described above. |
|
1990 |
|
1991 \oparg{max}{const}{int\_seq} |
|
1992 Specifies the sequence of maximum multiplicities to use for a given |
|
1993 (co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the |
|
1994 number of distinct values that it can construct. Nonsensical values (e.g., |
|
1995 \textit{max}~[]~$=$~2) are silently repaired. This option is only available for |
|
1996 datatypes equipped with several constructors. |
|
1997 |
|
1998 \opnodefault{max}{int\_seq} |
|
1999 Specifies the default sequence of maximum multiplicities to use for |
|
2000 (co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor |
|
2001 basis using the \textit{max}~\qty{const} option described above. |
|
2002 |
|
2003 \opsmart{binary\_ints}{unary\_ints} |
|
2004 Specifies whether natural numbers and integers should be encoded using a unary |
|
2005 or binary notation. In unary mode, the cardinality fully specifies the subset |
|
2006 used to approximate the type. For example: |
|
2007 % |
|
2008 $$\hbox{\begin{tabular}{@{}rll@{}}% |
|
2009 \textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\ |
|
2010 \textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\ |
|
2011 \textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$% |
|
2012 \end{tabular}}$$ |
|
2013 % |
|
2014 In general: |
|
2015 % |
|
2016 $$\hbox{\begin{tabular}{@{}rll@{}}% |
|
2017 \textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\ |
|
2018 \textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$% |
|
2019 \end{tabular}}$$ |
|
2020 % |
|
2021 In binary mode, the cardinality specifies the number of distinct values that can |
|
2022 be constructed. Each of these value is represented by a bit pattern whose length |
|
2023 is specified by the \textit{bits} (\S\ref{scope-of-search}) option. By default, |
|
2024 Nitpick attempts to choose the more appropriate encoding by inspecting the |
|
2025 formula at hand, preferring the binary notation for problems involving |
|
2026 multiplicative operators or large constants. |
|
2027 |
|
2028 \textbf{Warning:} For technical reasons, Nitpick always reverts to unary for |
|
2029 problems that refer to the types \textit{rat} or \textit{real} or the constants |
|
2030 \textit{Suc}, \textit{gcd}, or \textit{lcm}. |
|
2031 |
|
2032 {\small See also \textit{bits} (\S\ref{scope-of-search}) and |
|
2033 \textit{show\_datatypes} (\S\ref{output-format}).} |
|
2034 |
|
2035 \opdefault{bits}{int\_seq}{\upshape 1,2,3,4,6,8,10,12,14,16} |
|
2036 Specifies the number of bits to use to represent natural numbers and integers in |
|
2037 binary, excluding the sign bit. The minimum is 1 and the maximum is 31. |
|
2038 |
|
2039 {\small See also \textit{binary\_ints} (\S\ref{scope-of-search}).} |
|
2040 |
|
2041 \opargboolorsmart{wf}{const}{non\_wf} |
|
2042 Specifies whether the specified (co)in\-duc\-tively defined predicate is |
|
2043 well-founded. The option can take the following values: |
|
2044 |
|
2045 \begin{enum} |
|
2046 \item[\labelitemi] \textbf{\textit{true}:} Tentatively treat the (co)in\-duc\-tive |
|
2047 predicate as if it were well-founded. Since this is generally not sound when the |
|
2048 predicate is not well-founded, the counterexamples are tagged as ``quasi |
|
2049 genuine.'' |
|
2050 |
|
2051 \item[\labelitemi] \textbf{\textit{false}:} Treat the (co)in\-duc\-tive predicate |
|
2052 as if it were not well-founded. The predicate is then unrolled as prescribed by |
|
2053 the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter} |
|
2054 options. |
|
2055 |
|
2056 \item[\labelitemi] \textbf{\textit{smart}:} Try to prove that the inductive |
|
2057 predicate is well-founded using Isabelle's \textit{lexicographic\_order} and |
|
2058 \textit{size\_change} tactics. If this succeeds (or the predicate occurs with an |
|
2059 appropriate polarity in the formula to falsify), use an efficient fixed-point |
|
2060 equation as specification of the predicate; otherwise, unroll the predicates |
|
2061 according to the \textit{iter}~\qty{const} and \textit{iter} options. |
|
2062 \end{enum} |
|
2063 |
|
2064 \nopagebreak |
|
2065 {\small See also \textit{iter} (\S\ref{scope-of-search}), |
|
2066 \textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout} |
|
2067 (\S\ref{timeouts}).} |
|
2068 |
|
2069 \opsmart{wf}{non\_wf} |
|
2070 Specifies the default wellfoundedness setting to use. This can be overridden on |
|
2071 a per-predicate basis using the \textit{wf}~\qty{const} option above. |
|
2072 |
|
2073 \oparg{iter}{const}{int\_seq} |
|
2074 Specifies the sequence of iteration counts to use when unrolling a given |
|
2075 (co)in\-duc\-tive predicate. By default, unrolling is applied for inductive |
|
2076 predicates that occur negatively and coinductive predicates that occur |
|
2077 positively in the formula to falsify and that cannot be proved to be |
|
2078 well-founded, but this behavior is influenced by the \textit{wf} option. The |
|
2079 iteration counts are automatically bounded by the cardinality of the predicate's |
|
2080 domain. |
|
2081 |
|
2082 {\small See also \textit{wf} (\S\ref{scope-of-search}) and |
|
2083 \textit{star\_linear\_preds} (\S\ref{optimizations}).} |
|
2084 |
|
2085 \opdefault{iter}{int\_seq}{\upshape 0{,}1{,}2{,}4{,}8{,}12{,}16{,}20{,}24{,}28} |
|
2086 Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive |
|
2087 predicates. This can be overridden on a per-predicate basis using the |
|
2088 \textit{iter} \qty{const} option above. |
|
2089 |
|
2090 \opdefault{bisim\_depth}{int\_seq}{\upshape 9} |
|
2091 Specifies the sequence of iteration counts to use when unrolling the |
|
2092 bisimilarity predicate generated by Nitpick for coinductive datatypes. A value |
|
2093 of $-1$ means that no predicate is generated, in which case Nitpick performs an |
|
2094 after-the-fact check to see if the known coinductive datatype values are |
|
2095 bidissimilar. If two values are found to be bisimilar, the counterexample is |
|
2096 tagged as ``quasi genuine.'' The iteration counts are automatically bounded by |
|
2097 the sum of the cardinalities of the coinductive datatypes occurring in the |
|
2098 formula to falsify. |
|
2099 |
|
2100 \opargboolorsmart{box}{type}{dont\_box} |
|
2101 Specifies whether Nitpick should attempt to wrap (``box'') a given function or |
|
2102 product type in an isomorphic datatype internally. Boxing is an effective mean |
|
2103 to reduce the search space and speed up Nitpick, because the isomorphic datatype |
|
2104 is approximated by a subset of the possible function or pair values. |
|
2105 Like other drastic optimizations, it can also prevent the discovery of |
|
2106 counterexamples. The option can take the following values: |
|
2107 |
|
2108 \begin{enum} |
|
2109 \item[\labelitemi] \textbf{\textit{true}:} Box the specified type whenever |
|
2110 practicable. |
|
2111 \item[\labelitemi] \textbf{\textit{false}:} Never box the type. |
|
2112 \item[\labelitemi] \textbf{\textit{smart}:} Box the type only in contexts where it |
|
2113 is likely to help. For example, $n$-tuples where $n > 2$ and arguments to |
|
2114 higher-order functions are good candidates for boxing. |
|
2115 \end{enum} |
|
2116 |
|
2117 \nopagebreak |
|
2118 {\small See also \textit{finitize} (\S\ref{scope-of-search}), \textit{verbose} |
|
2119 (\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}).} |
|
2120 |
|
2121 \opsmart{box}{dont\_box} |
|
2122 Specifies the default boxing setting to use. This can be overridden on a |
|
2123 per-type basis using the \textit{box}~\qty{type} option described above. |
|
2124 |
|
2125 \opargboolorsmart{finitize}{type}{dont\_finitize} |
|
2126 Specifies whether Nitpick should attempt to finitize an infinite datatype. The |
|
2127 option can then take the following values: |
|
2128 |
|
2129 \begin{enum} |
|
2130 \item[\labelitemi] \textbf{\textit{true}:} Finitize the datatype. Since this is |
|
2131 unsound, counterexamples generated under these conditions are tagged as ``quasi |
|
2132 genuine.'' |
|
2133 \item[\labelitemi] \textbf{\textit{false}:} Don't attempt to finitize the datatype. |
|
2134 \item[\labelitemi] \textbf{\textit{smart}:} |
|
2135 If the datatype's constructors don't appear in the problem, perform a |
|
2136 monotonicity analysis to detect whether the datatype can be soundly finitized; |
|
2137 otherwise, don't finitize it. |
|
2138 \end{enum} |
|
2139 |
|
2140 \nopagebreak |
|
2141 {\small See also \textit{box} (\S\ref{scope-of-search}), \textit{mono} |
|
2142 (\S\ref{scope-of-search}), \textit{verbose} (\S\ref{output-format}), and |
|
2143 \textit{debug} (\S\ref{output-format}).} |
|
2144 |
|
2145 \opsmart{finitize}{dont\_finitize} |
|
2146 Specifies the default finitization setting to use. This can be overridden on a |
|
2147 per-type basis using the \textit{finitize}~\qty{type} option described above. |
|
2148 |
|
2149 \opargboolorsmart{mono}{type}{non\_mono} |
|
2150 Specifies whether the given type should be considered monotonic when enumerating |
|
2151 scopes and finitizing types. If the option is set to \textit{smart}, Nitpick |
|
2152 performs a monotonicity check on the type. Setting this option to \textit{true} |
|
2153 can reduce the number of scopes tried, but it can also diminish the chance of |
|
2154 finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}. The |
|
2155 option is implicitly set to \textit{true} for automatic runs. |
|
2156 |
|
2157 \nopagebreak |
|
2158 {\small See also \textit{card} (\S\ref{scope-of-search}), |
|
2159 \textit{finitize} (\S\ref{scope-of-search}), |
|
2160 \textit{merge\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose} |
|
2161 (\S\ref{output-format}).} |
|
2162 |
|
2163 \opsmart{mono}{non\_mono} |
|
2164 Specifies the default monotonicity setting to use. This can be overridden on a |
|
2165 per-type basis using the \textit{mono}~\qty{type} option described above. |
|
2166 |
|
2167 \opfalse{merge\_type\_vars}{dont\_merge\_type\_vars} |
|
2168 Specifies whether type variables with the same sort constraints should be |
|
2169 merged. Setting this option to \textit{true} can reduce the number of scopes |
|
2170 tried and the size of the generated Kodkod formulas, but it also diminishes the |
|
2171 theoretical chance of finding a counterexample. |
|
2172 |
|
2173 {\small See also \textit{mono} (\S\ref{scope-of-search}).} |
|
2174 |
|
2175 \opargbool{std}{type}{non\_std} |
|
2176 Specifies whether the given (recursive) datatype should be given standard |
|
2177 models. Nonstandard models are unsound but can help debug structural induction |
|
2178 proofs, as explained in \S\ref{inductive-properties}. |
|
2179 |
|
2180 \optrue{std}{non\_std} |
|
2181 Specifies the default standardness to use. This can be overridden on a per-type |
|
2182 basis using the \textit{std}~\qty{type} option described above. |
|
2183 \end{enum} |
|
2184 |
|
2185 \subsection{Output Format} |
|
2186 \label{output-format} |
|
2187 |
|
2188 \begin{enum} |
|
2189 \opfalse{verbose}{quiet} |
|
2190 Specifies whether the \textbf{nitpick} command should explain what it does. This |
|
2191 option is useful to determine which scopes are tried or which SAT solver is |
|
2192 used. This option is implicitly disabled for automatic runs. |
|
2193 |
|
2194 \opfalse{debug}{no\_debug} |
|
2195 Specifies whether Nitpick should display additional debugging information beyond |
|
2196 what \textit{verbose} already displays. Enabling \textit{debug} also enables |
|
2197 \textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug} |
|
2198 option is implicitly disabled for automatic runs. |
|
2199 |
|
2200 \nopagebreak |
|
2201 {\small See also \textit{overlord} (\S\ref{mode-of-operation}) and |
|
2202 \textit{batch\_size} (\S\ref{optimizations}).} |
|
2203 |
|
2204 \opfalse{show\_datatypes}{hide\_datatypes} |
|
2205 Specifies whether the subsets used to approximate (co)in\-duc\-tive data\-types should |
|
2206 be displayed as part of counterexamples. Such subsets are sometimes helpful when |
|
2207 investigating whether a potentially spurious counterexample is genuine, but |
|
2208 their potential for clutter is real. |
|
2209 |
|
2210 \optrue{show\_skolems}{hide\_skolem} |
|
2211 Specifies whether the values of Skolem constants should be displayed as part of |
|
2212 counterexamples. Skolem constants correspond to bound variables in the original |
|
2213 formula and usually help us to understand why the counterexample falsifies the |
|
2214 formula. |
|
2215 |
|
2216 \opfalse{show\_consts}{hide\_consts} |
|
2217 Specifies whether the values of constants occurring in the formula (including |
|
2218 its axioms) should be displayed along with any counterexample. These values are |
|
2219 sometimes helpful when investigating why a counterexample is |
|
2220 genuine, but they can clutter the output. |
|
2221 |
|
2222 \opnodefault{show\_all}{bool} |
|
2223 Abbreviation for \textit{show\_datatypes}, \textit{show\_skolems}, and |
|
2224 \textit{show\_consts}. |
|
2225 |
|
2226 \opdefault{max\_potential}{int}{\upshape 1} |
|
2227 Specifies the maximum number of potentially spurious counterexamples to display. |
|
2228 Setting this option to 0 speeds up the search for a genuine counterexample. This |
|
2229 option is implicitly set to 0 for automatic runs. If you set this option to a |
|
2230 value greater than 1, you will need an incremental SAT solver, such as |
|
2231 \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that many of |
|
2232 the counterexamples may be identical. |
|
2233 |
|
2234 \nopagebreak |
|
2235 {\small See also \textit{check\_potential} (\S\ref{authentication}) and |
|
2236 \textit{sat\_solver} (\S\ref{optimizations}).} |
|
2237 |
|
2238 \opdefault{max\_genuine}{int}{\upshape 1} |
|
2239 Specifies the maximum number of genuine counterexamples to display. If you set |
|
2240 this option to a value greater than 1, you will need an incremental SAT solver, |
|
2241 such as \textit{MiniSat\_JNI} (recommended) and \textit{SAT4J}. Be aware that |
|
2242 many of the counterexamples may be identical. |
|
2243 |
|
2244 \nopagebreak |
|
2245 {\small See also \textit{check\_genuine} (\S\ref{authentication}) and |
|
2246 \textit{sat\_solver} (\S\ref{optimizations}).} |
|
2247 |
|
2248 \opnodefault{eval}{term\_list} |
|
2249 Specifies the list of terms whose values should be displayed along with |
|
2250 counterexamples. This option suffers from an ``observer effect'': Nitpick might |
|
2251 find different counterexamples for different values of this option. |
|
2252 |
|
2253 \oparg{atoms}{type}{string\_list} |
|
2254 Specifies the names to use to refer to the atoms of the given type. By default, |
|
2255 Nitpick generates names of the form $a_1, \ldots, a_n$, where $a$ is the first |
|
2256 letter of the type's name. |
|
2257 |
|
2258 \opnodefault{atoms}{string\_list} |
|
2259 Specifies the default names to use to refer to atoms of any type. For example, |
|
2260 to call the three atoms of type ${'}a$ \textit{ichi}, \textit{ni}, and |
|
2261 \textit{san} instead of $a_1$, $a_2$, $a_3$, specify the option |
|
2262 ``\textit{atoms}~${'}a$ = \textit{ichi~ni~san}''. The default names can be |
|
2263 overridden on a per-type basis using the \textit{atoms}~\qty{type} option |
|
2264 described above. |
|
2265 |
|
2266 \oparg{format}{term}{int\_seq} |
|
2267 Specifies how to uncurry the value displayed for a variable or constant. |
|
2268 Uncurrying sometimes increases the readability of the output for high-arity |
|
2269 functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow |
|
2270 {'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow |
|
2271 {'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three |
|
2272 arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow |
|
2273 {'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list |
|
2274 of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an |
|
2275 $n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on; |
|
2276 arguments that are not accounted for are left alone, as if the specification had |
|
2277 been $1,\ldots,1,n_1,\ldots,n_k$. |
|
2278 |
|
2279 \opdefault{format}{int\_seq}{\upshape 1} |
|
2280 Specifies the default format to use. Irrespective of the default format, the |
|
2281 extra arguments to a Skolem constant corresponding to the outer bound variables |
|
2282 are kept separated from the remaining arguments, the \textbf{for} arguments of |
|
2283 an inductive definitions are kept separated from the remaining arguments, and |
|
2284 the iteration counter of an unrolled inductive definition is shown alone. The |
|
2285 default format can be overridden on a per-variable or per-constant basis using |
|
2286 the \textit{format}~\qty{term} option described above. |
|
2287 \end{enum} |
|
2288 |
|
2289 \subsection{Authentication} |
|
2290 \label{authentication} |
|
2291 |
|
2292 \begin{enum} |
|
2293 \opfalse{check\_potential}{trust\_potential} |
|
2294 Specifies whether potentially spurious counterexamples should be given to |
|
2295 Isabelle's \textit{auto} tactic to assess their validity. If a potentially |
|
2296 spurious counterexample is shown to be genuine, Nitpick displays a message to |
|
2297 this effect and terminates. |
|
2298 |
|
2299 \nopagebreak |
|
2300 {\small See also \textit{max\_potential} (\S\ref{output-format}).} |
|
2301 |
|
2302 \opfalse{check\_genuine}{trust\_genuine} |
|
2303 Specifies whether genuine and quasi genuine counterexamples should be given to |
|
2304 Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine'' |
|
2305 counterexample is shown to be spurious, the user is kindly asked to send a bug |
|
2306 report to the author at \authoremail. |
|
2307 |
|
2308 \nopagebreak |
|
2309 {\small See also \textit{max\_genuine} (\S\ref{output-format}).} |
|
2310 |
|
2311 \opnodefault{expect}{string} |
|
2312 Specifies the expected outcome, which must be one of the following: |
|
2313 |
|
2314 \begin{enum} |
|
2315 \item[\labelitemi] \textbf{\textit{genuine}:} Nitpick found a genuine counterexample. |
|
2316 \item[\labelitemi] \textbf{\textit{quasi\_genuine}:} Nitpick found a ``quasi |
|
2317 genuine'' counterexample (i.e., a counterexample that is genuine unless |
|
2318 it contradicts a missing axiom or a dangerous option was used inappropriately). |
|
2319 \item[\labelitemi] \textbf{\textit{potential}:} Nitpick found a potentially |
|
2320 spurious counterexample. |
|
2321 \item[\labelitemi] \textbf{\textit{none}:} Nitpick found no counterexample. |
|
2322 \item[\labelitemi] \textbf{\textit{unknown}:} Nitpick encountered some problem (e.g., |
|
2323 Kodkod ran out of memory). |
|
2324 \end{enum} |
|
2325 |
|
2326 Nitpick emits an error if the actual outcome differs from the expected outcome. |
|
2327 This option is useful for regression testing. |
|
2328 \end{enum} |
|
2329 |
|
2330 \subsection{Optimizations} |
|
2331 \label{optimizations} |
|
2332 |
|
2333 \def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}} |
|
2334 |
|
2335 \sloppy |
|
2336 |
|
2337 \begin{enum} |
|
2338 \opdefault{sat\_solver}{string}{smart} |
|
2339 Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend |
|
2340 to be faster than their Java counterparts, but they can be more difficult to |
|
2341 install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or |
|
2342 \textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1, |
|
2343 you will need an incremental SAT solver, such as \textit{MiniSat\_JNI} |
|
2344 (recommended) or \textit{SAT4J}. |
|
2345 |
|
2346 The supported solvers are listed below: |
|
2347 |
|
2348 \begin{enum} |
|
2349 |
|
2350 \item[\labelitemi] \textbf{\textit{CryptoMiniSat}:} CryptoMiniSat is the winner of |
|
2351 the 2010 SAT Race. To use CryptoMiniSat, set the environment variable |
|
2352 \texttt{CRYPTO\-MINISAT\_}\discretionary{}{}{}\texttt{HOME} to the directory that contains the \texttt{crypto\-minisat} |
|
2353 executable.% |
|
2354 \footnote{Important note for Cygwin users: The path must be specified using |
|
2355 native Windows syntax. Make sure to escape backslashes properly.% |
|
2356 \label{cygwin-paths}} |
|
2357 The \cpp{} sources and executables for Crypto\-Mini\-Sat are available at |
|
2358 \url{http://planete.inrialpes.fr/~soos/}\allowbreak\url{CryptoMiniSat2/index.php}. |
|
2359 Nitpick has been tested with version 2.51. |
|
2360 |
|
2361 \item[\labelitemi] \textbf{\textit{CryptoMiniSat\_JNI}:} The JNI (Java Native |
|
2362 Interface) version of CryptoMiniSat is bundled with Kodkodi and is precompiled |
|
2363 for Linux and Mac~OS~X. It is also available from the Kodkod web site |
|
2364 \cite{kodkod-2009}. |
|
2365 |
|
2366 \item[\labelitemi] \textbf{\textit{Lingeling\_JNI}:} |
|
2367 Lingeling is an efficient solver written in C. The JNI (Java Native Interface) |
|
2368 version of Lingeling is bundled with Kodkodi and is precompiled for Linux and |
|
2369 Mac~OS~X. It is also available from the Kodkod web site \cite{kodkod-2009}. |
|
2370 |
|
2371 \item[\labelitemi] \textbf{\textit{MiniSat}:} MiniSat is an efficient solver |
|
2372 written in \cpp{}. To use MiniSat, set the environment variable |
|
2373 \texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat} |
|
2374 executable.% |
|
2375 \footref{cygwin-paths} |
|
2376 The \cpp{} sources and executables for MiniSat are available at |
|
2377 \url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14 |
|
2378 and 2.2. |
|
2379 |
|
2380 \item[\labelitemi] \textbf{\textit{MiniSat\_JNI}:} The JNI |
|
2381 version of MiniSat is bundled with Kodkodi and is precompiled for Linux, |
|
2382 Mac~OS~X, and Windows (Cygwin). It is also available from the Kodkod web site |
|
2383 \cite{kodkod-2009}. Unlike the standard version of MiniSat, the JNI version can |
|
2384 be used incrementally. |
|
2385 |
|
2386 \item[\labelitemi] \textbf{\textit{zChaff}:} zChaff is an older solver written |
|
2387 in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to |
|
2388 the directory that contains the \texttt{zchaff} executable.% |
|
2389 \footref{cygwin-paths} |
|
2390 The \cpp{} sources and executables for zChaff are available at |
|
2391 \url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with |
|
2392 versions 2004-05-13, 2004-11-15, and 2007-03-12. |
|
2393 |
|
2394 \item[\labelitemi] \textbf{\textit{RSat}:} RSat is an efficient solver written in |
|
2395 \cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the |
|
2396 directory that contains the \texttt{rsat} executable.% |
|
2397 \footref{cygwin-paths} |
|
2398 The \cpp{} sources for RSat are available at |
|
2399 \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been tested with version |
|
2400 2.01. |
|
2401 |
|
2402 \item[\labelitemi] \textbf{\textit{BerkMin}:} BerkMin561 is an efficient solver |
|
2403 written in C. To use BerkMin, set the environment variable |
|
2404 \texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561} |
|
2405 executable.\footref{cygwin-paths} |
|
2406 The BerkMin executables are available at |
|
2407 \url{http://eigold.tripod.com/BerkMin.html}. |
|
2408 |
|
2409 \item[\labelitemi] \textbf{\textit{BerkMin\_Alloy}:} Variant of BerkMin that is |
|
2410 included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this |
|
2411 version of BerkMin, set the environment variable |
|
2412 \texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin} |
|
2413 executable.% |
|
2414 \footref{cygwin-paths} |
|
2415 |
|
2416 \item[\labelitemi] \textbf{\textit{SAT4J}:} SAT4J is a reasonably efficient solver |
|
2417 written in Java that can be used incrementally. It is bundled with Kodkodi and |
|
2418 requires no further installation or configuration steps. Do not attempt to |
|
2419 install the official SAT4J packages, because their API is incompatible with |
|
2420 Kodkod. |
|
2421 |
|
2422 \item[\labelitemi] \textbf{\textit{SAT4J\_Light}:} Variant of SAT4J that is |
|
2423 optimized for small problems. It can also be used incrementally. |
|
2424 |
|
2425 \item[\labelitemi] \textbf{\textit{smart}:} If \textit{sat\_solver} is set to |
|
2426 \textit{smart}, Nitpick selects the first solver among the above that is |
|
2427 recognized by Isabelle. If \textit{verbose} (\S\ref{output-format}) is enabled, |
|
2428 Nitpick displays which SAT solver was chosen. |
|
2429 \end{enum} |
|
2430 \fussy |
|
2431 |
|
2432 \opdefault{batch\_size}{smart\_int}{smart} |
|
2433 Specifies the maximum number of Kodkod problems that should be lumped together |
|
2434 when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems |
|
2435 together ensures that Kodkodi is launched less often, but it makes the verbose |
|
2436 output less readable and is sometimes detrimental to performance. If |
|
2437 \textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if |
|
2438 \textit{debug} (\S\ref{output-format}) is set and 50 otherwise. |
|
2439 |
|
2440 \optrue{destroy\_constrs}{dont\_destroy\_constrs} |
|
2441 Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should |
|
2442 be rewritten to use (automatically generated) discriminators and destructors. |
|
2443 This optimization can drastically reduce the size of the Boolean formulas given |
|
2444 to the SAT solver. |
|
2445 |
|
2446 \nopagebreak |
|
2447 {\small See also \textit{debug} (\S\ref{output-format}).} |
|
2448 |
|
2449 \optrue{specialize}{dont\_specialize} |
|
2450 Specifies whether functions invoked with static arguments should be specialized. |
|
2451 This optimization can drastically reduce the search space, especially for |
|
2452 higher-order functions. |
|
2453 |
|
2454 \nopagebreak |
|
2455 {\small See also \textit{debug} (\S\ref{output-format}) and |
|
2456 \textit{show\_consts} (\S\ref{output-format}).} |
|
2457 |
|
2458 \optrue{star\_linear\_preds}{dont\_star\_linear\_preds} |
|
2459 Specifies whether Nitpick should use Kodkod's transitive closure operator to |
|
2460 encode non-well-founded ``linear inductive predicates,'' i.e., inductive |
|
2461 predicates for which each the predicate occurs in at most one assumption of each |
|
2462 introduction rule. Using the reflexive transitive closure is in principle |
|
2463 equivalent to setting \textit{iter} to the cardinality of the predicate's |
|
2464 domain, but it is usually more efficient. |
|
2465 |
|
2466 {\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug} |
|
2467 (\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).} |
|
2468 |
|
2469 \opnodefault{whack}{term\_list} |
|
2470 Specifies a list of atomic terms (usually constants, but also free and schematic |
|
2471 variables) that should be taken as being $\unk$ (unknown). This can be useful to |
|
2472 reduce the size of the Kodkod problem if you can guess in advance that a |
|
2473 constant might not be needed to find a countermodel. |
|
2474 |
|
2475 {\small See also \textit{debug} (\S\ref{output-format}).} |
|
2476 |
|
2477 \opnodefault{need}{term\_list} |
|
2478 Specifies a list of datatype values (normally ground constructor terms) that |
|
2479 should be part of the subterm-closed subsets used to approximate datatypes. If |
|
2480 you know that a value must necessarily belong to the subset of representable |
|
2481 values that approximates a datatype, specifying it can speed up the search, |
|
2482 especially for high cardinalities. |
|
2483 %By default, Nitpick inspects the conjecture to infer needed datatype values. |
|
2484 |
|
2485 \opsmart{total\_consts}{partial\_consts} |
|
2486 Specifies whether constants occurring in the problem other than constructors can |
|
2487 be assumed to be considered total for the representable values that approximate |
|
2488 a datatype. This option is highly incomplete; it should be used only for |
|
2489 problems that do not construct datatype values explicitly. Since this option is |
|
2490 (in rare cases) unsound, counterexamples generated under these conditions are |
|
2491 tagged as ``quasi genuine.'' |
|
2492 |
|
2493 \opdefault{datatype\_sym\_break}{int}{\upshape 5} |
|
2494 Specifies an upper bound on the number of datatypes for which Nitpick generates |
|
2495 symmetry breaking predicates. Symmetry breaking can speed up the SAT solver |
|
2496 considerably, especially for unsatisfiable problems, but too much of it can slow |
|
2497 it down. |
|
2498 |
|
2499 \opdefault{kodkod\_sym\_break}{int}{\upshape 15} |
|
2500 Specifies an upper bound on the number of relations for which Kodkod generates |
|
2501 symmetry breaking predicates. Symmetry breaking can speed up the SAT solver |
|
2502 considerably, especially for unsatisfiable problems, but too much of it can slow |
|
2503 it down. |
|
2504 |
|
2505 \optrue{peephole\_optim}{no\_peephole\_optim} |
|
2506 Specifies whether Nitpick should simplify the generated Kodkod formulas using a |
|
2507 peephole optimizer. These optimizations can make a significant difference. |
|
2508 Unless you are tracking down a bug in Nitpick or distrust the peephole |
|
2509 optimizer, you should leave this option enabled. |
|
2510 |
|
2511 \opdefault{max\_threads}{int}{\upshape 0} |
|
2512 Specifies the maximum number of threads to use in Kodkod. If this option is set |
|
2513 to 0, Kodkod will compute an appropriate value based on the number of processor |
|
2514 cores available. The option is implicitly set to 1 for automatic runs. |
|
2515 |
|
2516 \nopagebreak |
|
2517 {\small See also \textit{batch\_size} (\S\ref{optimizations}) and |
|
2518 \textit{timeout} (\S\ref{timeouts}).} |
|
2519 \end{enum} |
|
2520 |
|
2521 \subsection{Timeouts} |
|
2522 \label{timeouts} |
|
2523 |
|
2524 \begin{enum} |
|
2525 \opdefault{timeout}{float\_or\_none}{\upshape 30} |
|
2526 Specifies the maximum number of seconds that the \textbf{nitpick} command should |
|
2527 spend looking for a counterexample. Nitpick tries to honor this constraint as |
|
2528 well as it can but offers no guarantees. For automatic runs, |
|
2529 \textit{timeout} is ignored; instead, Auto Quickcheck and Auto Nitpick share |
|
2530 a time slot whose length is specified by the ``Auto Counterexample Time |
|
2531 Limit'' option in Proof General. |
|
2532 |
|
2533 \nopagebreak |
|
2534 {\small See also \textit{max\_threads} (\S\ref{optimizations}).} |
|
2535 |
|
2536 \opdefault{tac\_timeout}{float\_or\_none}{\upshape 0.5} |
|
2537 Specifies the maximum number of seconds that should be used by internal |
|
2538 tactics---\textit{lexicographic\_order} and \textit{size\_change} when checking |
|
2539 whether a (co)in\-duc\-tive predicate is well-founded, \textit{auto} tactic when |
|
2540 checking a counterexample, or the monotonicity inference. Nitpick tries to honor |
|
2541 this constraint but offers no guarantees. |
|
2542 |
|
2543 \nopagebreak |
|
2544 {\small See also \textit{wf} (\S\ref{scope-of-search}), |
|
2545 \textit{mono} (\S\ref{scope-of-search}), |
|
2546 \textit{check\_potential} (\S\ref{authentication}), |
|
2547 and \textit{check\_genuine} (\S\ref{authentication}).} |
|
2548 \end{enum} |
|
2549 |
|
2550 \section{Attribute Reference} |
|
2551 \label{attribute-reference} |
|
2552 |
|
2553 Nitpick needs to consider the definitions of all constants occurring in a |
|
2554 formula in order to falsify it. For constants introduced using the |
|
2555 \textbf{definition} command, the definition is simply the associated |
|
2556 \textit{\_def} axiom. In contrast, instead of using the internal representation |
|
2557 of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and |
|
2558 \textbf{nominal\_primrec} packages, Nitpick relies on the more natural |
|
2559 equational specification entered by the user. |
|
2560 |
|
2561 Behind the scenes, Isabelle's built-in packages and theories rely on the |
|
2562 following attributes to affect Nitpick's behavior: |
|
2563 |
|
2564 \begin{enum} |
|
2565 \flushitem{\textit{nitpick\_unfold}} |
|
2566 |
|
2567 \nopagebreak |
|
2568 This attribute specifies an equation that Nitpick should use to expand a |
|
2569 constant. The equation should be logically equivalent to the constant's actual |
|
2570 definition and should be of the form |
|
2571 |
|
2572 \qquad $c~{?}x_1~\ldots~{?}x_n \,=\, t$, |
|
2573 |
|
2574 or |
|
2575 |
|
2576 \qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$, |
|
2577 |
|
2578 where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in |
|
2579 $t$. Each occurrence of $c$ in the problem is expanded to $\lambda x_1\,\ldots |
|
2580 x_n.\; t$. |
|
2581 |
|
2582 \flushitem{\textit{nitpick\_simp}} |
|
2583 |
|
2584 \nopagebreak |
|
2585 This attribute specifies the equations that constitute the specification of a |
|
2586 constant. The \textbf{primrec}, \textbf{function}, and |
|
2587 \textbf{nominal\_\allowbreak primrec} packages automatically attach this |
|
2588 attribute to their \textit{simps} rules. The equations must be of the form |
|
2589 |
|
2590 \qquad $c~t_1~\ldots\ t_n \;\bigl[{=}\; u\bigr]$ |
|
2591 |
|
2592 or |
|
2593 |
|
2594 \qquad $c~t_1~\ldots\ t_n \,\equiv\, u.$ |
|
2595 |
|
2596 \flushitem{\textit{nitpick\_psimp}} |
|
2597 |
|
2598 \nopagebreak |
|
2599 This attribute specifies the equations that constitute the partial specification |
|
2600 of a constant. The \textbf{function} package automatically attaches this |
|
2601 attribute to its \textit{psimps} rules. The conditional equations must be of the |
|
2602 form |
|
2603 |
|
2604 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \;\bigl[{=}\; u\bigr]$ |
|
2605 |
|
2606 or |
|
2607 |
|
2608 \qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,\equiv\, u$. |
|
2609 |
|
2610 \flushitem{\textit{nitpick\_choice\_spec}} |
|
2611 |
|
2612 \nopagebreak |
|
2613 This attribute specifies the (free-form) specification of a constant defined |
|
2614 using the \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command. |
|
2615 \end{enum} |
|
2616 |
|
2617 When faced with a constant, Nitpick proceeds as follows: |
|
2618 |
|
2619 \begin{enum} |
|
2620 \item[1.] If the \textit{nitpick\_simp} set associated with the constant |
|
2621 is not empty, Nitpick uses these rules as the specification of the constant. |
|
2622 |
|
2623 \item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with |
|
2624 the constant is not empty, it uses these rules as the specification of the |
|
2625 constant. |
|
2626 |
|
2627 \item[3.] Otherwise, if the constant was defined using the |
|
2628 \hbox{(\textbf{ax\_})}\allowbreak\textbf{specification} command and the |
|
2629 \textit{nitpick\_choice\_spec} set associated with the constant is not empty, it |
|
2630 uses these theorems as the specification of the constant. |
|
2631 |
|
2632 \item[4.] Otherwise, it looks up the definition of the constant. If the |
|
2633 \textit{nitpick\_unfold} set associated with the constant is not empty, it uses |
|
2634 the latest rule added to the set as the definition of the constant; otherwise it |
|
2635 uses the actual definition axiom. |
|
2636 |
|
2637 \begin{enum} |
|
2638 \item[1.] If the definition is of the form |
|
2639 |
|
2640 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$ |
|
2641 |
|
2642 or |
|
2643 |
|
2644 \qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{gfp}~(\lambda f.\; t).$ |
|
2645 |
|
2646 Nitpick assumes that the definition was made using a (co)inductive package |
|
2647 based on the user-specified introduction rules registered in Isabelle's internal |
|
2648 \textit{Spec\_Rules} table. The tool uses the introduction rules to ascertain |
|
2649 whether the definition is well-founded and the definition to generate a |
|
2650 fixed-point equation or an unrolled equation. |
|
2651 |
|
2652 \item[2.] If the definition is compact enough, the constant is \textsl{unfolded} |
|
2653 wherever it appears; otherwise, it is defined equationally, as with |
|
2654 the \textit{nitpick\_simp} attribute. |
|
2655 \end{enum} |
|
2656 \end{enum} |
|
2657 |
|
2658 As an illustration, consider the inductive definition |
|
2659 |
|
2660 \prew |
|
2661 \textbf{inductive}~\textit{odd}~\textbf{where} \\ |
|
2662 ``\textit{odd}~1'' $\,\mid$ \\ |
|
2663 ``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$'' |
|
2664 \postw |
|
2665 |
|
2666 By default, Nitpick uses the \textit{lfp}-based definition in conjunction with |
|
2667 the introduction rules. To override this, you can specify an alternative |
|
2668 definition as follows: |
|
2669 |
|
2670 \prew |
|
2671 \textbf{lemma} $\mathit{odd\_alt\_unfold}$ [\textit{nitpick\_unfold}]:\kern.4em ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$'' |
|
2672 \postw |
|
2673 |
|
2674 Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2 |
|
2675 = 1$. Alternatively, you can specify an equational specification of the constant: |
|
2676 |
|
2677 \prew |
|
2678 \textbf{lemma} $\mathit{odd\_simp}$ [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$'' |
|
2679 \postw |
|
2680 |
|
2681 Such tweaks should be done with great care, because Nitpick will assume that the |
|
2682 constant is completely defined by its equational specification. For example, if |
|
2683 you make ``$\textit{odd}~(2 * k + 1)$'' a \textit{nitpick\_simp} rule and neglect to provide rules to handle the $2 * k$ case, Nitpick will define |
|
2684 $\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug} |
|
2685 (\S\ref{output-format}) option is extremely useful to understand what is going |
|
2686 on when experimenting with \textit{nitpick\_} attributes. |
|
2687 |
|
2688 Because of its internal three-valued logic, Nitpick tends to lose a |
|
2689 lot of precision in the presence of partially specified constants. For example, |
|
2690 |
|
2691 \prew |
|
2692 \textbf{lemma} \textit{odd\_simp} [\textit{nitpick\_simp}]:\kern.4em ``$\textit{odd~x} = \lnot\, \textit{even}~x$'' |
|
2693 \postw |
|
2694 |
|
2695 is superior to |
|
2696 |
|
2697 \prew |
|
2698 \textbf{lemma} \textit{odd\_psimps} [\textit{nitpick\_simp}]: \\ |
|
2699 ``$\textit{even~x} \,\Longrightarrow\, \textit{odd~x} = \textit{False\/}$'' \\ |
|
2700 ``$\lnot\, \textit{even~x} \,\Longrightarrow\, \textit{odd~x} = \textit{True\/}$'' |
|
2701 \postw |
|
2702 |
|
2703 Because Nitpick sometimes unfolds definitions but never simplification rules, |
|
2704 you can ensure that a constant is defined explicitly using the |
|
2705 \textit{nitpick\_simp}. For example: |
|
2706 |
|
2707 \prew |
|
2708 \textbf{definition}~\textit{optimum} \textbf{where} [\textit{nitpick\_simp}]: \\ |
|
2709 ``$\textit{optimum}~t = |
|
2710 (\forall u.\; \textit{consistent}~u \mathrel{\land} \textit{alphabet}~t = \textit{alphabet}~u$ \\ |
|
2711 \phantom{``$\textit{optimum}~t = (\forall u.\;$}${\mathrel{\land}}\; \textit{freq}~t = \textit{freq}~u \longrightarrow |
|
2712 \textit{cost}~t \le \textit{cost}~u)$'' |
|
2713 \postw |
|
2714 |
|
2715 In some rare occasions, you might want to provide an inductive or coinductive |
|
2716 view on top of an existing constant $c$. The easiest way to achieve this is to |
|
2717 define a new constant $c'$ (co)inductively. Then prove that $c$ equals $c'$ |
|
2718 and let Nitpick know about it: |
|
2719 |
|
2720 \prew |
|
2721 \textbf{lemma} \textit{c\_alt\_unfold} [\textit{nitpick\_unfold}]:\kern.4em ``$c \equiv c'$\kern2pt '' |
|
2722 \postw |
|
2723 |
|
2724 This ensures that Nitpick will substitute $c'$ for $c$ and use the (co)inductive |
|
2725 definition. |
|
2726 |
|
2727 \section{Standard ML Interface} |
|
2728 \label{standard-ml-interface} |
|
2729 |
|
2730 Nitpick provides a rich Standard ML interface used mainly for internal purposes |
|
2731 and debugging. Among the most interesting functions exported by Nitpick are |
|
2732 those that let you invoke the tool programmatically and those that let you |
|
2733 register and unregister custom coinductive datatypes as well as term |
|
2734 postprocessors. |
|
2735 |
|
2736 \subsection{Invocation of Nitpick} |
|
2737 \label{invocation-of-nitpick} |
|
2738 |
|
2739 The \textit{Nitpick} structure offers the following functions for invoking your |
|
2740 favorite counterexample generator: |
|
2741 |
|
2742 \prew |
|
2743 $\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\ |
|
2744 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{mode} |
|
2745 \rightarrow \textit{int} \rightarrow \textit{int} \rightarrow \textit{int}$ \\ |
|
2746 $\hbox{}\quad{\rightarrow}\; (\textit{term} * \textit{term})~\textit{list} |
|
2747 \rightarrow \textit{term~list} \rightarrow \textit{term} \rightarrow \textit{string} * \textit{Proof.state}$ \\ |
|
2748 $\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\ |
|
2749 \hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{mode} \rightarrow \textit{int} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$ |
|
2750 \postw |
|
2751 |
|
2752 The return value is a new proof state paired with an outcome string |
|
2753 (``genuine'', ``quasi\_genuine'', ``potential'', ``none'', or ``unknown''). The |
|
2754 \textit{params} type is a large record that lets you set Nitpick's options. The |
|
2755 current default options can be retrieved by calling the following function |
|
2756 defined in the \textit{Nitpick\_Isar} structure: |
|
2757 |
|
2758 \prew |
|
2759 $\textbf{val}\,~\textit{default\_params} :\, |
|
2760 \textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$ |
|
2761 \postw |
|
2762 |
|
2763 The second argument lets you override option values before they are parsed and |
|
2764 put into a \textit{params} record. Here is an example where Nitpick is invoked |
|
2765 on subgoal $i$ of $n$ with no time limit: |
|
2766 |
|
2767 \prew |
|
2768 $\textbf{val}\,~\textit{params} = \textit{Nitpick\_Isar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout\/}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\ |
|
2769 $\textbf{val}\,~(\textit{outcome},\, \textit{state}') = {}$ \\ |
|
2770 $\hbox{}\quad\textit{Nitpick.pick\_nits\_in\_subgoal}~\textit{state}~\textit{params}~\textit{Nitpick.Normal}~\textit{i}~\textit{n}$ |
|
2771 \postw |
|
2772 |
|
2773 \let\antiq=\textrm |
|
2774 |
|
2775 \subsection{Registration of Coinductive Datatypes} |
|
2776 \label{registration-of-coinductive-datatypes} |
|
2777 |
|
2778 If you have defined a custom coinductive datatype, you can tell Nitpick about |
|
2779 it, so that it can use an efficient Kodkod axiomatization similar to the one it |
|
2780 uses for lazy lists. The interface for registering and unregistering coinductive |
|
2781 datatypes consists of the following pair of functions defined in the |
|
2782 \textit{Nitpick\_HOL} structure: |
|
2783 |
|
2784 \prew |
|
2785 $\textbf{val}\,~\textit{register\_codatatype\/} : {}$ \\ |
|
2786 $\hbox{}\quad\textit{morphism} \rightarrow \textit{typ} \rightarrow \textit{string} \rightarrow (\textit{string} \times \textit{typ})\;\textit{list} \rightarrow \textit{Context.generic} {}$ \\ |
|
2787 $\hbox{}\quad{\rightarrow}\; \textit{Context.generic}$ \\ |
|
2788 $\textbf{val}\,~\textit{unregister\_codatatype\/} : {}$ \\ |
|
2789 $\hbox{}\quad\textit{morphism} \rightarrow \textit{typ} \rightarrow \textit{Context.generic} \rightarrow \textit{Context.generic} {}$ |
|
2790 \postw |
|
2791 |
|
2792 The type $'a~\textit{llist}$ of lazy lists is already registered; had it |
|
2793 not been, you could have told Nitpick about it by adding the following line |
|
2794 to your theory file: |
|
2795 |
|
2796 \prew |
|
2797 $\textbf{declaration}~\,\{{*}$ \\ |
|
2798 $\hbox{}\quad\textit{Nitpick\_HOL.register\_codatatype}~@\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}$ \\ |
|
2799 $\hbox{}\qquad\quad @\{\antiq{const\_name}~ \textit{llist\_case}\}$ \\ |
|
2800 $\hbox{}\qquad\quad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])$ \\ |
|
2801 ${*}\}$ |
|
2802 \postw |
|
2803 |
|
2804 The \textit{register\_codatatype} function takes a coinductive datatype, its |
|
2805 case function, and the list of its constructors (in addition to the current |
|
2806 morphism and generic proof context). The case function must take its arguments |
|
2807 in the order that the constructors are listed. If no case function with the |
|
2808 correct signature is available, simply pass the empty string. |
|
2809 |
|
2810 On the other hand, if your goal is to cripple Nitpick, add the following line to |
|
2811 your theory file and try to check a few conjectures about lazy lists: |
|
2812 |
|
2813 \prew |
|
2814 $\textbf{declaration}~\,\{{*}$ \\ |
|
2815 $\hbox{}\quad\textit{Nitpick\_HOL.unregister\_codatatype}~@\{\antiq{typ}~``\kern1pt'a~\textit{llist\/}\textrm{''}\}$ \\ |
|
2816 ${*}\}$ |
|
2817 \postw |
|
2818 |
|
2819 Inductive datatypes can be registered as coinductive datatypes, given |
|
2820 appropriate coinductive constructors. However, doing so precludes |
|
2821 the use of the inductive constructors---Nitpick will generate an error if they |
|
2822 are needed. |
|
2823 |
|
2824 \subsection{Registration of Term Postprocessors} |
|
2825 \label{registration-of-term-postprocessors} |
|
2826 |
|
2827 It is possible to change the output of any term that Nitpick considers a |
|
2828 datatype by registering a term postprocessor. The interface for registering and |
|
2829 unregistering postprocessors consists of the following pair of functions defined |
|
2830 in the \textit{Nitpick\_Model} structure: |
|
2831 |
|
2832 \prew |
|
2833 $\textbf{type}\,~\textit{term\_postprocessor}\,~{=} {}$ \\ |
|
2834 $\hbox{}\quad\textit{Proof.context} \rightarrow \textit{string} \rightarrow (\textit{typ} \rightarrow \textit{term~list\/}) \rightarrow \textit{typ} \rightarrow \textit{term} \rightarrow \textit{term}$ \\ |
|
2835 $\textbf{val}\,~\textit{register\_term\_postprocessor} : {}$ \\ |
|
2836 $\hbox{}\quad\textit{typ} \rightarrow \textit{term\_postprocessor} \rightarrow \textit{morphism} \rightarrow \textit{Context.generic}$ \\ |
|
2837 $\hbox{}\quad{\rightarrow}\; \textit{Context.generic}$ \\ |
|
2838 $\textbf{val}\,~\textit{unregister\_term\_postprocessor} : {}$ \\ |
|
2839 $\hbox{}\quad\textit{typ} \rightarrow \textit{morphism} \rightarrow \textit{Context.generic} \rightarrow \textit{Context.generic}$ |
|
2840 \postw |
|
2841 |
|
2842 \S\ref{typedefs-quotient-types-records-rationals-and-reals} and |
|
2843 \texttt{src/HOL/Library/Multiset.thy} illustrate this feature in context. |
|
2844 |
|
2845 \section{Known Bugs and Limitations} |
|
2846 \label{known-bugs-and-limitations} |
|
2847 |
|
2848 Here are the known bugs and limitations in Nitpick at the time of writing: |
|
2849 |
|
2850 \begin{enum} |
|
2851 \item[\labelitemi] Underspecified functions defined using the \textbf{primrec}, |
|
2852 \textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead |
|
2853 Nitpick to generate spurious counterexamples for theorems that refer to values |
|
2854 for which the function is not defined. For example: |
|
2855 |
|
2856 \prew |
|
2857 \textbf{primrec} \textit{prec} \textbf{where} \\ |
|
2858 ``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount] |
|
2859 \textbf{lemma} ``$\textit{prec}~0 = \textit{undefined\/}$'' \\ |
|
2860 \textbf{nitpick} \\[2\smallskipamount] |
|
2861 \quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2: |
|
2862 \nopagebreak |
|
2863 \\[2\smallskipamount] |
|
2864 \hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount] |
|
2865 \textbf{by}~(\textit{auto simp}:~\textit{prec\_def}) |
|
2866 \postw |
|
2867 |
|
2868 Such theorems are generally considered bad style because they rely on the |
|
2869 internal representation of functions synthesized by Isabelle, an implementation |
|
2870 detail. |
|
2871 |
|
2872 \item[\labelitemi] Similarly, Nitpick might find spurious counterexamples for |
|
2873 theorems that rely on the use of the indefinite description operator internally |
|
2874 by \textbf{specification} and \textbf{quot\_type}. |
|
2875 |
|
2876 \item[\labelitemi] Axioms or definitions that restrict the possible values of the |
|
2877 \textit{undefined} constant or other partially specified built-in Isabelle |
|
2878 constants (e.g., \textit{Abs\_} and \textit{Rep\_} constants) are in general |
|
2879 ignored. Again, such nonconservative extensions are generally considered bad |
|
2880 style. |
|
2881 |
|
2882 \item[\labelitemi] Nitpick maintains a global cache of wellfoundedness conditions, |
|
2883 which can become invalid if you change the definition of an inductive predicate |
|
2884 that is registered in the cache. To clear the cache, |
|
2885 run Nitpick with the \textit{tac\_timeout} option set to a new value (e.g., |
|
2886 $0.51$). |
|
2887 |
|
2888 \item[\labelitemi] Nitpick produces spurious counterexamples when invoked after a |
|
2889 \textbf{guess} command in a structured proof. |
|
2890 |
|
2891 \item[\labelitemi] The \textit{nitpick\_xxx} attributes and the |
|
2892 \textit{Nitpick\_xxx.register\_yyy} functions can cause havoc if used |
|
2893 improperly. |
|
2894 |
|
2895 \item[\labelitemi] Although this has never been observed, arbitrary theorem |
|
2896 morphisms could possibly confuse Nitpick, resulting in spurious counterexamples. |
|
2897 |
|
2898 \item[\labelitemi] All constants, types, free variables, and schematic variables |
|
2899 whose names start with \textit{Nitpick}{.} are reserved for internal use. |
|
2900 \end{enum} |
|
2901 |
|
2902 \let\em=\sl |
|
2903 \bibliography{manual}{} |
|
2904 \bibliographystyle{abbrv} |
|
2905 |
|
2906 \end{document} |
|