merged
authorblanchet
Mon Oct 26 14:21:20 2009 +0100 (2009-10-26)
changeset 33203322d928d9f8f
parent 33188 3802b3b7845f
parent 33202 0183ab3ca7b4
child 33204 79bd3fbf5d61
merged
CONTRIBUTORS
src/HOL/IsaMakefile
     1.1 --- a/CONTRIBUTORS	Mon Oct 26 12:23:59 2009 +0100
     1.2 +++ b/CONTRIBUTORS	Mon Oct 26 14:21:20 2009 +0100
     1.3 @@ -7,6 +7,9 @@
     1.4  Contributions to this Isabelle version
     1.5  --------------------------------------
     1.6  
     1.7 +* October 2009: Jasmin Blanchette, TUM
     1.8 +  Nitpick: yet another counterexample generator for Isabelle/HOL
     1.9 +
    1.10  * October 2009: Sascha Boehme, TUM
    1.11    Extension of SMT method: proof-reconstruction for the SMT solver Z3.
    1.12  
     2.1 --- a/NEWS	Mon Oct 26 12:23:59 2009 +0100
     2.2 +++ b/NEWS	Mon Oct 26 14:21:20 2009 +0100
     2.3 @@ -50,6 +50,9 @@
     2.4  this method is proof-producing. Certificates are provided to
     2.5  avoid calling the external solvers solely for re-checking proofs.
     2.6  
     2.7 +* New counterexample generator tool "nitpick" based on the Kodkod
     2.8 +relational model finder.
     2.9 +
    2.10  * Reorganization of number theory:
    2.11    * former session NumberTheory now named Old_Number_Theory
    2.12    * new session Number_Theory by Jeremy Avigad; if possible, prefer this.
     3.1 --- a/doc-src/Dirs	Mon Oct 26 12:23:59 2009 +0100
     3.2 +++ b/doc-src/Dirs	Mon Oct 26 14:21:20 2009 +0100
     3.3 @@ -1,1 +1,1 @@
     3.4 -Intro Ref System Logics HOL ZF Inductive TutorialI IsarOverview IsarRef IsarImplementation Locales LaTeXsugar Classes Codegen Functions Main
     3.5 +Intro Ref System Logics HOL ZF Inductive TutorialI IsarOverview IsarRef IsarImplementation Locales LaTeXsugar Classes Codegen Functions Nitpick Main
     4.1 --- a/doc-src/Makefile.in	Mon Oct 26 12:23:59 2009 +0100
     4.2 +++ b/doc-src/Makefile.in	Mon Oct 26 14:21:20 2009 +0100
     4.3 @@ -45,6 +45,9 @@
     4.4  isabelle_zf.eps:
     4.5  	test -r isabelle_zf.eps || ln -s ../gfx/isabelle_zf.eps .
     4.6  
     4.7 +isabelle_nitpick.eps:
     4.8 +	test -r isabelle_nitpick.eps || ln -s ../gfx/isabelle_nitpick.eps .
     4.9 +
    4.10  
    4.11  isabelle.pdf:
    4.12  	test -r isabelle.pdf || ln -s ../gfx/isabelle.pdf .
    4.13 @@ -58,6 +61,9 @@
    4.14  isabelle_zf.pdf:
    4.15  	test -r isabelle_zf.pdf || ln -s ../gfx/isabelle_zf.pdf .
    4.16  
    4.17 +isabelle_nitpick.pdf:
    4.18 +	test -r isabelle_nitpick.pdf || ln -s ../gfx/isabelle_nitpick.pdf .
    4.19 +
    4.20  typedef.ps:
    4.21  	test -r typedef.ps || ln -s ../gfx/typedef.ps .
    4.22  
     5.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     5.2 +++ b/doc-src/Nitpick/Makefile	Mon Oct 26 14:21:20 2009 +0100
     5.3 @@ -0,0 +1,36 @@
     5.4 +#
     5.5 +# $Id$
     5.6 +#
     5.7 +
     5.8 +## targets
     5.9 +
    5.10 +default: dvi
    5.11 +
    5.12 +
    5.13 +## dependencies
    5.14 +
    5.15 +include ../Makefile.in
    5.16 +
    5.17 +NAME = nitpick
    5.18 +FILES = nitpick.tex ../iman.sty ../manual.bib
    5.19 +
    5.20 +dvi: $(NAME).dvi
    5.21 +
    5.22 +$(NAME).dvi: $(FILES) isabelle_nitpick.eps
    5.23 +	$(LATEX) $(NAME)
    5.24 +	$(BIBTEX) $(NAME)
    5.25 +	$(LATEX) $(NAME)
    5.26 +	$(LATEX) $(NAME)
    5.27 +	$(SEDINDEX) $(NAME)
    5.28 +	$(LATEX) $(NAME)
    5.29 +
    5.30 +pdf: $(NAME).pdf
    5.31 +
    5.32 +$(NAME).pdf: $(FILES) isabelle_nitpick.pdf
    5.33 +	$(PDFLATEX) $(NAME)
    5.34 +	$(BIBTEX) $(NAME)
    5.35 +	$(PDFLATEX) $(NAME)
    5.36 +	$(PDFLATEX) $(NAME)
    5.37 +	$(SEDINDEX) $(NAME)
    5.38 +	$(FIXBOOKMARKS) $(NAME).out
    5.39 +	$(PDFLATEX) $(NAME)
     6.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     6.2 +++ b/doc-src/Nitpick/nitpick.tex	Mon Oct 26 14:21:20 2009 +0100
     6.3 @@ -0,0 +1,2486 @@
     6.4 +\documentclass[a4paper,12pt]{article}
     6.5 +\usepackage[T1]{fontenc}
     6.6 +\usepackage{amsmath}
     6.7 +\usepackage{amssymb}
     6.8 +\usepackage[french,english]{babel}
     6.9 +\usepackage{color}
    6.10 +\usepackage{graphicx}
    6.11 +%\usepackage{mathpazo}
    6.12 +\usepackage{multicol}
    6.13 +\usepackage{stmaryrd}
    6.14 +%\usepackage[scaled=.85]{beramono}
    6.15 +\usepackage{../iman,../pdfsetup}
    6.16 +
    6.17 +%\oddsidemargin=4.6mm
    6.18 +%\evensidemargin=4.6mm
    6.19 +%\textwidth=150mm
    6.20 +%\topmargin=4.6mm
    6.21 +%\headheight=0mm
    6.22 +%\headsep=0mm
    6.23 +%\textheight=234mm
    6.24 +
    6.25 +\def\Colon{\mathord{:\mkern-1.5mu:}}
    6.26 +%\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}}
    6.27 +%\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}}
    6.28 +\def\lparr{\mathopen{(\mkern-4mu\mid}}
    6.29 +\def\rparr{\mathclose{\mid\mkern-4mu)}}
    6.30 +
    6.31 +\def\undef{\textit{undefined}}
    6.32 +\def\unk{{?}}
    6.33 +%\def\unr{\textit{others}}
    6.34 +\def\unr{\ldots}
    6.35 +\def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}}
    6.36 +\def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
    6.37 +
    6.38 +\hyphenation{Mini-Sat size-change First-Steps grand-parent nit-pick
    6.39 +counter-example counter-examples data-type data-types co-data-type 
    6.40 +co-data-types in-duc-tive co-in-duc-tive}
    6.41 +
    6.42 +\urlstyle{tt}
    6.43 +
    6.44 +\begin{document}
    6.45 +
    6.46 +\title{\includegraphics[scale=0.5]{isabelle_nitpick} \\[4ex]
    6.47 +Picking Nits \\[\smallskipamount]
    6.48 +\Large A User's Guide to Nitpick for Isabelle/HOL 2010}
    6.49 +\author{\hbox{} \\
    6.50 +Jasmin Christian Blanchette \\
    6.51 +{\normalsize Fakult\"at f\"ur Informatik, Technische Universit\"at M\"unchen} \\
    6.52 +\hbox{}}
    6.53 +
    6.54 +\maketitle
    6.55 +
    6.56 +\tableofcontents
    6.57 +
    6.58 +\setlength{\parskip}{.7em plus .2em minus .1em}
    6.59 +\setlength{\parindent}{0pt}
    6.60 +\setlength{\abovedisplayskip}{\parskip}
    6.61 +\setlength{\abovedisplayshortskip}{.9\parskip}
    6.62 +\setlength{\belowdisplayskip}{\parskip}
    6.63 +\setlength{\belowdisplayshortskip}{.9\parskip}
    6.64 +
    6.65 +% General-purpose enum environment with correct spacing
    6.66 +\newenvironment{enum}%
    6.67 +    {\begin{list}{}{%
    6.68 +        \setlength{\topsep}{.1\parskip}%
    6.69 +        \setlength{\partopsep}{.1\parskip}%
    6.70 +        \setlength{\itemsep}{\parskip}%
    6.71 +        \advance\itemsep by-\parsep}}
    6.72 +    {\end{list}}
    6.73 +
    6.74 +\def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin
    6.75 +\advance\rightskip by\leftmargin}
    6.76 +\def\post{\vskip0pt plus1ex\endgroup}
    6.77 +
    6.78 +\def\prew{\pre\advance\rightskip by-\leftmargin}
    6.79 +\def\postw{\post}
    6.80 +
    6.81 +\section{Introduction}
    6.82 +\label{introduction}
    6.83 +
    6.84 +Nitpick \cite{blanchette-nipkow-2009} is a counterexample generator for
    6.85 +Isabelle/HOL \cite{isa-tutorial} that is designed to handle formulas
    6.86 +combining (co)in\-duc\-tive datatypes, (co)in\-duc\-tively defined predicates, and
    6.87 +quantifiers. It builds on Kodkod \cite{torlak-jackson-2007}, a highly optimized
    6.88 +first-order relational model finder developed by the Software Design Group at
    6.89 +MIT. It is conceptually similar to Refute \cite{weber-2008}, from which it
    6.90 +borrows many ideas and code fragments, but it benefits from Kodkod's
    6.91 +optimizations and a new encoding scheme. The name Nitpick is shamelessly
    6.92 +appropriated from a now retired Alloy precursor.
    6.93 +
    6.94 +Nitpick is easy to use---you simply enter \textbf{nitpick} after a putative
    6.95 +theorem and wait a few seconds. Nonetheless, there are situations where knowing
    6.96 +how it works under the hood and how it reacts to various options helps
    6.97 +increase the test coverage. This manual also explains how to install the tool on
    6.98 +your workstation. Should the motivation fail you, think of the many hours of
    6.99 +hard work Nitpick will save you. Proving non-theorems is \textsl{hard work}.
   6.100 +
   6.101 +Another common use of Nitpick is to find out whether the axioms of a locale are
   6.102 +satisfiable, while the locale is being developed. To check this, it suffices to
   6.103 +write
   6.104 +
   6.105 +\prew
   6.106 +\textbf{lemma}~``$\textit{False}$'' \\
   6.107 +\textbf{nitpick}~[\textit{show\_all}]
   6.108 +\postw
   6.109 +
   6.110 +after the locale's \textbf{begin} keyword. To falsify \textit{False}, Nitpick
   6.111 +must find a model for the axioms. If it finds no model, we have an indication
   6.112 +that the axioms might be unsatisfiable.
   6.113 +
   6.114 +Nitpick requires the Kodkodi package for Isabelle as well as a Java 1.5 virtual
   6.115 +machine called \texttt{java}. The examples presented in this manual can be found
   6.116 +in Isabelle's \texttt{src/HOL/Nitpick\_Examples/Manual\_Nits.thy} theory.
   6.117 +
   6.118 +\newbox\boxA
   6.119 +\setbox\boxA=\hbox{\texttt{nospam}}
   6.120 +
   6.121 +The known bugs and limitations at the time of writing are listed in
   6.122 +\S\ref{known-bugs-and-limitations}. Comments and bug reports concerning Nitpick
   6.123 +or this manual should be directed to
   6.124 +\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
   6.125 +in.\allowbreak tum.\allowbreak de}.
   6.126 +
   6.127 +\vskip2.5\smallskipamount
   6.128 +
   6.129 +\textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
   6.130 +suggesting several textual improvements.
   6.131 +% and Perry James for reporting a typo.
   6.132 +
   6.133 +\section{First Steps}
   6.134 +\label{first-steps}
   6.135 +
   6.136 +This section introduces Nitpick by presenting small examples. If possible, you
   6.137 +should try out the examples on your workstation. Your theory file should start
   6.138 +the standard way:
   6.139 +
   6.140 +\prew
   6.141 +\textbf{theory}~\textit{Scratch} \\
   6.142 +\textbf{imports}~\textit{Main} \\
   6.143 +\textbf{begin}
   6.144 +\postw
   6.145 +
   6.146 +The results presented here were obtained using the JNI version of MiniSat and
   6.147 +with multithreading disabled to reduce nondeterminism. This was done by adding
   6.148 +the line
   6.149 +
   6.150 +\prew
   6.151 +\textbf{nitpick\_params} [\textit{sat\_solver}~= \textit{MiniSatJNI}, \,\textit{max\_threads}~= 1]
   6.152 +\postw
   6.153 +
   6.154 +after the \textbf{begin} keyword. The JNI version of MiniSat is bundled with
   6.155 +Kodkodi and is precompiled for the major platforms. Other SAT solvers can also
   6.156 +be installed, as explained in \S\ref{optimizations}. If you have already
   6.157 +configured SAT solvers in Isabelle (e.g., for Refute), these will also be
   6.158 +available to Nitpick.
   6.159 +
   6.160 +Throughout this manual, we will explicitly invoke the \textbf{nitpick} command.
   6.161 +Nitpick also provides an automatic mode that can be enabled by specifying
   6.162 +
   6.163 +\prew
   6.164 +\textbf{nitpick\_params} [\textit{auto}]
   6.165 +\postw
   6.166 +
   6.167 +at the beginning of the theory file. In this mode, Nitpick is run for up to 5
   6.168 +seconds (by default) on every newly entered theorem, much like Auto Quickcheck.
   6.169 +
   6.170 +\subsection{Propositional Logic}
   6.171 +\label{propositional-logic}
   6.172 +
   6.173 +Let's start with a trivial example from propositional logic:
   6.174 +
   6.175 +\prew
   6.176 +\textbf{lemma}~``$P \longleftrightarrow Q$'' \\
   6.177 +\textbf{nitpick}
   6.178 +\postw
   6.179 +
   6.180 +You should get the following output:
   6.181 +
   6.182 +\prew
   6.183 +\slshape
   6.184 +Nitpick found a counterexample: \\[2\smallskipamount]
   6.185 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.186 +\hbox{}\qquad\qquad $P = \textit{True}$ \\
   6.187 +\hbox{}\qquad\qquad $Q = \textit{False}$
   6.188 +\postw
   6.189 +
   6.190 +Nitpick can also be invoked on individual subgoals, as in the example below:
   6.191 +
   6.192 +\prew
   6.193 +\textbf{apply}~\textit{auto} \\[2\smallskipamount]
   6.194 +{\slshape goal (2 subgoals): \\
   6.195 +\ 1. $P\,\Longrightarrow\, Q$ \\
   6.196 +\ 2. $Q\,\Longrightarrow\, P$} \\[2\smallskipamount]
   6.197 +\textbf{nitpick}~1 \\[2\smallskipamount]
   6.198 +{\slshape Nitpick found a counterexample: \\[2\smallskipamount]
   6.199 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.200 +\hbox{}\qquad\qquad $P = \textit{True}$ \\
   6.201 +\hbox{}\qquad\qquad $Q = \textit{False}$} \\[2\smallskipamount]
   6.202 +\textbf{nitpick}~2 \\[2\smallskipamount]
   6.203 +{\slshape Nitpick found a counterexample: \\[2\smallskipamount]
   6.204 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.205 +\hbox{}\qquad\qquad $P = \textit{False}$ \\
   6.206 +\hbox{}\qquad\qquad $Q = \textit{True}$} \\[2\smallskipamount]
   6.207 +\textbf{oops}
   6.208 +\postw
   6.209 +
   6.210 +\subsection{Type Variables}
   6.211 +\label{type-variables}
   6.212 +
   6.213 +If you are left unimpressed by the previous example, don't worry. The next
   6.214 +one is more mind- and computer-boggling:
   6.215 +
   6.216 +\prew
   6.217 +\textbf{lemma} ``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
   6.218 +\postw
   6.219 +\pagebreak[2] %% TYPESETTING
   6.220 +
   6.221 +The putative lemma involves the definite description operator, {THE}, presented
   6.222 +in section 5.10.1 of the Isabelle tutorial \cite{isa-tutorial}. The
   6.223 +operator is defined by the axiom $(\textrm{THE}~x.\; x = a) = a$. The putative
   6.224 +lemma is merely asserting the indefinite description operator axiom with {THE}
   6.225 +substituted for {SOME}.
   6.226 +
   6.227 +The free variable $x$ and the bound variable $y$ have type $'a$. For formulas
   6.228 +containing type variables, Nitpick enumerates the possible domains for each type
   6.229 +variable, up to a given cardinality (8 by default), looking for a finite
   6.230 +countermodel:
   6.231 +
   6.232 +\prew
   6.233 +\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
   6.234 +\slshape
   6.235 +Trying 8 scopes: \nopagebreak \\
   6.236 +\hbox{}\qquad \textit{card}~$'a$~= 1; \\
   6.237 +\hbox{}\qquad \textit{card}~$'a$~= 2; \\
   6.238 +\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
   6.239 +\hbox{}\qquad \textit{card}~$'a$~= 8. \\[2\smallskipamount]
   6.240 +Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
   6.241 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.242 +\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
   6.243 +\hbox{}\qquad\qquad $x = a_3$ \\[2\smallskipamount]
   6.244 +Total time: 580 ms.
   6.245 +\postw
   6.246 +
   6.247 +Nitpick found a counterexample in which $'a$ has cardinality 3. (For
   6.248 +cardinalities 1 and 2, the formula holds.) In the counterexample, the three
   6.249 +values of type $'a$ are written $a_1$, $a_2$, and $a_3$.
   6.250 +
   6.251 +The message ``Trying $n$ scopes: {\ldots}''\ is shown only if the option
   6.252 +\textit{verbose} is enabled. You can specify \textit{verbose} each time you
   6.253 +invoke \textbf{nitpick}, or you can set it globally using the command
   6.254 +
   6.255 +\prew
   6.256 +\textbf{nitpick\_params} [\textit{verbose}]
   6.257 +\postw
   6.258 +
   6.259 +This command also displays the current default values for all of the options
   6.260 +supported by Nitpick. The options are listed in \S\ref{option-reference}.
   6.261 +
   6.262 +\subsection{Constants}
   6.263 +\label{constants}
   6.264 +
   6.265 +By just looking at Nitpick's output, it might not be clear why the
   6.266 +counterexample in \S\ref{type-variables} is genuine. Let's invoke Nitpick again,
   6.267 +this time telling it to show the values of the constants that occur in the
   6.268 +formula:
   6.269 +
   6.270 +\prew
   6.271 +\textbf{lemma}~``$P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$'' \\
   6.272 +\textbf{nitpick}~[\textit{show\_consts}] \\[2\smallskipamount]
   6.273 +\slshape
   6.274 +Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
   6.275 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.276 +\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
   6.277 +\hbox{}\qquad\qquad $x = a_3$ \\
   6.278 +\hbox{}\qquad Constant: \nopagebreak \\
   6.279 +\hbox{}\qquad\qquad $\textit{The}~\textsl{fallback} = a_1$
   6.280 +\postw
   6.281 +
   6.282 +We can see more clearly now. Since the predicate $P$ isn't true for a unique
   6.283 +value, $\textrm{THE}~y.\;P~y$ can denote any value of type $'a$, even
   6.284 +$a_1$. Since $P~a_1$ is false, the entire formula is falsified.
   6.285 +
   6.286 +As an optimization, Nitpick's preprocessor introduced the special constant
   6.287 +``\textit{The} fallback'' corresponding to $\textrm{THE}~y.\;P~y$ (i.e.,
   6.288 +$\mathit{The}~(\lambda y.\;P~y)$) when there doesn't exist a unique $y$
   6.289 +satisfying $P~y$. We disable this optimization by passing the
   6.290 +\textit{full\_descrs} option:
   6.291 +
   6.292 +\prew
   6.293 +\textbf{nitpick}~[\textit{full\_descrs},\, \textit{show\_consts}] \\[2\smallskipamount]
   6.294 +\slshape
   6.295 +Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
   6.296 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.297 +\hbox{}\qquad\qquad $P = \{a_2,\, a_3\}$ \\
   6.298 +\hbox{}\qquad\qquad $x = a_3$ \\
   6.299 +\hbox{}\qquad Constant: \nopagebreak \\
   6.300 +\hbox{}\qquad\qquad $\hbox{\slshape THE}~y.\;P~y = a_1$
   6.301 +\postw
   6.302 +
   6.303 +As the result of another optimization, Nitpick directly assigned a value to the
   6.304 +subterm $\textrm{THE}~y.\;P~y$, rather than to the \textit{The} constant. If we
   6.305 +disable this second optimization by using the command
   6.306 +
   6.307 +\prew
   6.308 +\textbf{nitpick}~[\textit{dont\_specialize},\, \textit{full\_descrs},\,
   6.309 +\textit{show\_consts}]
   6.310 +\postw
   6.311 +
   6.312 +we finally get \textit{The}:
   6.313 +
   6.314 +\prew
   6.315 +\slshape Constant: \nopagebreak \\
   6.316 +\hbox{}\qquad $\mathit{The} = \undef{}
   6.317 +    (\!\begin{aligned}[t]%
   6.318 +    & \{\} := a_3,\> \{a_3\} := a_3,\> \{a_2\} := a_2, \\[-2pt] %% TYPESETTING
   6.319 +    & \{a_2, a_3\} := a_1,\> \{a_1\} := a_1,\> \{a_1, a_3\} := a_3, \\[-2pt]
   6.320 +    & \{a_1, a_2\} := a_3,\> \{a_1, a_2, a_3\} := a_3)\end{aligned}$
   6.321 +\postw
   6.322 +
   6.323 +Notice that $\textit{The}~(\lambda y.\;P~y) = \textit{The}~\{a_2, a_3\} = a_1$,
   6.324 +just like before.\footnote{The \undef{} symbol's presence is explained as
   6.325 +follows: In higher-order logic, any function can be built from the undefined
   6.326 +function using repeated applications of the function update operator $f(x :=
   6.327 +y)$, just like any list can be built from the empty list using $x \mathbin{\#}
   6.328 +xs$.}
   6.329 +
   6.330 +Our misadventures with THE suggest adding `$\exists!x{.}$' (``there exists a
   6.331 +unique $x$ such that'') at the front of our putative lemma's assumption:
   6.332 +
   6.333 +\prew
   6.334 +\textbf{lemma}~``$\exists {!}x.\; P~x\,\Longrightarrow\, P~(\textrm{THE}~y.\;P~y)$''
   6.335 +\postw
   6.336 +
   6.337 +The fix appears to work:
   6.338 +
   6.339 +\prew
   6.340 +\textbf{nitpick} \\[2\smallskipamount]
   6.341 +\slshape Nitpick found no counterexample.
   6.342 +\postw
   6.343 +
   6.344 +We can further increase our confidence in the formula by exhausting all
   6.345 +cardinalities up to 50:
   6.346 +
   6.347 +\prew
   6.348 +\textbf{nitpick} [\textit{card} $'a$~= 1--50]\footnote{The symbol `--'
   6.349 +can be entered as \texttt{-} (hyphen) or
   6.350 +\texttt{\char`\\\char`\<midarrow\char`\>}.} \\[2\smallskipamount]
   6.351 +\slshape Nitpick found no counterexample.
   6.352 +\postw
   6.353 +
   6.354 +Let's see if Sledgehammer \cite{sledgehammer-2009} can find a proof:
   6.355 +
   6.356 +\prew
   6.357 +\textbf{sledgehammer} \\[2\smallskipamount]
   6.358 +{\slshape Sledgehammer: external prover ``$e$'' for subgoal 1: \\
   6.359 +$\exists{!}x.\; P~x\,\Longrightarrow\, P~(\hbox{\slshape THE}~y.\; P~y)$ \\
   6.360 +Try this command: \textrm{apply}~(\textit{metis~the\_equality})} \\[2\smallskipamount]
   6.361 +\textbf{apply}~(\textit{metis~the\_equality\/}) \nopagebreak \\[2\smallskipamount]
   6.362 +{\slshape No subgoals!}% \\[2\smallskipamount]
   6.363 +%\textbf{done}
   6.364 +\postw
   6.365 +
   6.366 +This must be our lucky day.
   6.367 +
   6.368 +\subsection{Skolemization}
   6.369 +\label{skolemization}
   6.370 +
   6.371 +Are all invertible functions onto? Let's find out:
   6.372 +
   6.373 +\prew
   6.374 +\textbf{lemma} ``$\exists g.\; \forall x.~g~(f~x) = x
   6.375 + \,\Longrightarrow\, \forall y.\; \exists x.~y = f~x$'' \\
   6.376 +\textbf{nitpick} \\[2\smallskipamount]
   6.377 +\slshape
   6.378 +Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\[2\smallskipamount]
   6.379 +\hbox{}\qquad Free variable: \nopagebreak \\
   6.380 +\hbox{}\qquad\qquad $f = \undef{}(b_1 := a_1)$ \\
   6.381 +\hbox{}\qquad Skolem constants: \nopagebreak \\
   6.382 +\hbox{}\qquad\qquad $g = \undef{}(a_1 := b_1,\> a_2 := b_1)$ \\
   6.383 +\hbox{}\qquad\qquad $y = a_2$
   6.384 +\postw
   6.385 +
   6.386 +Although $f$ is the only free variable occurring in the formula, Nitpick also
   6.387 +displays values for the bound variables $g$ and $y$. These values are available
   6.388 +to Nitpick because it performs skolemization as a preprocessing step.
   6.389 +
   6.390 +In the previous example, skolemization only affected the outermost quantifiers.
   6.391 +This is not always the case, as illustrated below:
   6.392 +
   6.393 +\prew
   6.394 +\textbf{lemma} ``$\exists x.\; \forall f.\; f~x = x$'' \\
   6.395 +\textbf{nitpick} \\[2\smallskipamount]
   6.396 +\slshape
   6.397 +Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
   6.398 +\hbox{}\qquad Skolem constant: \nopagebreak \\
   6.399 +\hbox{}\qquad\qquad $\lambda x.\; f =
   6.400 +    \undef{}(\!\begin{aligned}[t]
   6.401 +    & a_1 := \undef{}(a_1 := a_2,\> a_2 := a_1), \\[-2pt]
   6.402 +    & a_2 := \undef{}(a_1 := a_1,\> a_2 := a_1))\end{aligned}$
   6.403 +\postw
   6.404 +
   6.405 +The variable $f$ is bound within the scope of $x$; therefore, $f$ depends on
   6.406 +$x$, as suggested by the notation $\lambda x.\,f$. If $x = a_1$, then $f$ is the
   6.407 +function that maps $a_1$ to $a_2$ and vice versa; otherwise, $x = a_2$ and $f$
   6.408 +maps both $a_1$ and $a_2$ to $a_1$. In both cases, $f~x \not= x$.
   6.409 +
   6.410 +The source of the Skolem constants is sometimes more obscure:
   6.411 +
   6.412 +\prew
   6.413 +\textbf{lemma} ``$\mathit{refl}~r\,\Longrightarrow\, \mathit{sym}~r$'' \\
   6.414 +\textbf{nitpick} \\[2\smallskipamount]
   6.415 +\slshape
   6.416 +Nitpick found a counterexample for \textit{card} $'a$~= 2: \\[2\smallskipamount]
   6.417 +\hbox{}\qquad Free variable: \nopagebreak \\
   6.418 +\hbox{}\qquad\qquad $r = \{(a_1, a_1),\, (a_2, a_1),\, (a_2, a_2)\}$ \\
   6.419 +\hbox{}\qquad Skolem constants: \nopagebreak \\
   6.420 +\hbox{}\qquad\qquad $\mathit{sym}.x = a_2$ \\
   6.421 +\hbox{}\qquad\qquad $\mathit{sym}.y = a_1$
   6.422 +\postw
   6.423 +
   6.424 +What happened here is that Nitpick expanded the \textit{sym} constant to its
   6.425 +definition:
   6.426 +
   6.427 +\prew
   6.428 +$\mathit{sym}~r \,\equiv\,
   6.429 + \forall x\> y.\,\> (x, y) \in r \longrightarrow (y, x) \in r.$
   6.430 +\postw
   6.431 +
   6.432 +As their names suggest, the Skolem constants $\mathit{sym}.x$ and
   6.433 +$\mathit{sym}.y$ are simply the bound variables $x$ and $y$
   6.434 +from \textit{sym}'s definition.
   6.435 +
   6.436 +Although skolemization is a useful optimization, you can disable it by invoking
   6.437 +Nitpick with \textit{dont\_skolemize}. See \S\ref{optimizations} for details.
   6.438 +
   6.439 +\subsection{Natural Numbers and Integers}
   6.440 +\label{natural-numbers-and-integers}
   6.441 +
   6.442 +Because of the axiom of infinity, the type \textit{nat} does not admit any
   6.443 +finite models. To deal with this, Nitpick considers prefixes $\{0,\, 1,\,
   6.444 +\ldots,\, K - 1\}$ of \textit{nat} (where $K = \textit{card}~\textit{nat}$) and
   6.445 +maps all other numbers to the undefined value ($\unk$). The type \textit{int} is
   6.446 +handled in a similar way: If $K = \textit{card}~\textit{int}$, the subset of
   6.447 +\textit{int} known to Nitpick is $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor
   6.448 +K/2 \rfloor\}$. Undefined values lead to a three-valued logic.
   6.449 +
   6.450 +Here is an example involving \textit{int}:
   6.451 +
   6.452 +\prew
   6.453 +\textbf{lemma} ``$\lbrakk i \le j;\> n \le (m{\Colon}\mathit{int})\rbrakk \,\Longrightarrow\, i * n + j * m \le i * m + j * n$'' \\
   6.454 +\textbf{nitpick} \\[2\smallskipamount]
   6.455 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
   6.456 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.457 +\hbox{}\qquad\qquad $i = 0$ \\
   6.458 +\hbox{}\qquad\qquad $j = 1$ \\
   6.459 +\hbox{}\qquad\qquad $m = 1$ \\
   6.460 +\hbox{}\qquad\qquad $n = 0$
   6.461 +\postw
   6.462 +
   6.463 +With infinite types, we don't always have the luxury of a genuine counterexample
   6.464 +and must often content ourselves with a potential one. The tedious task of
   6.465 +finding out whether the potential counterexample is in fact genuine can be
   6.466 +outsourced to \textit{auto} by passing the option \textit{check\_potential}. For
   6.467 +example:
   6.468 +
   6.469 +\prew
   6.470 +\textbf{lemma} ``$\forall n.\; \textit{Suc}~n \mathbin{\not=} n \,\Longrightarrow\, P$'' \\
   6.471 +\textbf{nitpick} [\textit{card~nat}~= 100,\, \textit{check\_potential}] \\[2\smallskipamount]
   6.472 +\slshape Nitpick found a potential counterexample: \\[2\smallskipamount]
   6.473 +\hbox{}\qquad Free variable: \nopagebreak \\
   6.474 +\hbox{}\qquad\qquad $P = \textit{False}$ \\[2\smallskipamount]
   6.475 +Confirmation by ``\textit{auto}'': The above counterexample is genuine.
   6.476 +\postw
   6.477 +
   6.478 +You might wonder why the counterexample is first reported as potential. The root
   6.479 +of the problem is that the bound variable in $\forall n.\; \textit{Suc}~n
   6.480 +\mathbin{\not=} n$ ranges over an infinite type. If Nitpick finds an $n$ such
   6.481 +that $\textit{Suc}~n \mathbin{=} n$, it evaluates the assumption to
   6.482 +\textit{False}; but otherwise, it does not know anything about values of $n \ge
   6.483 +\textit{card~nat}$ and must therefore evaluate the assumption to $\unk$, not
   6.484 +\textit{True}. Since the assumption can never be satisfied, the putative lemma
   6.485 +can never be falsified.
   6.486 +
   6.487 +Incidentally, if you distrust the so-called genuine counterexamples, you can
   6.488 +enable \textit{check\_\allowbreak genuine} to verify them as well. However, be
   6.489 +aware that \textit{auto} will often fail to prove that the counterexample is
   6.490 +genuine or spurious.
   6.491 +
   6.492 +Some conjectures involving elementary number theory make Nitpick look like a
   6.493 +giant with feet of clay:
   6.494 +
   6.495 +\prew
   6.496 +\textbf{lemma} ``$P~\textit{Suc}$'' \\
   6.497 +\textbf{nitpick} [\textit{card} = 1--6] \\[2\smallskipamount]
   6.498 +\slshape
   6.499 +Nitpick found no counterexample.
   6.500 +\postw
   6.501 +
   6.502 +For any cardinality $k$, \textit{Suc} is the partial function $\{0 \mapsto 1,\,
   6.503 +1 \mapsto 2,\, \ldots,\, k - 1 \mapsto \unk\}$, which evaluates to $\unk$ when
   6.504 +it is passed as argument to $P$. As a result, $P~\textit{Suc}$ is always $\unk$.
   6.505 +The next example is similar:
   6.506 +
   6.507 +\prew
   6.508 +\textbf{lemma} ``$P~(\textit{op}~{+}\Colon
   6.509 +\textit{nat}\mathbin{\Rightarrow}\textit{nat}\mathbin{\Rightarrow}\textit{nat})$'' \\
   6.510 +\textbf{nitpick} [\textit{card nat} = 1] \\[2\smallskipamount]
   6.511 +{\slshape Nitpick found a counterexample:} \\[2\smallskipamount]
   6.512 +\hbox{}\qquad Free variable: \nopagebreak \\
   6.513 +\hbox{}\qquad\qquad $P = \{\}$ \\[2\smallskipamount]
   6.514 +\textbf{nitpick} [\textit{card nat} = 2] \\[2\smallskipamount]
   6.515 +{\slshape Nitpick found no counterexample.}
   6.516 +\postw
   6.517 +
   6.518 +The problem here is that \textit{op}~+ is total when \textit{nat} is taken to be
   6.519 +$\{0\}$ but becomes partial as soon as we add $1$, because $1 + 1 \notin \{0,
   6.520 +1\}$.
   6.521 +
   6.522 +Because numbers are infinite and are approximated using a three-valued logic,
   6.523 +there is usually no need to systematically enumerate domain sizes. If Nitpick
   6.524 +cannot find a genuine counterexample for \textit{card~nat}~= $k$, it is very
   6.525 +unlikely that one could be found for smaller domains. (The $P~(\textit{op}~{+})$
   6.526 +example above is an exception to this principle.) Nitpick nonetheless enumerates
   6.527 +all cardinalities from 1 to 8 for \textit{nat}, mainly because smaller
   6.528 +cardinalities are fast to handle and give rise to simpler counterexamples. This
   6.529 +is explained in more detail in \S\ref{scope-monotonicity}.
   6.530 +
   6.531 +\subsection{Inductive Datatypes}
   6.532 +\label{inductive-datatypes}
   6.533 +
   6.534 +Like natural numbers and integers, inductive datatypes with recursive
   6.535 +constructors admit no finite models and must be approximated by a subterm-closed
   6.536 +subset. For example, using a cardinality of 10 for ${'}a~\textit{list}$,
   6.537 +Nitpick looks for all counterexamples that can be built using at most 10
   6.538 +different lists.
   6.539 +
   6.540 +Let's see with an example involving \textit{hd} (which returns the first element
   6.541 +of a list) and $@$ (which concatenates two lists):
   6.542 +
   6.543 +\prew
   6.544 +\textbf{lemma} ``$\textit{hd}~(\textit{xs} \mathbin{@} [y, y]) = \textit{hd}~\textit{xs}$'' \\
   6.545 +\textbf{nitpick} \\[2\smallskipamount]
   6.546 +\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
   6.547 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.548 +\hbox{}\qquad\qquad $\textit{xs} = []$ \\
   6.549 +\hbox{}\qquad\qquad $\textit{y} = a_3$
   6.550 +\postw
   6.551 +
   6.552 +To see why the counterexample is genuine, we enable \textit{show\_consts}
   6.553 +and \textit{show\_\allowbreak datatypes}:
   6.554 +
   6.555 +\prew
   6.556 +{\slshape Datatype:} \\
   6.557 +\hbox{}\qquad $'a$~\textit{list}~= $\{[],\, [a_3, a_3],\, [a_3],\, \unr\}$ \\
   6.558 +{\slshape Constants:} \\
   6.559 +\hbox{}\qquad $\lambda x_1.\; x_1 \mathbin{@} [y, y] = \undef([] := [a_3, a_3],\> [a_3, a_3] := \unk,\> [a_3] := \unk)$ \\
   6.560 +\hbox{}\qquad $\textit{hd} = \undef([] := a_2,\> [a_3, a_3] := a_3,\> [a_3] := a_3)$
   6.561 +\postw
   6.562 +
   6.563 +Since $\mathit{hd}~[]$ is undefined in the logic, it may be given any value,
   6.564 +including $a_2$.
   6.565 +
   6.566 +The second constant, $\lambda x_1.\; x_1 \mathbin{@} [y, y]$, is simply the
   6.567 +append operator whose second argument is fixed to be $[y, y]$. Appending $[a_3,
   6.568 +a_3]$ to $[a_3]$ would normally give $[a_3, a_3, a_3]$, but this value is not
   6.569 +representable in the subset of $'a$~\textit{list} considered by Nitpick, which
   6.570 +is shown under the ``Datatype'' heading; hence the result is $\unk$. Similarly,
   6.571 +appending $[a_3, a_3]$ to itself gives $\unk$.
   6.572 +
   6.573 +Given \textit{card}~$'a = 3$ and \textit{card}~$'a~\textit{list} = 3$, Nitpick
   6.574 +considers the following subsets:
   6.575 +
   6.576 +\kern-.5\smallskipamount %% TYPESETTING
   6.577 +
   6.578 +\prew
   6.579 +\begin{multicols}{3}
   6.580 +$\{[],\, [a_1],\, [a_2]\}$; \\
   6.581 +$\{[],\, [a_1],\, [a_3]\}$; \\
   6.582 +$\{[],\, [a_2],\, [a_3]\}$; \\
   6.583 +$\{[],\, [a_1],\, [a_1, a_1]\}$; \\
   6.584 +$\{[],\, [a_1],\, [a_2, a_1]\}$; \\
   6.585 +$\{[],\, [a_1],\, [a_3, a_1]\}$; \\
   6.586 +$\{[],\, [a_2],\, [a_1, a_2]\}$; \\
   6.587 +$\{[],\, [a_2],\, [a_2, a_2]\}$; \\
   6.588 +$\{[],\, [a_2],\, [a_3, a_2]\}$; \\
   6.589 +$\{[],\, [a_3],\, [a_1, a_3]\}$; \\
   6.590 +$\{[],\, [a_3],\, [a_2, a_3]\}$; \\
   6.591 +$\{[],\, [a_3],\, [a_3, a_3]\}$.
   6.592 +\end{multicols}
   6.593 +\postw
   6.594 +
   6.595 +\kern-2\smallskipamount %% TYPESETTING
   6.596 +
   6.597 +All subterm-closed subsets of $'a~\textit{list}$ consisting of three values
   6.598 +are listed and only those. As an example of a non-subterm-closed subset,
   6.599 +consider $\mathcal{S} = \{[],\, [a_1],\,\allowbreak [a_1, a_3]\}$, and observe
   6.600 +that $[a_1, a_3]$ (i.e., $a_1 \mathbin{\#} [a_3]$) has $[a_3] \notin
   6.601 +\mathcal{S}$ as a subterm.
   6.602 +
   6.603 +Here's another m\"ochtegern-lemma that Nitpick can refute without a blink:
   6.604 +
   6.605 +\prew
   6.606 +\textbf{lemma} ``$\lbrakk \textit{length}~\textit{xs} = 1;\> \textit{length}~\textit{ys} = 1
   6.607 +\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$''
   6.608 +\\
   6.609 +\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
   6.610 +\slshape Nitpick found a counterexample for \textit{card} $'a$~= 3: \\[2\smallskipamount]
   6.611 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.612 +\hbox{}\qquad\qquad $\textit{xs} = [a_2]$ \\
   6.613 +\hbox{}\qquad\qquad $\textit{ys} = [a_3]$ \\
   6.614 +\hbox{}\qquad Datatypes: \\
   6.615 +\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
   6.616 +\hbox{}\qquad\qquad $'a$~\textit{list} = $\{[],\, [a_3],\, [a_2],\, \unr\}$
   6.617 +\postw
   6.618 +
   6.619 +Because datatypes are approximated using a three-valued logic, there is usually
   6.620 +no need to systematically enumerate cardinalities: If Nitpick cannot find a
   6.621 +genuine counterexample for \textit{card}~$'a~\textit{list}$~= 10, it is very
   6.622 +unlikely that one could be found for smaller cardinalities.
   6.623 +
   6.624 +\subsection{Typedefs, Records, Rationals, and Reals}
   6.625 +\label{typedefs-records-rationals-and-reals}
   6.626 +
   6.627 +Nitpick generally treats types declared using \textbf{typedef} as datatypes
   6.628 +whose single constructor is the corresponding \textit{Abs\_\kern.1ex} function.
   6.629 +For example:
   6.630 +
   6.631 +\prew
   6.632 +\textbf{typedef}~\textit{three} = ``$\{0\Colon\textit{nat},\, 1,\, 2\}$'' \\
   6.633 +\textbf{by}~\textit{blast} \\[2\smallskipamount]
   6.634 +\textbf{definition}~$A \mathbin{\Colon} \textit{three}$ \textbf{where} ``\kern-.1em$A \,\equiv\, \textit{Abs\_\allowbreak three}~0$'' \\
   6.635 +\textbf{definition}~$B \mathbin{\Colon} \textit{three}$ \textbf{where} ``$B \,\equiv\, \textit{Abs\_three}~1$'' \\
   6.636 +\textbf{definition}~$C \mathbin{\Colon} \textit{three}$ \textbf{where} ``$C \,\equiv\, \textit{Abs\_three}~2$'' \\[2\smallskipamount]
   6.637 +\textbf{lemma} ``$\lbrakk P~A;\> P~B\rbrakk \,\Longrightarrow\, P~x$'' \\
   6.638 +\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
   6.639 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
   6.640 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.641 +\hbox{}\qquad\qquad $P = \{\Abs{1},\, \Abs{0}\}$ \\
   6.642 +\hbox{}\qquad\qquad $x = \Abs{2}$ \\
   6.643 +\hbox{}\qquad Datatypes: \\
   6.644 +\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, \unr\}$ \\
   6.645 +\hbox{}\qquad\qquad $\textit{three} = \{\Abs{2},\, \Abs{1},\, \Abs{0},\, \unr\}$
   6.646 +\postw
   6.647 +
   6.648 +%% MARK
   6.649 +In the output above, $\Abs{n}$ abbreviates $\textit{Abs\_three}~n$.
   6.650 +
   6.651 +%% MARK
   6.652 +Records, which are implemented as \textbf{typedef}s behind the scenes, are
   6.653 +handled in much the same way:
   6.654 +
   6.655 +\prew
   6.656 +\textbf{record} \textit{point} = \\
   6.657 +\hbox{}\quad $\textit{Xcoord} \mathbin{\Colon} \textit{int}$ \\
   6.658 +\hbox{}\quad $\textit{Ycoord} \mathbin{\Colon} \textit{int}$ \\[2\smallskipamount]
   6.659 +\textbf{lemma} ``$\textit{Xcoord}~(p\Colon\textit{point}) = \textit{Xcoord}~(q\Colon\textit{point})$'' \\
   6.660 +\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
   6.661 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
   6.662 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.663 +\hbox{}\qquad\qquad $p = \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr$ \\
   6.664 +\hbox{}\qquad\qquad $q = \lparr\textit{Xcoord} = 1,\> \textit{Ycoord} = 1\rparr$ \\
   6.665 +\hbox{}\qquad Datatypes: \\
   6.666 +\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, \unr\}$ \\
   6.667 +\hbox{}\qquad\qquad $\textit{point} = \{\lparr\textit{Xcoord} = 1,\>
   6.668 +\textit{Ycoord} = 1\rparr,\> \lparr\textit{Xcoord} = 0,\> \textit{Ycoord} = 0\rparr,\, \unr\}$\kern-1pt %% QUIET
   6.669 +\postw
   6.670 +
   6.671 +Finally, Nitpick provides rudimentary support for rationals and reals using a
   6.672 +similar approach:
   6.673 +
   6.674 +\prew
   6.675 +\textbf{lemma} ``$4 * x + 3 * (y\Colon\textit{real}) \not= 1/2$'' \\
   6.676 +\textbf{nitpick} [\textit{show\_datatypes}] \\[2\smallskipamount]
   6.677 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
   6.678 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.679 +\hbox{}\qquad\qquad $x = 1/2$ \\
   6.680 +\hbox{}\qquad\qquad $y = -1/2$ \\
   6.681 +\hbox{}\qquad Datatypes: \\
   6.682 +\hbox{}\qquad\qquad $\textit{nat} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, \unr\}$ \\
   6.683 +\hbox{}\qquad\qquad $\textit{int} = \{0,\, 1,\, 2,\, 3,\, 4,\, -3,\, -2,\, -1,\, \unr\}$ \\
   6.684 +\hbox{}\qquad\qquad $\textit{real} = \{1,\, 0,\, 4,\, -3/2,\, 3,\, 2,\, 1/2,\, -1/2,\, \unr\}$
   6.685 +\postw
   6.686 +
   6.687 +\subsection{Inductive and Coinductive Predicates}
   6.688 +\label{inductive-and-coinductive-predicates}
   6.689 +
   6.690 +Inductively defined predicates (and sets) are particularly problematic for
   6.691 +counterexample generators. They can make Quickcheck~\cite{berghofer-nipkow-2004}
   6.692 +loop forever and Refute~\cite{weber-2008} run out of resources. The crux of
   6.693 +the problem is that they are defined using a least fixed point construction.
   6.694 +
   6.695 +Nitpick's philosophy is that not all inductive predicates are equal. Consider
   6.696 +the \textit{even} predicate below:
   6.697 +
   6.698 +\prew
   6.699 +\textbf{inductive}~\textit{even}~\textbf{where} \\
   6.700 +``\textit{even}~0'' $\,\mid$ \\
   6.701 +``\textit{even}~$n\,\Longrightarrow\, \textit{even}~(\textit{Suc}~(\textit{Suc}~n))$''
   6.702 +\postw
   6.703 +
   6.704 +This predicate enjoys the desirable property of being well-founded, which means
   6.705 +that the introduction rules don't give rise to infinite chains of the form
   6.706 +
   6.707 +\prew
   6.708 +$\cdots\,\Longrightarrow\, \textit{even}~k''
   6.709 +       \,\Longrightarrow\, \textit{even}~k'
   6.710 +       \,\Longrightarrow\, \textit{even}~k.$
   6.711 +\postw
   6.712 +
   6.713 +For \textit{even}, this is obvious: Any chain ending at $k$ will be of length
   6.714 +$k/2 + 1$:
   6.715 +
   6.716 +\prew
   6.717 +$\textit{even}~0\,\Longrightarrow\, \textit{even}~2\,\Longrightarrow\, \cdots
   6.718 +       \,\Longrightarrow\, \textit{even}~(k - 2)
   6.719 +       \,\Longrightarrow\, \textit{even}~k.$
   6.720 +\postw
   6.721 +
   6.722 +Wellfoundedness is desirable because it enables Nitpick to use a very efficient
   6.723 +fixed point computation.%
   6.724 +\footnote{If an inductive predicate is
   6.725 +well-founded, then it has exactly one fixed point, which is simultaneously the
   6.726 +least and the greatest fixed point. In these circumstances, the computation of
   6.727 +the least fixed point amounts to the computation of an arbitrary fixed point,
   6.728 +which can be performed using a straightforward recursive equation.}
   6.729 +Moreover, Nitpick can prove wellfoundedness of most well-founded predicates,
   6.730 +just as Isabelle's \textbf{function} package usually discharges termination
   6.731 +proof obligations automatically.
   6.732 +
   6.733 +Let's try an example:
   6.734 +
   6.735 +\prew
   6.736 +\textbf{lemma} ``$\exists n.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
   6.737 +\textbf{nitpick}~[\textit{card nat}~= 100,\, \textit{verbose}] \\[2\smallskipamount]
   6.738 +\slshape The inductive predicate ``\textit{even}'' was proved well-founded.
   6.739 +Nitpick can compute it efficiently. \\[2\smallskipamount]
   6.740 +Trying 1 scope: \\
   6.741 +\hbox{}\qquad \textit{card nat}~= 100. \\[2\smallskipamount]
   6.742 +Nitpick found a potential counterexample for \textit{card nat}~= 100: \\[2\smallskipamount]
   6.743 +\hbox{}\qquad Empty assignment \\[2\smallskipamount]
   6.744 +Nitpick could not find a better counterexample. \\[2\smallskipamount]
   6.745 +Total time: 2274 ms.
   6.746 +\postw
   6.747 +
   6.748 +No genuine counterexample is possible because Nitpick cannot rule out the
   6.749 +existence of a natural number $n \ge 100$ such that both $\textit{even}~n$ and
   6.750 +$\textit{even}~(\textit{Suc}~n)$ are true. To help Nitpick, we can bound the
   6.751 +existential quantifier:
   6.752 +
   6.753 +\prew
   6.754 +\textbf{lemma} ``$\exists n \mathbin{\le} 99.\; \textit{even}~n \mathrel{\land} \textit{even}~(\textit{Suc}~n)$'' \\
   6.755 +\textbf{nitpick}~[\textit{card nat}~= 100] \\[2\smallskipamount]
   6.756 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
   6.757 +\hbox{}\qquad Empty assignment
   6.758 +\postw
   6.759 +
   6.760 +So far we were blessed by the wellfoundedness of \textit{even}. What happens if
   6.761 +we use the following definition instead?
   6.762 +
   6.763 +\prew
   6.764 +\textbf{inductive} $\textit{even}'$ \textbf{where} \\
   6.765 +``$\textit{even}'~(0{\Colon}\textit{nat})$'' $\,\mid$ \\
   6.766 +``$\textit{even}'~2$'' $\,\mid$ \\
   6.767 +``$\lbrakk\textit{even}'~m;\> \textit{even}'~n\rbrakk \,\Longrightarrow\, \textit{even}'~(m + n)$''
   6.768 +\postw
   6.769 +
   6.770 +This definition is not well-founded: From $\textit{even}'~0$ and
   6.771 +$\textit{even}'~0$, we can derive that $\textit{even}'~0$. Nonetheless, the
   6.772 +predicates $\textit{even}$ and $\textit{even}'$ are equivalent.
   6.773 +
   6.774 +Let's check a property involving $\textit{even}'$. To make up for the
   6.775 +foreseeable computational hurdles entailed by non-wellfoundedness, we decrease
   6.776 +\textit{nat}'s cardinality to a mere 10:
   6.777 +
   6.778 +\prew
   6.779 +\textbf{lemma}~``$\exists n \in \{0, 2, 4, 6, 8\}.\;
   6.780 +\lnot\;\textit{even}'~n$'' \\
   6.781 +\textbf{nitpick}~[\textit{card nat}~= 10,\, \textit{verbose},\, \textit{show\_consts}] \\[2\smallskipamount]
   6.782 +\slshape
   6.783 +The inductive predicate ``$\textit{even}'\!$'' could not be proved well-founded.
   6.784 +Nitpick might need to unroll it. \\[2\smallskipamount]
   6.785 +Trying 6 scopes: \\
   6.786 +\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 0; \\
   6.787 +\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 1; \\
   6.788 +\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2; \\
   6.789 +\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 4; \\
   6.790 +\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 8; \\
   6.791 +\hbox{}\qquad \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 9. \\[2\smallskipamount]
   6.792 +Nitpick found a counterexample for \textit{card nat}~= 10 and \textit{iter} $\textit{even}'$~= 2: \\[2\smallskipamount]
   6.793 +\hbox{}\qquad Constant: \nopagebreak \\
   6.794 +\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
   6.795 +& 2 := \{0, 2, 4, 6, 8, 1^\Q, 3^\Q, 5^\Q, 7^\Q, 9^\Q\}, \\[-2pt]
   6.796 +& 1 := \{0, 2, 4, 1^\Q, 3^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\}, \\[-2pt]
   6.797 +& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$ \\[2\smallskipamount]
   6.798 +Total time: 1140 ms.
   6.799 +\postw
   6.800 +
   6.801 +Nitpick's output is very instructive. First, it tells us that the predicate is
   6.802 +unrolled, meaning that it is computed iteratively from the empty set. Then it
   6.803 +lists six scopes specifying different bounds on the numbers of iterations:\ 0,
   6.804 +1, 2, 4, 8, and~9.
   6.805 +
   6.806 +The output also shows how each iteration contributes to $\textit{even}'$. The
   6.807 +notation $\lambda i.\; \textit{even}'$ indicates that the value of the
   6.808 +predicate depends on an iteration counter. Iteration 0 provides the basis
   6.809 +elements, $0$ and $2$. Iteration 1 contributes $4$ ($= 2 + 2$). Iteration 2
   6.810 +throws $6$ ($= 2 + 4 = 4 + 2$) and $8$ ($= 4 + 4$) into the mix. Further
   6.811 +iterations would not contribute any new elements.
   6.812 +
   6.813 +Some values are marked with superscripted question
   6.814 +marks~(`\lower.2ex\hbox{$^\Q$}'). These are the elements for which the
   6.815 +predicate evaluates to $\unk$. Thus, $\textit{even}'$ evaluates to either
   6.816 +\textit{True} or $\unk$, never \textit{False}.
   6.817 +
   6.818 +When unrolling a predicate, Nitpick tries 0, 1, 2, 4, 8, 12, 16, and 24
   6.819 +iterations. However, these numbers are bounded by the cardinality of the
   6.820 +predicate's domain. With \textit{card~nat}~= 10, no more than 9 iterations are
   6.821 +ever needed to compute the value of a \textit{nat} predicate. You can specify
   6.822 +the number of iterations using the \textit{iter} option, as explained in
   6.823 +\S\ref{scope-of-search}.
   6.824 +
   6.825 +In the next formula, $\textit{even}'$ occurs both positively and negatively:
   6.826 +
   6.827 +\prew
   6.828 +\textbf{lemma} ``$\textit{even}'~(n - 2) \,\Longrightarrow\, \textit{even}'~n$'' \\
   6.829 +\textbf{nitpick} [\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
   6.830 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
   6.831 +\hbox{}\qquad Free variable: \nopagebreak \\
   6.832 +\hbox{}\qquad\qquad $n = 1$ \\
   6.833 +\hbox{}\qquad Constants: \nopagebreak \\
   6.834 +\hbox{}\qquad\qquad $\lambda i.\; \textit{even}'$ = $\undef(\!\begin{aligned}[t]
   6.835 +& 0 := \{0, 2, 1^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q\})\end{aligned}$  \\
   6.836 +\hbox{}\qquad\qquad $\textit{even}' \subseteq \{0, 2, 4, 6, 8, \unr\}$
   6.837 +\postw
   6.838 +
   6.839 +Notice the special constraint $\textit{even}' \subseteq \{0,\, 2,\, 4,\, 6,\,
   6.840 +8,\, \unr\}$ in the output, whose right-hand side represents an arbitrary
   6.841 +fixed point (not necessarily the least one). It is used to falsify
   6.842 +$\textit{even}'~n$. In contrast, the unrolled predicate is used to satisfy
   6.843 +$\textit{even}'~(n - 2)$.
   6.844 +
   6.845 +Coinductive predicates are handled dually. For example:
   6.846 +
   6.847 +\prew
   6.848 +\textbf{coinductive} \textit{nats} \textbf{where} \\
   6.849 +``$\textit{nats}~(x\Colon\textit{nat}) \,\Longrightarrow\, \textit{nats}~x$'' \\[2\smallskipamount]
   6.850 +\textbf{lemma} ``$\textit{nats} = \{0, 1, 2, 3, 4\}$'' \\
   6.851 +\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
   6.852 +\slshape Nitpick found a counterexample:
   6.853 +\\[2\smallskipamount]
   6.854 +\hbox{}\qquad Constants: \nopagebreak \\
   6.855 +\hbox{}\qquad\qquad $\lambda i.\; \textit{nats} = \undef(0 := \{\!\begin{aligned}[t]
   6.856 +& 0^\Q, 1^\Q, 2^\Q, 3^\Q, 4^\Q, 5^\Q, 6^\Q, 7^\Q, 8^\Q, 9^\Q, \\[-2pt]
   6.857 +& \unr\})\end{aligned}$ \\
   6.858 +\hbox{}\qquad\qquad $nats \supseteq \{9, 5^\Q, 6^\Q, 7^\Q, 8^\Q, \unr\}$
   6.859 +\postw
   6.860 +
   6.861 +As a special case, Nitpick uses Kodkod's transitive closure operator to encode
   6.862 +negative occurrences of non-well-founded ``linear inductive predicates,'' i.e.,
   6.863 +inductive predicates for which each the predicate occurs in at most one
   6.864 +assumption of each introduction rule. For example:
   6.865 +
   6.866 +\prew
   6.867 +\textbf{inductive} \textit{odd} \textbf{where} \\
   6.868 +``$\textit{odd}~1$'' $\,\mid$ \\
   6.869 +``$\lbrakk \textit{odd}~m;\>\, \textit{even}~n\rbrakk \,\Longrightarrow\, \textit{odd}~(m + n)$'' \\[2\smallskipamount]
   6.870 +\textbf{lemma}~``$\textit{odd}~n \,\Longrightarrow\, \textit{odd}~(n - 2)$'' \\
   6.871 +\textbf{nitpick}~[\textit{card nat} = 10,\, \textit{show\_consts}] \\[2\smallskipamount]
   6.872 +\slshape Nitpick found a counterexample:
   6.873 +\\[2\smallskipamount]
   6.874 +\hbox{}\qquad Free variable: \nopagebreak \\
   6.875 +\hbox{}\qquad\qquad $n = 1$ \\
   6.876 +\hbox{}\qquad Constants: \nopagebreak \\
   6.877 +\hbox{}\qquad\qquad $\textit{even} = \{0, 2, 4, 6, 8, \unr\}$ \\
   6.878 +\hbox{}\qquad\qquad $\textit{odd}_{\textsl{base}} = \{1, \unr\}$ \\
   6.879 +\hbox{}\qquad\qquad $\textit{odd}_{\textsl{step}} = \!
   6.880 +\!\begin{aligned}[t]
   6.881 +  & \{(0, 0), (0, 2), (0, 4), (0, 6), (0, 8), (1, 1), (1, 3), (1, 5), \\[-2pt]
   6.882 +  & \phantom{\{} (1, 7), (1, 9), (2, 2), (2, 4), (2, 6), (2, 8), (3, 3),
   6.883 +       (3, 5), \\[-2pt]
   6.884 +  & \phantom{\{} (3, 7), (3, 9), (4, 4), (4, 6), (4, 8), (5, 5), (5, 7), (5, 9), \\[-2pt]
   6.885 +  & \phantom{\{} (6, 6), (6, 8), (7, 7), (7, 9), (8, 8), (9, 9), \unr\}\end{aligned}$ \\
   6.886 +\hbox{}\qquad\qquad $\textit{odd} \subseteq \{1, 3, 5, 7, 9, 8^\Q, \unr\}$
   6.887 +\postw
   6.888 +
   6.889 +\noindent
   6.890 +In the output, $\textit{odd}_{\textrm{base}}$ represents the base elements and
   6.891 +$\textit{odd}_{\textrm{step}}$ is a transition relation that computes new
   6.892 +elements from known ones. The set $\textit{odd}$ consists of all the values
   6.893 +reachable through the reflexive transitive closure of
   6.894 +$\textit{odd}_{\textrm{step}}$ starting with any element from
   6.895 +$\textit{odd}_{\textrm{base}}$, namely 1, 3, 5, 7, and 9. Using Kodkod's
   6.896 +transitive closure to encode linear predicates is normally either more thorough
   6.897 +or more efficient than unrolling (depending on the value of \textit{iter}), but
   6.898 +for those cases where it isn't you can disable it by passing the
   6.899 +\textit{dont\_star\_linear\_preds} option.
   6.900 +
   6.901 +\subsection{Coinductive Datatypes}
   6.902 +\label{coinductive-datatypes}
   6.903 +
   6.904 +While Isabelle regrettably lacks a high-level mechanism for defining coinductive
   6.905 +datatypes, the \textit{Coinductive\_List} theory provides a coinductive ``lazy
   6.906 +list'' datatype, $'a~\textit{llist}$, defined the hard way. Nitpick supports
   6.907 +these lazy lists seamlessly and provides a hook, described in
   6.908 +\S\ref{registration-of-coinductive-datatypes}, to register custom coinductive
   6.909 +datatypes.
   6.910 +
   6.911 +(Co)intuitively, a coinductive datatype is similar to an inductive datatype but
   6.912 +allows infinite objects. Thus, the infinite lists $\textit{ps}$ $=$ $[a, a, a,
   6.913 +\ldots]$, $\textit{qs}$ $=$ $[a, b, a, b, \ldots]$, and $\textit{rs}$ $=$ $[0,
   6.914 +1, 2, 3, \ldots]$ can be defined as lazy lists using the
   6.915 +$\textit{LNil}\mathbin{\Colon}{'}a~\textit{llist}$ and
   6.916 +$\textit{LCons}\mathbin{\Colon}{'}a \mathbin{\Rightarrow} {'}a~\textit{llist}
   6.917 +\mathbin{\Rightarrow} {'}a~\textit{llist}$ constructors.
   6.918 +
   6.919 +Although it is otherwise no friend of infinity, Nitpick can find counterexamples
   6.920 +involving cyclic lists such as \textit{ps} and \textit{qs} above as well as
   6.921 +finite lists:
   6.922 +
   6.923 +\prew
   6.924 +\textbf{lemma} ``$\textit{xs} \not= \textit{LCons}~a~\textit{xs}$'' \\
   6.925 +\textbf{nitpick} \\[2\smallskipamount]
   6.926 +\slshape Nitpick found a counterexample for {\itshape card}~$'a$ = 1: \\[2\smallskipamount]
   6.927 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.928 +\hbox{}\qquad\qquad $\textit{a} = a_1$ \\
   6.929 +\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_1~\omega$
   6.930 +\postw
   6.931 +
   6.932 +The notation $\textrm{THE}~\omega.\; \omega = t(\omega)$ stands
   6.933 +for the infinite term $t(t(t(\ldots)))$. Hence, \textit{xs} is simply the
   6.934 +infinite list $[a_1, a_1, a_1, \ldots]$.
   6.935 +
   6.936 +The next example is more interesting:
   6.937 +
   6.938 +\prew
   6.939 +\textbf{lemma}~``$\lbrakk\textit{xs} = \textit{LCons}~a~\textit{xs};\>\,
   6.940 +\textit{ys} = \textit{iterates}~(\lambda b.\> a)~b\rbrakk \,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
   6.941 +\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
   6.942 +\slshape The type ``\kern1pt$'a$'' passed the monotonicity test. Nitpick might be able to skip
   6.943 +some scopes. \\[2\smallskipamount]
   6.944 +Trying 8 scopes: \\
   6.945 +\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 1,
   6.946 +and \textit{bisim\_depth}~= 0. \\
   6.947 +\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
   6.948 +\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} ``\kern1pt$'a~\textit{list}$''~= 8,
   6.949 +and \textit{bisim\_depth}~= 7. \\[2\smallskipamount]
   6.950 +Nitpick found a counterexample for {\itshape card}~$'a$ = 2,
   6.951 +\textit{card}~``\kern1pt$'a~\textit{list}$''~= 2, and \textit{bisim\_\allowbreak
   6.952 +depth}~= 1:
   6.953 +\\[2\smallskipamount]
   6.954 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.955 +\hbox{}\qquad\qquad $\textit{a} = a_2$ \\
   6.956 +\hbox{}\qquad\qquad $\textit{b} = a_1$ \\
   6.957 +\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
   6.958 +\hbox{}\qquad\qquad $\textit{ys} = \textit{LCons}~a_1~(\textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega)$ \\[2\smallskipamount]
   6.959 +Total time: 726 ms.
   6.960 +\postw
   6.961 +
   6.962 +The lazy list $\textit{xs}$ is simply $[a_2, a_2, a_2, \ldots]$, whereas
   6.963 +$\textit{ys}$ is $[a_1, a_2, a_2, a_2, \ldots]$, i.e., a lasso-shaped list with
   6.964 +$[a_1]$ as its stem and $[a_2]$ as its cycle. In general, the list segment
   6.965 +within the scope of the {THE} binder corresponds to the lasso's cycle, whereas
   6.966 +the segment leading to the binder is the stem.
   6.967 +
   6.968 +A salient property of coinductive datatypes is that two objects are considered
   6.969 +equal if and only if they lead to the same observations. For example, the lazy
   6.970 +lists $\textrm{THE}~\omega.\; \omega =
   6.971 +\textit{LCons}~a~(\textit{LCons}~b~\omega)$ and
   6.972 +$\textit{LCons}~a~(\textrm{THE}~\omega.\; \omega =
   6.973 +\textit{LCons}~b~(\textit{LCons}~a~\omega))$ are identical, because both lead
   6.974 +to the sequence of observations $a$, $b$, $a$, $b$, \hbox{\ldots} (or,
   6.975 +equivalently, both encode the infinite list $[a, b, a, b, \ldots]$). This
   6.976 +concept of equality for coinductive datatypes is called bisimulation and is
   6.977 +defined coinductively.
   6.978 +
   6.979 +Internally, Nitpick encodes the coinductive bisimilarity predicate as part of
   6.980 +the Kodkod problem to ensure that distinct objects lead to different
   6.981 +observations. This precaution is somewhat expensive and often unnecessary, so it
   6.982 +can be disabled by setting the \textit{bisim\_depth} option to $-1$. The
   6.983 +bisimilarity check is then performed \textsl{after} the counterexample has been
   6.984 +found to ensure correctness. If this after-the-fact check fails, the
   6.985 +counterexample is tagged as ``likely genuine'' and Nitpick recommends to try
   6.986 +again with \textit{bisim\_depth} set to a nonnegative integer. Disabling the
   6.987 +check for the previous example saves approximately 150~milli\-seconds; the speed
   6.988 +gains can be more significant for larger scopes.
   6.989 +
   6.990 +The next formula illustrates the need for bisimilarity (either as a Kodkod
   6.991 +predicate or as an after-the-fact check) to prevent spurious counterexamples:
   6.992 +
   6.993 +\prew
   6.994 +\textbf{lemma} ``$\lbrakk xs = \textit{LCons}~a~\textit{xs};\>\, \textit{ys} = \textit{LCons}~a~\textit{ys}\rbrakk
   6.995 +\,\Longrightarrow\, \textit{xs} = \textit{ys}$'' \\
   6.996 +\textbf{nitpick} [\textit{bisim\_depth} = $-1$,\, \textit{show\_datatypes}] \\[2\smallskipamount]
   6.997 +\slshape Nitpick found a likely genuine counterexample for $\textit{card}~'a$ = 2: \\[2\smallskipamount]
   6.998 +\hbox{}\qquad Free variables: \nopagebreak \\
   6.999 +\hbox{}\qquad\qquad $a = a_2$ \\
  6.1000 +\hbox{}\qquad\qquad $\textit{xs} = \textsl{THE}~\omega.\; \omega =
  6.1001 +\textit{LCons}~a_2~\omega$ \\
  6.1002 +\hbox{}\qquad\qquad $\textit{ys} = \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega$ \\
  6.1003 +\hbox{}\qquad Codatatype:\strut \nopagebreak \\
  6.1004 +\hbox{}\qquad\qquad $'a~\textit{llist} =
  6.1005 +\{\!\begin{aligned}[t]
  6.1006 +  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega, \\[-2pt]
  6.1007 +  & \textsl{THE}~\omega.\; \omega = \textit{LCons}~a_2~\omega,\> \unr\}\end{aligned}$
  6.1008 +\\[2\smallskipamount]
  6.1009 +Try again with ``\textit{bisim\_depth}'' set to a nonnegative value to confirm
  6.1010 +that the counterexample is genuine. \\[2\smallskipamount]
  6.1011 +{\upshape\textbf{nitpick}} \\[2\smallskipamount]
  6.1012 +\slshape Nitpick found no counterexample.
  6.1013 +\postw
  6.1014 +
  6.1015 +In the first \textbf{nitpick} invocation, the after-the-fact check discovered 
  6.1016 +that the two known elements of type $'a~\textit{llist}$ are bisimilar.
  6.1017 +
  6.1018 +A compromise between leaving out the bisimilarity predicate from the Kodkod
  6.1019 +problem and performing the after-the-fact check is to specify a lower
  6.1020 +nonnegative \textit{bisim\_depth} value than the default one provided by
  6.1021 +Nitpick. In general, a value of $K$ means that Nitpick will require all lists to
  6.1022 +be distinguished from each other by their prefixes of length $K$. Be aware that
  6.1023 +setting $K$ to a too low value can overconstrain Nitpick, preventing it from
  6.1024 +finding any counterexamples.
  6.1025 +
  6.1026 +\subsection{Boxing}
  6.1027 +\label{boxing}
  6.1028 +
  6.1029 +Nitpick normally maps function and product types directly to the corresponding
  6.1030 +Kodkod concepts. As a consequence, if $'a$ has cardinality 3 and $'b$ has
  6.1031 +cardinality 4, then $'a \times {'}b$ has cardinality 12 ($= 4 \times 3$) and $'a
  6.1032 +\Rightarrow {'}b$ has cardinality 64 ($= 4^3$). In some circumstances, it pays
  6.1033 +off to treat these types in the same way as plain datatypes, by approximating
  6.1034 +them by a subset of a given cardinality. This technique is called ``boxing'' and
  6.1035 +is particularly useful for functions passed as arguments to other functions, for
  6.1036 +high-arity functions, and for large tuples. Under the hood, boxing involves
  6.1037 +wrapping occurrences of the types $'a \times {'}b$ and $'a \Rightarrow {'}b$ in
  6.1038 +isomorphic datatypes, as can be seen by enabling the \textit{debug} option.
  6.1039 +
  6.1040 +To illustrate boxing, we consider a formalization of $\lambda$-terms represented
  6.1041 +using de Bruijn's notation:
  6.1042 +
  6.1043 +\prew
  6.1044 +\textbf{datatype} \textit{tm} = \textit{Var}~\textit{nat}~$\mid$~\textit{Lam}~\textit{tm} $\mid$ \textit{App~tm~tm}
  6.1045 +\postw
  6.1046 +
  6.1047 +The $\textit{lift}~t~k$ function increments all variables with indices greater
  6.1048 +than or equal to $k$ by one:
  6.1049 +
  6.1050 +\prew
  6.1051 +\textbf{primrec} \textit{lift} \textbf{where} \\
  6.1052 +``$\textit{lift}~(\textit{Var}~j)~k = \textit{Var}~(\textrm{if}~j < k~\textrm{then}~j~\textrm{else}~j + 1)$'' $\mid$ \\
  6.1053 +``$\textit{lift}~(\textit{Lam}~t)~k = \textit{Lam}~(\textit{lift}~t~(k + 1))$'' $\mid$ \\
  6.1054 +``$\textit{lift}~(\textit{App}~t~u)~k = \textit{App}~(\textit{lift}~t~k)~(\textit{lift}~u~k)$''
  6.1055 +\postw
  6.1056 +
  6.1057 +The $\textit{loose}~t~k$ predicate returns \textit{True} if and only if
  6.1058 +term $t$ has a loose variable with index $k$ or more:
  6.1059 +
  6.1060 +\prew
  6.1061 +\textbf{primrec}~\textit{loose} \textbf{where} \\
  6.1062 +``$\textit{loose}~(\textit{Var}~j)~k = (j \ge k)$'' $\mid$ \\
  6.1063 +``$\textit{loose}~(\textit{Lam}~t)~k = \textit{loose}~t~(\textit{Suc}~k)$'' $\mid$ \\
  6.1064 +``$\textit{loose}~(\textit{App}~t~u)~k = (\textit{loose}~t~k \mathrel{\lor} \textit{loose}~u~k)$''
  6.1065 +\postw
  6.1066 +
  6.1067 +Next, the $\textit{subst}~\sigma~t$ function applies the substitution $\sigma$
  6.1068 +on $t$:
  6.1069 +
  6.1070 +\prew
  6.1071 +\textbf{primrec}~\textit{subst} \textbf{where} \\
  6.1072 +``$\textit{subst}~\sigma~(\textit{Var}~j) = \sigma~j$'' $\mid$ \\
  6.1073 +``$\textit{subst}~\sigma~(\textit{Lam}~t) = {}$\phantom{''} \\
  6.1074 +\phantom{``}$\textit{Lam}~(\textit{subst}~(\lambda n.\> \textrm{case}~n~\textrm{of}~0 \Rightarrow \textit{Var}~0 \mid \textit{Suc}~m \Rightarrow \textit{lift}~(\sigma~m)~1)~t)$'' $\mid$ \\
  6.1075 +``$\textit{subst}~\sigma~(\textit{App}~t~u) = \textit{App}~(\textit{subst}~\sigma~t)~(\textit{subst}~\sigma~u)$''
  6.1076 +\postw
  6.1077 +
  6.1078 +A substitution is a function that maps variable indices to terms. Observe that
  6.1079 +$\sigma$ is a function passed as argument and that Nitpick can't optimize it
  6.1080 +away, because the recursive call for the \textit{Lam} case involves an altered
  6.1081 +version. Also notice the \textit{lift} call, which increments the variable
  6.1082 +indices when moving under a \textit{Lam}.
  6.1083 +
  6.1084 +A reasonable property to expect of substitution is that it should leave closed
  6.1085 +terms unchanged. Alas, even this simple property does not hold:
  6.1086 +
  6.1087 +\pre
  6.1088 +\textbf{lemma}~``$\lnot\,\textit{loose}~t~0 \,\Longrightarrow\, \textit{subst}~\sigma~t = t$'' \\
  6.1089 +\textbf{nitpick} [\textit{verbose}] \\[2\smallskipamount]
  6.1090 +\slshape
  6.1091 +Trying 8 scopes: \nopagebreak \\
  6.1092 +\hbox{}\qquad \textit{card~nat}~= 1, \textit{card tm}~= 1, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 1; \\
  6.1093 +\hbox{}\qquad \textit{card~nat}~= 2, \textit{card tm}~= 2, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 2; \\
  6.1094 +\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
  6.1095 +\hbox{}\qquad \textit{card~nat}~= 8, \textit{card tm}~= 8, and \textit{card} ``$\textit{nat} \Rightarrow \textit{tm}$'' = 8. \\[2\smallskipamount]
  6.1096 +Nitpick found a counterexample for \textit{card~nat}~= 6, \textit{card~tm}~= 6,
  6.1097 +and \textit{card}~``$\textit{nat} \Rightarrow \textit{tm}$''~= 6: \\[2\smallskipamount]
  6.1098 +\hbox{}\qquad Free variables: \nopagebreak \\
  6.1099 +\hbox{}\qquad\qquad $\sigma = \undef(\!\begin{aligned}[t]
  6.1100 +& 0 := \textit{Var}~0,\>
  6.1101 +  1 := \textit{Var}~0,\>
  6.1102 +  2 := \textit{Var}~0, \\[-2pt]
  6.1103 +& 3 := \textit{Var}~0,\>
  6.1104 +  4 := \textit{Var}~0,\>
  6.1105 +  5 := \textit{Var}~0)\end{aligned}$ \\
  6.1106 +\hbox{}\qquad\qquad $t = \textit{Lam}~(\textit{Lam}~(\textit{Var}~1))$ \\[2\smallskipamount]
  6.1107 +Total time: $4679$ ms.
  6.1108 +\postw
  6.1109 +
  6.1110 +Using \textit{eval}, we find out that $\textit{subst}~\sigma~t =
  6.1111 +\textit{Lam}~(\textit{Lam}~(\textit{Var}~0))$. Using the traditional
  6.1112 +$\lambda$-term notation, $t$~is
  6.1113 +$\lambda x\, y.\> x$ whereas $\textit{subst}~\sigma~t$ is $\lambda x\, y.\> y$.
  6.1114 +The bug is in \textit{subst}: The $\textit{lift}~(\sigma~m)~1$ call should be
  6.1115 +replaced with $\textit{lift}~(\sigma~m)~0$.
  6.1116 +
  6.1117 +An interesting aspect of Nitpick's verbose output is that it assigned inceasing
  6.1118 +cardinalities from 1 to 8 to the type $\textit{nat} \Rightarrow \textit{tm}$.
  6.1119 +For the formula of interest, knowing 6 values of that type was enough to find
  6.1120 +the counterexample. Without boxing, $46\,656$ ($= 6^6$) values must be
  6.1121 +considered, a hopeless undertaking:
  6.1122 +
  6.1123 +\prew
  6.1124 +\textbf{nitpick} [\textit{dont\_box}] \\[2\smallskipamount]
  6.1125 +{\slshape Nitpick ran out of time after checking 4 of 8 scopes.}
  6.1126 +\postw
  6.1127 +
  6.1128 +{\looseness=-1
  6.1129 +Boxing can be enabled or disabled globally or on a per-type basis using the
  6.1130 +\textit{box} option. Moreover, setting the cardinality of a function or
  6.1131 +product type implicitly enables boxing for that type. Nitpick usually performs
  6.1132 +reasonable choices about which types should be boxed, but option tweaking
  6.1133 +sometimes helps.
  6.1134 +
  6.1135 +}
  6.1136 +
  6.1137 +\subsection{Scope Monotonicity}
  6.1138 +\label{scope-monotonicity}
  6.1139 +
  6.1140 +The \textit{card} option (together with \textit{iter}, \textit{bisim\_depth},
  6.1141 +and \textit{max}) controls which scopes are actually tested. In general, to
  6.1142 +exhaust all models below a certain cardinality bound, the number of scopes that
  6.1143 +Nitpick must consider increases exponentially with the number of type variables
  6.1144 +(and \textbf{typedecl}'d types) occurring in the formula. Given the default
  6.1145 +cardinality specification of 1--8, no fewer than $8^4 = 4096$ scopes must be
  6.1146 +considered for a formula involving $'a$, $'b$, $'c$, and $'d$.
  6.1147 +
  6.1148 +Fortunately, many formulas exhibit a property called \textsl{scope
  6.1149 +monotonicity}, meaning that if the formula is falsifiable for a given scope,
  6.1150 +it is also falsifiable for all larger scopes \cite[p.~165]{jackson-2006}.
  6.1151 +
  6.1152 +Consider the formula
  6.1153 +
  6.1154 +\prew
  6.1155 +\textbf{lemma}~``$\textit{length~xs} = \textit{length~ys} \,\Longrightarrow\, \textit{rev}~(\textit{zip~xs~ys}) = \textit{zip~xs}~(\textit{rev~ys})$''
  6.1156 +\postw
  6.1157 +
  6.1158 +where \textit{xs} is of type $'a~\textit{list}$ and \textit{ys} is of type
  6.1159 +$'b~\textit{list}$. A priori, Nitpick would need to consider 512 scopes to
  6.1160 +exhaust the specification \textit{card}~= 1--8. However, our intuition tells us
  6.1161 +that any counterexample found with a small scope would still be a counterexample
  6.1162 +in a larger scope---by simply ignoring the fresh $'a$ and $'b$ values provided
  6.1163 +by the larger scope. Nitpick comes to the same conclusion after a careful
  6.1164 +inspection of the formula and the relevant definitions:
  6.1165 +
  6.1166 +\prew
  6.1167 +\textbf{nitpick}~[\textit{verbose}] \\[2\smallskipamount]
  6.1168 +\slshape
  6.1169 +The types ``\kern1pt$'a$'' and ``\kern1pt$'b$'' passed the monotonicity test.
  6.1170 +Nitpick might be able to skip some scopes.
  6.1171 + \\[2\smallskipamount]
  6.1172 +Trying 8 scopes: \\
  6.1173 +\hbox{}\qquad \textit{card} $'a$~= 1, \textit{card} $'b$~= 1,
  6.1174 +\textit{card} \textit{nat}~= 1, \textit{card} ``$('a \times {'}b)$
  6.1175 +\textit{list}''~= 1, \\
  6.1176 +\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 1, and
  6.1177 +\textit{card} ``\kern1pt$'b$ \textit{list}''~= 1. \\
  6.1178 +\hbox{}\qquad \textit{card} $'a$~= 2, \textit{card} $'b$~= 2,
  6.1179 +\textit{card} \textit{nat}~= 2, \textit{card} ``$('a \times {'}b)$
  6.1180 +\textit{list}''~= 2, \\
  6.1181 +\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 2, and
  6.1182 +\textit{card} ``\kern1pt$'b$ \textit{list}''~= 2. \\
  6.1183 +\hbox{}\qquad $\qquad\vdots$ \\[.5\smallskipamount]
  6.1184 +\hbox{}\qquad \textit{card} $'a$~= 8, \textit{card} $'b$~= 8,
  6.1185 +\textit{card} \textit{nat}~= 8, \textit{card} ``$('a \times {'}b)$
  6.1186 +\textit{list}''~= 8, \\
  6.1187 +\hbox{}\qquad\quad \textit{card} ``\kern1pt$'a$ \textit{list}''~= 8, and
  6.1188 +\textit{card} ``\kern1pt$'b$ \textit{list}''~= 8.
  6.1189 +\\[2\smallskipamount]
  6.1190 +Nitpick found a counterexample for
  6.1191 +\textit{card} $'a$~= 5, \textit{card} $'b$~= 5,
  6.1192 +\textit{card} \textit{nat}~= 5, \textit{card} ``$('a \times {'}b)$
  6.1193 +\textit{list}''~= 5, \textit{card} ``\kern1pt$'a$ \textit{list}''~= 5, and
  6.1194 +\textit{card} ``\kern1pt$'b$ \textit{list}''~= 5:
  6.1195 +\\[2\smallskipamount]
  6.1196 +\hbox{}\qquad Free variables: \nopagebreak \\
  6.1197 +\hbox{}\qquad\qquad $\textit{xs} = [a_4, a_5]$ \\
  6.1198 +\hbox{}\qquad\qquad $\textit{ys} = [b_3, b_3]$ \\[2\smallskipamount]
  6.1199 +Total time: 1636 ms.
  6.1200 +\postw
  6.1201 +
  6.1202 +In theory, it should be sufficient to test a single scope:
  6.1203 +
  6.1204 +\prew
  6.1205 +\textbf{nitpick}~[\textit{card}~= 8]
  6.1206 +\postw
  6.1207 +
  6.1208 +However, this is often less efficient in practice and may lead to overly complex
  6.1209 +counterexamples.
  6.1210 +
  6.1211 +If the monotonicity check fails but we believe that the formula is monotonic (or
  6.1212 +we don't mind missing some counterexamples), we can pass the
  6.1213 +\textit{mono} option. To convince yourself that this option is risky,
  6.1214 +simply consider this example from \S\ref{skolemization}:
  6.1215 +
  6.1216 +\prew
  6.1217 +\textbf{lemma} ``$\exists g.\; \forall x\Colon 'b.~g~(f~x) = x
  6.1218 + \,\Longrightarrow\, \forall y\Colon {'}a.\; \exists x.~y = f~x$'' \\
  6.1219 +\textbf{nitpick} [\textit{mono}] \\[2\smallskipamount]
  6.1220 +{\slshape Nitpick found no counterexample.} \\[2\smallskipamount]
  6.1221 +\textbf{nitpick} \\[2\smallskipamount]
  6.1222 +\slshape
  6.1223 +Nitpick found a counterexample for \textit{card} $'a$~= 2 and \textit{card} $'b$~=~1: \\
  6.1224 +\hbox{}\qquad $\vdots$
  6.1225 +\postw
  6.1226 +
  6.1227 +(It turns out the formula holds if and only if $\textit{card}~'a \le
  6.1228 +\textit{card}~'b$.) Although this is rarely advisable, the automatic
  6.1229 +monotonicity checks can be disabled by passing \textit{non\_mono}
  6.1230 +(\S\ref{optimizations}).
  6.1231 +
  6.1232 +As insinuated in \S\ref{natural-numbers-and-integers} and
  6.1233 +\S\ref{inductive-datatypes}, \textit{nat}, \textit{int}, and inductive datatypes
  6.1234 +are normally monotonic and treated as such. The same is true for record types,
  6.1235 +\textit{rat}, \textit{real}, and some \textbf{typedef}'d types. Thus, given the
  6.1236 +cardinality specification 1--8, a formula involving \textit{nat}, \textit{int},
  6.1237 +\textit{int~list}, \textit{rat}, and \textit{rat~list} will lead Nitpick to
  6.1238 +consider only 8~scopes instead of $32\,768$.
  6.1239 +
  6.1240 +\section{Case Studies}
  6.1241 +\label{case-studies}
  6.1242 +
  6.1243 +As a didactic device, the previous section focused mostly on toy formulas whose
  6.1244 +validity can easily be assessed just by looking at the formula. We will now
  6.1245 +review two somewhat more realistic case studies that are within Nitpick's
  6.1246 +reach:\ a context-free grammar modeled by mutually inductive sets and a
  6.1247 +functional implementation of AA trees. The results presented in this
  6.1248 +section were produced with the following settings:
  6.1249 +
  6.1250 +\prew
  6.1251 +\textbf{nitpick\_params} [\textit{max\_potential}~= 0,\, \textit{max\_threads} = 2]
  6.1252 +\postw
  6.1253 +
  6.1254 +\subsection{A Context-Free Grammar}
  6.1255 +\label{a-context-free-grammar}
  6.1256 +
  6.1257 +Our first case study is taken from section 7.4 in the Isabelle tutorial
  6.1258 +\cite{isa-tutorial}. The following grammar, originally due to Hopcroft and
  6.1259 +Ullman, produces all strings with an equal number of $a$'s and $b$'s:
  6.1260 +
  6.1261 +\prew
  6.1262 +\begin{tabular}{@{}r@{$\;\,$}c@{$\;\,$}l@{}}
  6.1263 +$S$ & $::=$ & $\epsilon \mid bA \mid aB$ \\
  6.1264 +$A$ & $::=$ & $aS \mid bAA$ \\
  6.1265 +$B$ & $::=$ & $bS \mid aBB$
  6.1266 +\end{tabular}
  6.1267 +\postw
  6.1268 +
  6.1269 +The intuition behind the grammar is that $A$ generates all string with one more
  6.1270 +$a$ than $b$'s and $B$ generates all strings with one more $b$ than $a$'s.
  6.1271 +
  6.1272 +The alphabet consists exclusively of $a$'s and $b$'s:
  6.1273 +
  6.1274 +\prew
  6.1275 +\textbf{datatype} \textit{alphabet}~= $a$ $\mid$ $b$
  6.1276 +\postw
  6.1277 +
  6.1278 +Strings over the alphabet are represented by \textit{alphabet list}s.
  6.1279 +Nonterminals in the grammar become sets of strings. The production rules
  6.1280 +presented above can be expressed as a mutually inductive definition:
  6.1281 +
  6.1282 +\prew
  6.1283 +\textbf{inductive\_set} $S$ \textbf{and} $A$ \textbf{and} $B$ \textbf{where} \\
  6.1284 +\textit{R1}:\kern.4em ``$[] \in S$'' $\,\mid$ \\
  6.1285 +\textit{R2}:\kern.4em ``$w \in A\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
  6.1286 +\textit{R3}:\kern.4em ``$w \in B\,\Longrightarrow\, a \mathbin{\#} w \in S$'' $\,\mid$ \\
  6.1287 +\textit{R4}:\kern.4em ``$w \in S\,\Longrightarrow\, a \mathbin{\#} w \in A$'' $\,\mid$ \\
  6.1288 +\textit{R5}:\kern.4em ``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$'' $\,\mid$ \\
  6.1289 +\textit{R6}:\kern.4em ``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
  6.1290 +\postw
  6.1291 +
  6.1292 +The conversion of the grammar into the inductive definition was done manually by
  6.1293 +Joe Blow, an underpaid undergraduate student. As a result, some errors might
  6.1294 +have sneaked in.
  6.1295 +
  6.1296 +Debugging faulty specifications is at the heart of Nitpick's \textsl{raison
  6.1297 +d'\^etre}. A good approach is to state desirable properties of the specification
  6.1298 +(here, that $S$ is exactly the set of strings over $\{a, b\}$ with as many $a$'s
  6.1299 +as $b$'s) and check them with Nitpick. If the properties are correctly stated,
  6.1300 +counterexamples will point to bugs in the specification. For our grammar
  6.1301 +example, we will proceed in two steps, separating the soundness and the
  6.1302 +completeness of the set $S$. First, soundness:
  6.1303 +
  6.1304 +\prew
  6.1305 +\textbf{theorem}~\textit{S\_sound}: \\
  6.1306 +``$w \in S \longrightarrow \textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
  6.1307 +  \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]$'' \\
  6.1308 +\textbf{nitpick} \\[2\smallskipamount]
  6.1309 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
  6.1310 +\hbox{}\qquad Free variable: \nopagebreak \\
  6.1311 +\hbox{}\qquad\qquad $w = [b]$
  6.1312 +\postw
  6.1313 +
  6.1314 +It would seem that $[b] \in S$. How could this be? An inspection of the
  6.1315 +introduction rules reveals that the only rule with a right-hand side of the form
  6.1316 +$b \mathbin{\#} {\ldots} \in S$ that could have introduced $[b]$ into $S$ is
  6.1317 +\textit{R5}:
  6.1318 +
  6.1319 +\prew
  6.1320 +``$w \in S\,\Longrightarrow\, b \mathbin{\#} w \in S$''
  6.1321 +\postw
  6.1322 +
  6.1323 +On closer inspection, we can see that this rule is wrong. To match the
  6.1324 +production $B ::= bS$, the second $S$ should be a $B$. We fix the typo and try
  6.1325 +again:
  6.1326 +
  6.1327 +\prew
  6.1328 +\textbf{nitpick} \\[2\smallskipamount]
  6.1329 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
  6.1330 +\hbox{}\qquad Free variable: \nopagebreak \\
  6.1331 +\hbox{}\qquad\qquad $w = [a, a, b]$
  6.1332 +\postw
  6.1333 +
  6.1334 +Some detective work is necessary to find out what went wrong here. To get $[a,
  6.1335 +a, b] \in S$, we need $[a, b] \in B$ by \textit{R3}, which in turn can only come
  6.1336 +from \textit{R6}:
  6.1337 +
  6.1338 +\prew
  6.1339 +``$\lbrakk v \in B;\> v \in B\rbrakk \,\Longrightarrow\, a \mathbin{\#} v \mathbin{@} w \in B$''
  6.1340 +\postw
  6.1341 +
  6.1342 +Now, this formula must be wrong: The same assumption occurs twice, and the
  6.1343 +variable $w$ is unconstrained. Clearly, one of the two occurrences of $v$ in
  6.1344 +the assumptions should have been a $w$.
  6.1345 +
  6.1346 +With the correction made, we don't get any counterexample from Nitpick. Let's
  6.1347 +move on and check completeness:
  6.1348 +
  6.1349 +\prew
  6.1350 +\textbf{theorem}~\textit{S\_complete}: \\
  6.1351 +``$\textit{length}~[x\mathbin{\leftarrow} w.\; x = a] =
  6.1352 +   \textit{length}~[x\mathbin{\leftarrow} w.\; x = b]
  6.1353 +  \longrightarrow w \in S$'' \\
  6.1354 +\textbf{nitpick} \\[2\smallskipamount]
  6.1355 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
  6.1356 +\hbox{}\qquad Free variable: \nopagebreak \\
  6.1357 +\hbox{}\qquad\qquad $w = [b, b, a, a]$
  6.1358 +\postw
  6.1359 +
  6.1360 +Apparently, $[b, b, a, a] \notin S$, even though it has the same numbers of
  6.1361 +$a$'s and $b$'s. But since our inductive definition passed the soundness check,
  6.1362 +the introduction rules we have are probably correct. Perhaps we simply lack an
  6.1363 +introduction rule. Comparing the grammar with the inductive definition, our
  6.1364 +suspicion is confirmed: Joe Blow simply forgot the production $A ::= bAA$,
  6.1365 +without which the grammar cannot generate two or more $b$'s in a row. So we add
  6.1366 +the rule
  6.1367 +
  6.1368 +\prew
  6.1369 +``$\lbrakk v \in A;\> w \in A\rbrakk \,\Longrightarrow\, b \mathbin{\#} v \mathbin{@} w \in A$''
  6.1370 +\postw
  6.1371 +
  6.1372 +With this last change, we don't get any counterexamples from Nitpick for either
  6.1373 +soundness or completeness. We can even generalize our result to cover $A$ and
  6.1374 +$B$ as well:
  6.1375 +
  6.1376 +\prew
  6.1377 +\textbf{theorem} \textit{S\_A\_B\_sound\_and\_complete}: \\
  6.1378 +``$w \in S \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b]$'' \\
  6.1379 +``$w \in A \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] + 1$'' \\
  6.1380 +``$w \in B \longleftrightarrow \textit{length}~[x \mathbin{\leftarrow} w.\; x = b] = \textit{length}~[x \mathbin{\leftarrow} w.\; x = a] + 1$'' \\
  6.1381 +\textbf{nitpick} \\[2\smallskipamount]
  6.1382 +\slshape Nitpick found no counterexample.
  6.1383 +\postw
  6.1384 +
  6.1385 +\subsection{AA Trees}
  6.1386 +\label{aa-trees}
  6.1387 +
  6.1388 +AA trees are a kind of balanced trees discovered by Arne Andersson that provide
  6.1389 +similar performance to red-black trees, but with a simpler implementation
  6.1390 +\cite{andersson-1993}. They can be used to store sets of elements equipped with
  6.1391 +a total order $<$. We start by defining the datatype and some basic extractor
  6.1392 +functions:
  6.1393 +
  6.1394 +\prew
  6.1395 +\textbf{datatype} $'a$~\textit{tree} = $\Lambda$ $\mid$ $N$ ``\kern1pt$'a\Colon \textit{linorder}$'' \textit{nat} ``\kern1pt$'a$ \textit{tree}'' ``\kern1pt$'a$ \textit{tree}''  \\[2\smallskipamount]
  6.1396 +\textbf{primrec} \textit{data} \textbf{where} \\
  6.1397 +``$\textit{data}~\Lambda = \undef$'' $\,\mid$ \\
  6.1398 +``$\textit{data}~(N~x~\_~\_~\_) = x$'' \\[2\smallskipamount]
  6.1399 +\textbf{primrec} \textit{dataset} \textbf{where} \\
  6.1400 +``$\textit{dataset}~\Lambda = \{\}$'' $\,\mid$ \\
  6.1401 +``$\textit{dataset}~(N~x~\_~t~u) = \{x\} \cup \textit{dataset}~t \mathrel{\cup} \textit{dataset}~u$'' \\[2\smallskipamount]
  6.1402 +\textbf{primrec} \textit{level} \textbf{where} \\
  6.1403 +``$\textit{level}~\Lambda = 0$'' $\,\mid$ \\
  6.1404 +``$\textit{level}~(N~\_~k~\_~\_) = k$'' \\[2\smallskipamount]
  6.1405 +\textbf{primrec} \textit{left} \textbf{where} \\
  6.1406 +``$\textit{left}~\Lambda = \Lambda$'' $\,\mid$ \\
  6.1407 +``$\textit{left}~(N~\_~\_~t~\_) = t$'' \\[2\smallskipamount]
  6.1408 +\textbf{primrec} \textit{right} \textbf{where} \\
  6.1409 +``$\textit{right}~\Lambda = \Lambda$'' $\,\mid$ \\
  6.1410 +``$\textit{right}~(N~\_~\_~\_~u) = u$''
  6.1411 +\postw
  6.1412 +
  6.1413 +The wellformedness criterion for AA trees is fairly complex. Wikipedia states it
  6.1414 +as follows \cite{wikipedia-2009-aa-trees}:
  6.1415 +
  6.1416 +\kern.2\parskip %% TYPESETTING
  6.1417 +
  6.1418 +\pre
  6.1419 +Each node has a level field, and the following invariants must remain true for
  6.1420 +the tree to be valid:
  6.1421 +
  6.1422 +\raggedright
  6.1423 +
  6.1424 +\kern-.4\parskip %% TYPESETTING
  6.1425 +
  6.1426 +\begin{enum}
  6.1427 +\item[]
  6.1428 +\begin{enum}
  6.1429 +\item[1.] The level of a leaf node is one.
  6.1430 +\item[2.] The level of a left child is strictly less than that of its parent.
  6.1431 +\item[3.] The level of a right child is less than or equal to that of its parent.
  6.1432 +\item[4.] The level of a right grandchild is strictly less than that of its grandparent.
  6.1433 +\item[5.] Every node of level greater than one must have two children.
  6.1434 +\end{enum}
  6.1435 +\end{enum}
  6.1436 +\post
  6.1437 +
  6.1438 +\kern.4\parskip %% TYPESETTING
  6.1439 +
  6.1440 +The \textit{wf} predicate formalizes this description:
  6.1441 +
  6.1442 +\prew
  6.1443 +\textbf{primrec} \textit{wf} \textbf{where} \\
  6.1444 +``$\textit{wf}~\Lambda = \textit{True}$'' $\,\mid$ \\
  6.1445 +``$\textit{wf}~(N~\_~k~t~u) =$ \\
  6.1446 +\phantom{``}$(\textrm{if}~t = \Lambda~\textrm{then}$ \\
  6.1447 +\phantom{``$(\quad$}$k = 1 \mathrel{\land} (u = \Lambda \mathrel{\lor} (\textit{level}~u = 1 \mathrel{\land} \textit{left}~u = \Lambda \mathrel{\land} \textit{right}~u = \Lambda))$ \\
  6.1448 +\phantom{``$($}$\textrm{else}$ \\
  6.1449 +\hbox{}\phantom{``$(\quad$}$\textit{wf}~t \mathrel{\land} \textit{wf}~u
  6.1450 +\mathrel{\land} u \not= \Lambda \mathrel{\land} \textit{level}~t < k
  6.1451 +\mathrel{\land} \textit{level}~u \le k$ \\
  6.1452 +\hbox{}\phantom{``$(\quad$}${\land}\; \textit{level}~(\textit{right}~u) < k)$''
  6.1453 +\postw
  6.1454 +
  6.1455 +Rebalancing the tree upon insertion and removal of elements is performed by two
  6.1456 +auxiliary functions called \textit{skew} and \textit{split}, defined below:
  6.1457 +
  6.1458 +\prew
  6.1459 +\textbf{primrec} \textit{skew} \textbf{where} \\
  6.1460 +``$\textit{skew}~\Lambda = \Lambda$'' $\,\mid$ \\
  6.1461 +``$\textit{skew}~(N~x~k~t~u) = {}$ \\
  6.1462 +\phantom{``}$(\textrm{if}~t \not= \Lambda \mathrel{\land} k =
  6.1463 +\textit{level}~t~\textrm{then}$ \\
  6.1464 +\phantom{``(\quad}$N~(\textit{data}~t)~k~(\textit{left}~t)~(N~x~k~
  6.1465 +(\textit{right}~t)~u)$ \\
  6.1466 +\phantom{``(}$\textrm{else}$ \\
  6.1467 +\phantom{``(\quad}$N~x~k~t~u)$''
  6.1468 +\postw
  6.1469 +
  6.1470 +\prew
  6.1471 +\textbf{primrec} \textit{split} \textbf{where} \\
  6.1472 +``$\textit{split}~\Lambda = \Lambda$'' $\,\mid$ \\
  6.1473 +``$\textit{split}~(N~x~k~t~u) = {}$ \\
  6.1474 +\phantom{``}$(\textrm{if}~u \not= \Lambda \mathrel{\land} k =
  6.1475 +\textit{level}~(\textit{right}~u)~\textrm{then}$ \\
  6.1476 +\phantom{``(\quad}$N~(\textit{data}~u)~(\textit{Suc}~k)~
  6.1477 +(N~x~k~t~(\textit{left}~u))~(\textit{right}~u)$ \\
  6.1478 +\phantom{``(}$\textrm{else}$ \\
  6.1479 +\phantom{``(\quad}$N~x~k~t~u)$''
  6.1480 +\postw
  6.1481 +
  6.1482 +Performing a \textit{skew} or a \textit{split} should have no impact on the set
  6.1483 +of elements stored in the tree:
  6.1484 +
  6.1485 +\prew
  6.1486 +\textbf{theorem}~\textit{dataset\_skew\_split}:\\
  6.1487 +``$\textit{dataset}~(\textit{skew}~t) = \textit{dataset}~t$'' \\
  6.1488 +``$\textit{dataset}~(\textit{split}~t) = \textit{dataset}~t$'' \\
  6.1489 +\textbf{nitpick} \\[2\smallskipamount]
  6.1490 +{\slshape Nitpick ran out of time after checking 7 of 8 scopes.}
  6.1491 +\postw
  6.1492 +
  6.1493 +Furthermore, applying \textit{skew} or \textit{split} to a well-formed tree
  6.1494 +should not alter the tree:
  6.1495 +
  6.1496 +\prew
  6.1497 +\textbf{theorem}~\textit{wf\_skew\_split}:\\
  6.1498 +``$\textit{wf}~t\,\Longrightarrow\, \textit{skew}~t = t$'' \\
  6.1499 +``$\textit{wf}~t\,\Longrightarrow\, \textit{split}~t = t$'' \\
  6.1500 +\textbf{nitpick} \\[2\smallskipamount]
  6.1501 +{\slshape Nitpick found no counterexample.}
  6.1502 +\postw
  6.1503 +
  6.1504 +Insertion is implemented recursively. It preserves the sort order:
  6.1505 +
  6.1506 +\prew
  6.1507 +\textbf{primrec}~\textit{insort} \textbf{where} \\
  6.1508 +``$\textit{insort}~\Lambda~x = N~x~1~\Lambda~\Lambda$'' $\,\mid$ \\
  6.1509 +``$\textit{insort}~(N~y~k~t~u)~x =$ \\
  6.1510 +\phantom{``}$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~(\textrm{if}~x < y~\textrm{then}~\textit{insort}~t~x~\textrm{else}~t)$ \\
  6.1511 +\phantom{``$({*}~(\textit{split} \circ \textit{skew})~{*})~(N~y~k~$}$(\textrm{if}~x > y~\textrm{then}~\textit{insort}~u~x~\textrm{else}~u))$''
  6.1512 +\postw
  6.1513 +
  6.1514 +Notice that we deliberately commented out the application of \textit{skew} and
  6.1515 +\textit{split}. Let's see if this causes any problems:
  6.1516 +
  6.1517 +\prew
  6.1518 +\textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
  6.1519 +\textbf{nitpick} \\[2\smallskipamount]
  6.1520 +\slshape Nitpick found a counterexample for \textit{card} $'a$ = 4: \\[2\smallskipamount]
  6.1521 +\hbox{}\qquad Free variables: \nopagebreak \\
  6.1522 +\hbox{}\qquad\qquad $t = N~a_3~1~\Lambda~\Lambda$ \\
  6.1523 +\hbox{}\qquad\qquad $x = a_4$ \\[2\smallskipamount]
  6.1524 +Hint: Maybe you forgot a type constraint?
  6.1525 +\postw
  6.1526 +
  6.1527 +It's hard to see why this is a counterexample. The hint is of no help here. To
  6.1528 +improve readability, we will restrict the theorem to \textit{nat}, so that we
  6.1529 +don't need to look up the value of the $\textit{op}~{<}$ constant to find out
  6.1530 +which element is smaller than the other. In addition, we will tell Nitpick to
  6.1531 +display the value of $\textit{insort}~t~x$ using the \textit{eval} option. This
  6.1532 +gives
  6.1533 +
  6.1534 +\prew
  6.1535 +\textbf{theorem} \textit{wf\_insort\_nat}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~(x\Colon\textit{nat}))$'' \\
  6.1536 +\textbf{nitpick} [\textit{eval} = ``$\textit{insort}~t~x$''] \\[2\smallskipamount]
  6.1537 +\slshape Nitpick found a counterexample: \\[2\smallskipamount]
  6.1538 +\hbox{}\qquad Free variables: \nopagebreak \\
  6.1539 +\hbox{}\qquad\qquad $t = N~1~1~\Lambda~\Lambda$ \\
  6.1540 +\hbox{}\qquad\qquad $x = 0$ \\
  6.1541 +\hbox{}\qquad Evaluated term: \\
  6.1542 +\hbox{}\qquad\qquad $\textit{insort}~t~x = N~1~1~(N~0~1~\Lambda~\Lambda)~\Lambda$
  6.1543 +\postw
  6.1544 +
  6.1545 +Nitpick's output reveals that the element $0$ was added as a left child of $1$,
  6.1546 +where both have a level of 1. This violates the second AA tree invariant, which
  6.1547 +states that a left child's level must be less than its parent's. This shouldn't
  6.1548 +come as a surprise, considering that we commented out the tree rebalancing code.
  6.1549 +Reintroducing the code seems to solve the problem:
  6.1550 +
  6.1551 +\prew
  6.1552 +\textbf{theorem}~\textit{wf\_insort}:\kern.4em ``$\textit{wf}~t\,\Longrightarrow\, \textit{wf}~(\textit{insort}~t~x)$'' \\
  6.1553 +\textbf{nitpick} \\[2\smallskipamount]
  6.1554 +{\slshape Nitpick ran out of time after checking 6 of 8 scopes.}
  6.1555 +\postw
  6.1556 +
  6.1557 +Insertion should transform the set of elements represented by the tree in the
  6.1558 +obvious way:
  6.1559 +
  6.1560 +\prew
  6.1561 +\textbf{theorem} \textit{dataset\_insort}:\kern.4em
  6.1562 +``$\textit{dataset}~(\textit{insort}~t~x) = \{x\} \cup \textit{dataset}~t$'' \\
  6.1563 +\textbf{nitpick} \\[2\smallskipamount]
  6.1564 +{\slshape Nitpick ran out of time after checking 5 of 8 scopes.}
  6.1565 +\postw
  6.1566 +
  6.1567 +We could continue like this and sketch a complete theory of AA trees without
  6.1568 +performing a single proof. Once the definitions and main theorems are in place
  6.1569 +and have been thoroughly tested using Nitpick, we could start working on the
  6.1570 +proofs. Developing theories this way usually saves time, because faulty theorems
  6.1571 +and definitions are discovered much earlier in the process.
  6.1572 +
  6.1573 +\section{Option Reference}
  6.1574 +\label{option-reference}
  6.1575 +
  6.1576 +\def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
  6.1577 +\def\qty#1{$\left<\textit{#1}\right>$}
  6.1578 +\def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
  6.1579 +\def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
  6.1580 +\def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
  6.1581 +\def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
  6.1582 +\def\ops#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
  6.1583 +\def\opt#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
  6.1584 +\def\opu#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
  6.1585 +\def\opusmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
  6.1586 +
  6.1587 +Nitpick's behavior can be influenced by various options, which can be specified
  6.1588 +in brackets after the \textbf{nitpick} command. Default values can be set
  6.1589 +using \textbf{nitpick\_\allowbreak params}. For example:
  6.1590 +
  6.1591 +\prew
  6.1592 +\textbf{nitpick\_params} [\textit{verbose}, \,\textit{timeout} = 60$\,s$]
  6.1593 +\postw
  6.1594 +
  6.1595 +The options are categorized as follows:\ mode of operation
  6.1596 +(\S\ref{mode-of-operation}), scope of search (\S\ref{scope-of-search}), output
  6.1597 +format (\S\ref{output-format}), automatic counterexample checks
  6.1598 +(\S\ref{authentication}), optimizations
  6.1599 +(\S\ref{optimizations}), and timeouts (\S\ref{timeouts}).
  6.1600 +
  6.1601 +The number of options can be overwhelming at first glance. Do not let that worry
  6.1602 +you: Nitpick's defaults have been chosen so that it almost always does the right
  6.1603 +thing, and the most important options have been covered in context in
  6.1604 +\S\ref{first-steps}.
  6.1605 +
  6.1606 +The descriptions below refer to the following syntactic quantities:
  6.1607 +
  6.1608 +\begin{enum}
  6.1609 +\item[$\bullet$] \qtybf{string}: A string.
  6.1610 +\item[$\bullet$] \qtybf{bool}: \textit{true} or \textit{false}.
  6.1611 +\item[$\bullet$] \qtybf{bool\_or\_smart}: \textit{true}, \textit{false}, or \textit{smart}.
  6.1612 +\item[$\bullet$] \qtybf{int}: An integer. Negative integers are prefixed with a hyphen.
  6.1613 +\item[$\bullet$] \qtybf{int\_or\_smart}: An integer or \textit{smart}.
  6.1614 +\item[$\bullet$] \qtybf{int\_range}: An integer (e.g., 3) or a range
  6.1615 +of nonnegative integers (e.g., $1$--$4$). The range symbol `--' can be entered as \texttt{-} (hyphen) or \texttt{\char`\\\char`\<midarrow\char`\>}.
  6.1616 +
  6.1617 +\item[$\bullet$] \qtybf{int\_seq}: A comma-separated sequence of ranges of integers (e.g.,~1{,}3{,}\allowbreak6--8).
  6.1618 +\item[$\bullet$] \qtybf{time}: An integer followed by $\textit{min}$ (minutes), $s$ (seconds), or \textit{ms}
  6.1619 +(milliseconds), or the keyword \textit{none} ($\infty$ years).
  6.1620 +\item[$\bullet$] \qtybf{const}: The name of a HOL constant.
  6.1621 +\item[$\bullet$] \qtybf{term}: A HOL term (e.g., ``$f~x$'').
  6.1622 +\item[$\bullet$] \qtybf{term\_list}: A space-separated list of HOL terms (e.g.,
  6.1623 +``$f~x$''~``$g~y$'').
  6.1624 +\item[$\bullet$] \qtybf{type}: A HOL type.
  6.1625 +\end{enum}
  6.1626 +
  6.1627 +Default values are indicated in square brackets. Boolean options have a negated
  6.1628 +counterpart (e.g., \textit{auto} vs.\ \textit{no\_auto}). When setting Boolean
  6.1629 +options, ``= \textit{true}'' may be omitted.
  6.1630 +
  6.1631 +\subsection{Mode of Operation}
  6.1632 +\label{mode-of-operation}
  6.1633 +
  6.1634 +\begin{enum}
  6.1635 +\opfalse{auto}{no\_auto}
  6.1636 +Specifies whether Nitpick should be run automatically on newly entered theorems.
  6.1637 +For automatic runs, \textit{user\_axioms} (\S\ref{mode-of-operation}) and
  6.1638 +\textit{assms} (\S\ref{mode-of-operation}) are implicitly enabled,
  6.1639 +\textit{blocking} (\S\ref{mode-of-operation}), \textit{verbose}
  6.1640 +(\S\ref{output-format}), and \textit{debug} (\S\ref{output-format}) are
  6.1641 +disabled, \textit{max\_potential} (\S\ref{output-format}) is taken to be 0, and
  6.1642 +\textit{auto\_timeout} (\S\ref{timeouts}) is used as the time limit instead of
  6.1643 +\textit{timeout} (\S\ref{timeouts}). The output is also more concise.
  6.1644 +
  6.1645 +\nopagebreak
  6.1646 +{\small See also \textit{auto\_timeout} (\S\ref{timeouts}).}
  6.1647 +
  6.1648 +\optrue{blocking}{non\_blocking}
  6.1649 +Specifies whether the \textbf{nitpick} command should operate synchronously.
  6.1650 +The asynchronous (non-blocking) mode lets the user start proving the putative
  6.1651 +theorem while Nitpick looks for a counterexample, but it can also be more
  6.1652 +confusing. For technical reasons, automatic runs currently always block.
  6.1653 +
  6.1654 +\nopagebreak
  6.1655 +{\small See also \textit{auto} (\S\ref{mode-of-operation}).}
  6.1656 +
  6.1657 +\optrue{falsify}{satisfy}
  6.1658 +Specifies whether Nitpick should look for falsifying examples (countermodels) or
  6.1659 +satisfying examples (models). This manual assumes throughout that
  6.1660 +\textit{falsify} is enabled.
  6.1661 +
  6.1662 +\opsmart{user\_axioms}{no\_user\_axioms}
  6.1663 +Specifies whether the user-defined axioms (specified using 
  6.1664 +\textbf{axiomatization} and \textbf{axioms}) should be considered. If the option
  6.1665 +is set to \textit{smart}, Nitpick performs an ad hoc axiom selection based on
  6.1666 +the constants that occur in the formula to falsify. The option is implicitly set
  6.1667 +to \textit{true} for automatic runs.
  6.1668 +
  6.1669 +\textbf{Warning:} If the option is set to \textit{true}, Nitpick might
  6.1670 +nonetheless ignore some polymorphic axioms. Counterexamples generated under
  6.1671 +these conditions are tagged as ``likely genuine.'' The \textit{debug}
  6.1672 +(\S\ref{output-format}) option can be used to find out which axioms were
  6.1673 +considered.
  6.1674 +
  6.1675 +\nopagebreak
  6.1676 +{\small See also \textit{auto} (\S\ref{mode-of-operation}), \textit{assms}
  6.1677 +(\S\ref{mode-of-operation}), and \textit{debug} (\S\ref{output-format}).}
  6.1678 +
  6.1679 +\optrue{assms}{no\_assms}
  6.1680 +Specifies whether the relevant assumptions in structured proof should be
  6.1681 +considered. The option is implicitly enabled for automatic runs.
  6.1682 +
  6.1683 +\nopagebreak
  6.1684 +{\small See also \textit{auto} (\S\ref{mode-of-operation})
  6.1685 +and \textit{user\_axioms} (\S\ref{mode-of-operation}).}
  6.1686 +
  6.1687 +\opfalse{overlord}{no\_overlord}
  6.1688 +Specifies whether Nitpick should put its temporary files in
  6.1689 +\texttt{\$ISABELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
  6.1690 +debugging Nitpick but also unsafe if several instances of the tool are run
  6.1691 +simultaneously.
  6.1692 +
  6.1693 +\nopagebreak
  6.1694 +{\small See also \textit{debug} (\S\ref{output-format}).}
  6.1695 +\end{enum}
  6.1696 +
  6.1697 +\subsection{Scope of Search}
  6.1698 +\label{scope-of-search}
  6.1699 +
  6.1700 +\begin{enum}
  6.1701 +\opu{card}{type}{int\_seq}
  6.1702 +Specifies the sequence of cardinalities to use for a given type. For
  6.1703 +\textit{nat} and \textit{int}, the cardinality fully specifies the subset used
  6.1704 +to approximate the type. For example:
  6.1705 +%
  6.1706 +$$\hbox{\begin{tabular}{@{}rll@{}}%
  6.1707 +\textit{card nat} = 4 & induces & $\{0,\, 1,\, 2,\, 3\}$ \\
  6.1708 +\textit{card int} = 4 & induces & $\{-1,\, 0,\, +1,\, +2\}$ \\
  6.1709 +\textit{card int} = 5 & induces & $\{-2,\, -1,\, 0,\, +1,\, +2\}.$%
  6.1710 +\end{tabular}}$$
  6.1711 +%
  6.1712 +In general:
  6.1713 +%
  6.1714 +$$\hbox{\begin{tabular}{@{}rll@{}}%
  6.1715 +\textit{card nat} = $K$ & induces & $\{0,\, \ldots,\, K - 1\}$ \\
  6.1716 +\textit{card int} = $K$ & induces & $\{-\lceil K/2 \rceil + 1,\, \ldots,\, +\lfloor K/2 \rfloor\}.$%
  6.1717 +\end{tabular}}$$
  6.1718 +%
  6.1719 +For free types, and often also for \textbf{typedecl}'d types, it usually makes
  6.1720 +sense to specify cardinalities as a range of the form \textit{$1$--$n$}.
  6.1721 +Although function and product types are normally mapped directly to the
  6.1722 +corresponding Kodkod concepts, setting
  6.1723 +the cardinality of such types is also allowed and implicitly enables ``boxing''
  6.1724 +for them, as explained in the description of the \textit{box}~\qty{type}
  6.1725 +and \textit{box} (\S\ref{scope-of-search}) options.
  6.1726 +
  6.1727 +\nopagebreak
  6.1728 +{\small See also \textit{mono} (\S\ref{scope-of-search}).}
  6.1729 +
  6.1730 +\opt{card}{int\_seq}{$\mathbf{1}$--$\mathbf{8}$}
  6.1731 +Specifies the default sequence of cardinalities to use. This can be overridden
  6.1732 +on a per-type basis using the \textit{card}~\qty{type} option described above.
  6.1733 +
  6.1734 +\opu{max}{const}{int\_seq}
  6.1735 +Specifies the sequence of maximum multiplicities to use for a given
  6.1736 +(co)in\-duc\-tive datatype constructor. A constructor's multiplicity is the
  6.1737 +number of distinct values that it can construct. Nonsensical values (e.g.,
  6.1738 +\textit{max}~[]~$=$~2) are silently repaired. This option is only available for
  6.1739 +datatypes equipped with several constructors.
  6.1740 +
  6.1741 +\ops{max}{int\_seq}
  6.1742 +Specifies the default sequence of maximum multiplicities to use for
  6.1743 +(co)in\-duc\-tive datatype constructors. This can be overridden on a per-constructor
  6.1744 +basis using the \textit{max}~\qty{const} option described above.
  6.1745 +
  6.1746 +\opusmart{wf}{const}{non\_wf}
  6.1747 +Specifies whether the specified (co)in\-duc\-tively defined predicate is
  6.1748 +well-founded. The option can take the following values:
  6.1749 +
  6.1750 +\begin{enum}
  6.1751 +\item[$\bullet$] \textbf{\textit{true}}: Tentatively treat the (co)in\-duc\-tive
  6.1752 +predicate as if it were well-founded. Since this is generally not sound when the
  6.1753 +predicate is not well-founded, the counterexamples are tagged as ``likely
  6.1754 +genuine.''
  6.1755 +
  6.1756 +\item[$\bullet$] \textbf{\textit{false}}: Treat the (co)in\-duc\-tive predicate
  6.1757 +as if it were not well-founded. The predicate is then unrolled as prescribed by
  6.1758 +the \textit{star\_linear\_preds}, \textit{iter}~\qty{const}, and \textit{iter}
  6.1759 +options.
  6.1760 +
  6.1761 +\item[$\bullet$] \textbf{\textit{smart}}: Try to prove that the inductive
  6.1762 +predicate is well-founded using Isabelle's \textit{lexicographic\_order} and
  6.1763 +\textit{sizechange} tactics. If this succeeds (or the predicate occurs with an
  6.1764 +appropriate polarity in the formula to falsify), use an efficient fixed point
  6.1765 +equation as specification of the predicate; otherwise, unroll the predicates
  6.1766 +according to the \textit{iter}~\qty{const} and \textit{iter} options.
  6.1767 +\end{enum}
  6.1768 +
  6.1769 +\nopagebreak
  6.1770 +{\small See also \textit{iter} (\S\ref{scope-of-search}),
  6.1771 +\textit{star\_linear\_preds} (\S\ref{optimizations}), and \textit{tac\_timeout}
  6.1772 +(\S\ref{timeouts}).}
  6.1773 +
  6.1774 +\opsmart{wf}{non\_wf}
  6.1775 +Specifies the default wellfoundedness setting to use. This can be overridden on
  6.1776 +a per-predicate basis using the \textit{wf}~\qty{const} option above.
  6.1777 +
  6.1778 +\opu{iter}{const}{int\_seq}
  6.1779 +Specifies the sequence of iteration counts to use when unrolling a given
  6.1780 +(co)in\-duc\-tive predicate. By default, unrolling is applied for inductive
  6.1781 +predicates that occur negatively and coinductive predicates that occur
  6.1782 +positively in the formula to falsify and that cannot be proved to be
  6.1783 +well-founded, but this behavior is influenced by the \textit{wf} option. The
  6.1784 +iteration counts are automatically bounded by the cardinality of the predicate's
  6.1785 +domain.
  6.1786 +
  6.1787 +{\small See also \textit{wf} (\S\ref{scope-of-search}) and
  6.1788 +\textit{star\_linear\_preds} (\S\ref{optimizations}).}
  6.1789 +
  6.1790 +\opt{iter}{int\_seq}{$\mathbf{1{,}2{,}4{,}8{,}12{,}16{,}24{,}32}$}
  6.1791 +Specifies the sequence of iteration counts to use when unrolling (co)in\-duc\-tive
  6.1792 +predicates. This can be overridden on a per-predicate basis using the
  6.1793 +\textit{iter} \qty{const} option above.
  6.1794 +
  6.1795 +\opt{bisim\_depth}{int\_seq}{$\mathbf{7}$}
  6.1796 +Specifies the sequence of iteration counts to use when unrolling the
  6.1797 +bisimilarity predicate generated by Nitpick for coinductive datatypes. A value
  6.1798 +of $-1$ means that no predicate is generated, in which case Nitpick performs an
  6.1799 +after-the-fact check to see if the known coinductive datatype values are
  6.1800 +bidissimilar. If two values are found to be bisimilar, the counterexample is
  6.1801 +tagged as ``likely genuine.'' The iteration counts are automatically bounded by
  6.1802 +the sum of the cardinalities of the coinductive datatypes occurring in the
  6.1803 +formula to falsify.
  6.1804 +
  6.1805 +\opusmart{box}{type}{dont\_box}
  6.1806 +Specifies whether Nitpick should attempt to wrap (``box'') a given function or
  6.1807 +product type in an isomorphic datatype internally. Boxing is an effective mean
  6.1808 +to reduce the search space and speed up Nitpick, because the isomorphic datatype
  6.1809 +is approximated by a subset of the possible function or pair values;
  6.1810 +like other drastic optimizations, it can also prevent the discovery of
  6.1811 +counterexamples. The option can take the following values:
  6.1812 +
  6.1813 +\begin{enum}
  6.1814 +\item[$\bullet$] \textbf{\textit{true}}: Box the specified type whenever
  6.1815 +practicable.
  6.1816 +\item[$\bullet$] \textbf{\textit{false}}: Never box the type.
  6.1817 +\item[$\bullet$] \textbf{\textit{smart}}: Box the type only in contexts where it
  6.1818 +is likely to help. For example, $n$-tuples where $n > 2$ and arguments to
  6.1819 +higher-order functions are good candidates for boxing.
  6.1820 +\end{enum}
  6.1821 +
  6.1822 +Setting the \textit{card}~\qty{type} option for a function or product type
  6.1823 +implicitly enables boxing for that type.
  6.1824 +
  6.1825 +\nopagebreak
  6.1826 +{\small See also \textit{verbose} (\S\ref{output-format})
  6.1827 +and \textit{debug} (\S\ref{output-format}).}
  6.1828 +
  6.1829 +\opsmart{box}{dont\_box}
  6.1830 +Specifies the default boxing setting to use. This can be overridden on a
  6.1831 +per-type basis using the \textit{box}~\qty{type} option described above.
  6.1832 +
  6.1833 +\opusmart{mono}{type}{non\_mono}
  6.1834 +Specifies whether the specified type should be considered monotonic when
  6.1835 +enumerating scopes. If the option is set to \textit{smart}, Nitpick performs a
  6.1836 +monotonicity check on the type. Setting this option to \textit{true} can reduce
  6.1837 +the number of scopes tried, but it also diminishes the theoretical chance of
  6.1838 +finding a counterexample, as demonstrated in \S\ref{scope-monotonicity}.
  6.1839 +
  6.1840 +\nopagebreak
  6.1841 +{\small See also \textit{card} (\S\ref{scope-of-search}),
  6.1842 +\textit{coalesce\_type\_vars} (\S\ref{scope-of-search}), and \textit{verbose}
  6.1843 +(\S\ref{output-format}).}
  6.1844 +
  6.1845 +\opsmart{mono}{non\_box}
  6.1846 +Specifies the default monotonicity setting to use. This can be overridden on a
  6.1847 +per-type basis using the \textit{mono}~\qty{type} option described above.
  6.1848 +
  6.1849 +\opfalse{coalesce\_type\_vars}{dont\_coalesce\_type\_vars}
  6.1850 +Specifies whether type variables with the same sort constraints should be
  6.1851 +merged. Setting this option to \textit{true} can reduce the number of scopes
  6.1852 +tried and the size of the generated Kodkod formulas, but it also diminishes the
  6.1853 +theoretical chance of finding a counterexample.
  6.1854 +
  6.1855 +{\small See also \textit{mono} (\S\ref{scope-of-search}).}
  6.1856 +\end{enum}
  6.1857 +
  6.1858 +\subsection{Output Format}
  6.1859 +\label{output-format}
  6.1860 +
  6.1861 +\begin{enum}
  6.1862 +\opfalse{verbose}{quiet}
  6.1863 +Specifies whether the \textbf{nitpick} command should explain what it does. This
  6.1864 +option is useful to determine which scopes are tried or which SAT solver is
  6.1865 +used. This option is implicitly disabled for automatic runs.
  6.1866 +
  6.1867 +\nopagebreak
  6.1868 +{\small See also \textit{auto} (\S\ref{mode-of-operation}).}
  6.1869 +
  6.1870 +\opfalse{debug}{no\_debug}
  6.1871 +Specifies whether Nitpick should display additional debugging information beyond
  6.1872 +what \textit{verbose} already displays. Enabling \textit{debug} also enables
  6.1873 +\textit{verbose} and \textit{show\_all} behind the scenes. The \textit{debug}
  6.1874 +option is implicitly disabled for automatic runs.
  6.1875 +
  6.1876 +\nopagebreak
  6.1877 +{\small See also \textit{auto} (\S\ref{mode-of-operation}), \textit{overlord}
  6.1878 +(\S\ref{mode-of-operation}), and \textit{batch\_size} (\S\ref{optimizations}).}
  6.1879 +
  6.1880 +\optrue{show\_skolems}{hide\_skolem}
  6.1881 +Specifies whether the values of Skolem constants should be displayed as part of
  6.1882 +counterexamples. Skolem constants correspond to bound variables in the original
  6.1883 +formula and usually help us to understand why the counterexample falsifies the
  6.1884 +formula.
  6.1885 +
  6.1886 +\nopagebreak
  6.1887 +{\small See also \textit{skolemize} (\S\ref{optimizations}).}
  6.1888 +
  6.1889 +\opfalse{show\_datatypes}{hide\_datatypes}
  6.1890 +Specifies whether the subsets used to approximate (co)in\-duc\-tive datatypes should
  6.1891 +be displayed as part of counterexamples. Such subsets are sometimes helpful when
  6.1892 +investigating whether a potential counterexample is genuine or spurious, but
  6.1893 +their potential for clutter is real.
  6.1894 +
  6.1895 +\opfalse{show\_consts}{hide\_consts}
  6.1896 +Specifies whether the values of constants occurring in the formula (including
  6.1897 +its axioms) should be displayed along with any counterexample. These values are
  6.1898 +sometimes helpful when investigating why a counterexample is
  6.1899 +genuine, but they can clutter the output.
  6.1900 +
  6.1901 +\opfalse{show\_all}{dont\_show\_all}
  6.1902 +Enabling this option effectively enables \textit{show\_skolems},
  6.1903 +\textit{show\_datatypes}, and \textit{show\_consts}.
  6.1904 +
  6.1905 +\opt{max\_potential}{int}{$\mathbf{1}$}
  6.1906 +Specifies the maximum number of potential counterexamples to display. Setting
  6.1907 +this option to 0 speeds up the search for a genuine counterexample. This option
  6.1908 +is implicitly set to 0 for automatic runs. If you set this option to a value
  6.1909 +greater than 1, you will need an incremental SAT solver: For efficiency, it is
  6.1910 +recommended to install the JNI version of MiniSat and set \textit{sat\_solver} =
  6.1911 +\textit{MiniSatJNI}. Also be aware that many of the counterexamples may look
  6.1912 +identical, unless the \textit{show\_all} (\S\ref{output-format}) option is
  6.1913 +enabled.
  6.1914 +
  6.1915 +\nopagebreak
  6.1916 +{\small See also \textit{auto} (\S\ref{mode-of-operation}),
  6.1917 +\textit{check\_potential} (\S\ref{authentication}), and
  6.1918 +\textit{sat\_solver} (\S\ref{optimizations}).}
  6.1919 +
  6.1920 +\opt{max\_genuine}{int}{$\mathbf{1}$}
  6.1921 +Specifies the maximum number of genuine counterexamples to display. If you set
  6.1922 +this option to a value greater than 1, you will need an incremental SAT solver:
  6.1923 +For efficiency, it is recommended to install the JNI version of MiniSat and set
  6.1924 +\textit{sat\_solver} = \textit{MiniSatJNI}. Also be aware that many of the
  6.1925 +counterexamples may look identical, unless the \textit{show\_all}
  6.1926 +(\S\ref{output-format}) option is enabled.
  6.1927 +
  6.1928 +\nopagebreak
  6.1929 +{\small See also \textit{check\_genuine} (\S\ref{authentication}) and
  6.1930 +\textit{sat\_solver} (\S\ref{optimizations}).}
  6.1931 +
  6.1932 +\ops{eval}{term\_list}
  6.1933 +Specifies the list of terms whose values should be displayed along with
  6.1934 +counterexamples. This option suffers from an ``observer effect'': Nitpick might
  6.1935 +find different counterexamples for different values of this option.
  6.1936 +
  6.1937 +\opu{format}{term}{int\_seq}
  6.1938 +Specifies how to uncurry the value displayed for a variable or constant.
  6.1939 +Uncurrying sometimes increases the readability of the output for high-arity
  6.1940 +functions. For example, given the variable $y \mathbin{\Colon} {'a}\Rightarrow
  6.1941 +{'b}\Rightarrow {'c}\Rightarrow {'d}\Rightarrow {'e}\Rightarrow {'f}\Rightarrow
  6.1942 +{'g}$, setting \textit{format}~$y$ = 3 tells Nitpick to group the last three
  6.1943 +arguments, as if the type had been ${'a}\Rightarrow {'b}\Rightarrow
  6.1944 +{'c}\Rightarrow {'d}\times {'e}\times {'f}\Rightarrow {'g}$. In general, a list
  6.1945 +of values $n_1,\ldots,n_k$ tells Nitpick to show the last $n_k$ arguments as an
  6.1946 +$n_k$-tuple, the previous $n_{k-1}$ arguments as an $n_{k-1}$-tuple, and so on;
  6.1947 +arguments that are not accounted for are left alone, as if the specification had
  6.1948 +been $1,\ldots,1,n_1,\ldots,n_k$.
  6.1949 +
  6.1950 +\nopagebreak
  6.1951 +{\small See also \textit{uncurry} (\S\ref{optimizations}).}
  6.1952 +
  6.1953 +\opt{format}{int\_seq}{$\mathbf{1}$}
  6.1954 +Specifies the default format to use. Irrespective of the default format, the
  6.1955 +extra arguments to a Skolem constant corresponding to the outer bound variables
  6.1956 +are kept separated from the remaining arguments, the \textbf{for} arguments of
  6.1957 +an inductive definitions are kept separated from the remaining arguments, and
  6.1958 +the iteration counter of an unrolled inductive definition is shown alone. The
  6.1959 +default format can be overridden on a per-variable or per-constant basis using
  6.1960 +the \textit{format}~\qty{term} option described above.
  6.1961 +\end{enum}
  6.1962 +
  6.1963 +%% MARK: Authentication
  6.1964 +\subsection{Authentication}
  6.1965 +\label{authentication}
  6.1966 +
  6.1967 +\begin{enum}
  6.1968 +\opfalse{check\_potential}{trust\_potential}
  6.1969 +Specifies whether potential counterexamples should be given to Isabelle's
  6.1970 +\textit{auto} tactic to assess their validity. If a potential counterexample is
  6.1971 +shown to be genuine, Nitpick displays a message to this effect and terminates.
  6.1972 +
  6.1973 +\nopagebreak
  6.1974 +{\small See also \textit{max\_potential} (\S\ref{output-format}) and
  6.1975 +\textit{auto\_timeout} (\S\ref{timeouts}).}
  6.1976 +
  6.1977 +\opfalse{check\_genuine}{trust\_genuine}
  6.1978 +Specifies whether genuine and likely genuine counterexamples should be given to
  6.1979 +Isabelle's \textit{auto} tactic to assess their validity. If a ``genuine''
  6.1980 +counterexample is shown to be spurious, the user is kindly asked to send a bug
  6.1981 +report to the author at
  6.1982 +\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@in.tum.de}.
  6.1983 +
  6.1984 +\nopagebreak
  6.1985 +{\small See also \textit{max\_genuine} (\S\ref{output-format}) and
  6.1986 +\textit{auto\_timeout} (\S\ref{timeouts}).}
  6.1987 +
  6.1988 +\ops{expect}{string}
  6.1989 +Specifies the expected outcome, which must be one of the following:
  6.1990 +
  6.1991 +\begin{enum}
  6.1992 +\item[$\bullet$] \textbf{\textit{genuine}}: Nitpick found a genuine counterexample.
  6.1993 +\item[$\bullet$] \textbf{\textit{likely\_genuine}}: Nitpick found a ``likely
  6.1994 +genuine'' counterexample (i.e., a counterexample that is genuine unless
  6.1995 +it contradicts a missing axiom or a dangerous option was used inappropriately).
  6.1996 +\item[$\bullet$] \textbf{\textit{potential}}: Nitpick found a potential counterexample.
  6.1997 +\item[$\bullet$] \textbf{\textit{none}}: Nitpick found no counterexample.
  6.1998 +\item[$\bullet$] \textbf{\textit{unknown}}: Nitpick encountered some problem (e.g.,
  6.1999 +Kodkod ran out of memory).
  6.2000 +\end{enum}
  6.2001 +
  6.2002 +Nitpick emits an error if the actual outcome differs from the expected outcome.
  6.2003 +This option is useful for regression testing.
  6.2004 +\end{enum}
  6.2005 +
  6.2006 +\subsection{Optimizations}
  6.2007 +\label{optimizations}
  6.2008 +
  6.2009 +\def\cpp{C\nobreak\raisebox{.1ex}{+}\nobreak\raisebox{.1ex}{+}}
  6.2010 +
  6.2011 +\sloppy
  6.2012 +
  6.2013 +\begin{enum}
  6.2014 +\opt{sat\_solver}{string}{smart}
  6.2015 +Specifies which SAT solver to use. SAT solvers implemented in C or \cpp{} tend
  6.2016 +to be faster than their Java counterparts, but they can be more difficult to
  6.2017 +install. Also, if you set the \textit{max\_potential} (\S\ref{output-format}) or
  6.2018 +\textit{max\_genuine} (\S\ref{output-format}) option to a value greater than 1,
  6.2019 +you will need an incremental SAT solver, such as \textit{MiniSatJNI}
  6.2020 +(recommended) or \textit{SAT4J}.
  6.2021 +
  6.2022 +The supported solvers are listed below:
  6.2023 +
  6.2024 +\begin{enum}
  6.2025 +
  6.2026 +\item[$\bullet$] \textbf{\textit{MiniSat}}: MiniSat is an efficient solver
  6.2027 +written in \cpp{}. To use MiniSat, set the environment variable
  6.2028 +\texttt{MINISAT\_HOME} to the directory that contains the \texttt{minisat}
  6.2029 +executable. The \cpp{} sources and executables for MiniSat are available at
  6.2030 +\url{http://minisat.se/MiniSat.html}. Nitpick has been tested with versions 1.14
  6.2031 +and 2.0 beta (2007-07-21).
  6.2032 +
  6.2033 +\item[$\bullet$] \textbf{\textit{MiniSatJNI}}: The JNI (Java Native Interface)
  6.2034 +version of MiniSat is bundled in \texttt{nativesolver.\allowbreak tgz}, which
  6.2035 +you will find on Kodkod's web site \cite{kodkod-2009}. Unlike the standard
  6.2036 +version of MiniSat, the JNI version can be used incrementally.
  6.2037 +
  6.2038 +\item[$\bullet$] \textbf{\textit{PicoSAT}}: PicoSAT is an efficient solver
  6.2039 +written in C. It is bundled with Kodkodi and requires no further installation or
  6.2040 +configuration steps. Alternatively, you can install a standard version of
  6.2041 +PicoSAT and set the environment variable \texttt{PICOSAT\_HOME} to the directory
  6.2042 +that contains the \texttt{picosat} executable. The C sources for PicoSAT are
  6.2043 +available at \url{http://fmv.jku.at/picosat/} and are also bundled with Kodkodi.
  6.2044 +Nitpick has been tested with version 913.
  6.2045 +
  6.2046 +\item[$\bullet$] \textbf{\textit{zChaff}}: zChaff is an efficient solver written
  6.2047 +in \cpp{}. To use zChaff, set the environment variable \texttt{ZCHAFF\_HOME} to
  6.2048 +the directory that contains the \texttt{zchaff} executable. The \cpp{} sources
  6.2049 +and executables for zChaff are available at
  6.2050 +\url{http://www.princeton.edu/~chaff/zchaff.html}. Nitpick has been tested with
  6.2051 +versions 2004-05-13, 2004-11-15, and 2007-03-12.
  6.2052 +
  6.2053 +\item[$\bullet$] \textbf{\textit{zChaffJNI}}: The JNI version of zChaff is
  6.2054 +bundled in \texttt{native\-solver.\allowbreak tgz}, which you will find on
  6.2055 +Kodkod's web site \cite{kodkod-2009}.
  6.2056 +
  6.2057 +\item[$\bullet$] \textbf{\textit{RSat}}: RSat is an efficient solver written in
  6.2058 +\cpp{}. To use RSat, set the environment variable \texttt{RSAT\_HOME} to the
  6.2059 +directory that contains the \texttt{rsat} executable. The \cpp{} sources for
  6.2060 +RSat are available at \url{http://reasoning.cs.ucla.edu/rsat/}. Nitpick has been
  6.2061 +tested with version 2.01.
  6.2062 +
  6.2063 +\item[$\bullet$] \textbf{\textit{BerkMin}}: BerkMin561 is an efficient solver
  6.2064 +written in C. To use BerkMin, set the environment variable
  6.2065 +\texttt{BERKMIN\_HOME} to the directory that contains the \texttt{BerkMin561}
  6.2066 +executable. The BerkMin executables are available at
  6.2067 +\url{http://eigold.tripod.com/BerkMin.html}.
  6.2068 +
  6.2069 +\item[$\bullet$] \textbf{\textit{BerkMinAlloy}}: Variant of BerkMin that is
  6.2070 +included with Alloy 4 and calls itself ``sat56'' in its banner text. To use this
  6.2071 +version of BerkMin, set the environment variable
  6.2072 +\texttt{BERKMINALLOY\_HOME} to the directory that contains the \texttt{berkmin}
  6.2073 +executable.
  6.2074 +
  6.2075 +\item[$\bullet$] \textbf{\textit{Jerusat}}: Jerusat 1.3 is an efficient solver
  6.2076 +written in C. To use Jerusat, set the environment variable
  6.2077 +\texttt{JERUSAT\_HOME} to the directory that contains the \texttt{Jerusat1.3}
  6.2078 +executable. The C sources for Jerusat are available at
  6.2079 +\url{http://www.cs.tau.ac.il/~ale1/Jerusat1.3.tgz}.
  6.2080 +
  6.2081 +\item[$\bullet$] \textbf{\textit{SAT4J}}: SAT4J is a reasonably efficient solver
  6.2082 +written in Java that can be used incrementally. It is bundled with Kodkodi and
  6.2083 +requires no further installation or configuration steps. Do not attempt to
  6.2084 +install the official SAT4J packages, because their API is incompatible with
  6.2085 +Kodkod.
  6.2086 +
  6.2087 +\item[$\bullet$] \textbf{\textit{SAT4JLight}}: Variant of SAT4J that is
  6.2088 +optimized for small problems. It can also be used incrementally.
  6.2089 +
  6.2090 +\item[$\bullet$] \textbf{\textit{HaifaSat}}: HaifaSat 1.0 beta is an
  6.2091 +experimental solver written in \cpp. To use HaifaSat, set the environment
  6.2092 +variable \texttt{HAIFASAT\_\allowbreak HOME} to the directory that contains the
  6.2093 +\texttt{HaifaSat} executable. The \cpp{} sources for HaifaSat are available at
  6.2094 +\url{http://cs.technion.ac.il/~gershman/HaifaSat.htm}.
  6.2095 +
  6.2096 +\item[$\bullet$] \textbf{\textit{smart}}: If \textit{sat\_solver} is set to
  6.2097 +\textit{smart}, Nitpick selects the first solver among MiniSat, PicoSAT, zChaff,
  6.2098 +RSat, BerkMin, BerkMinAlloy, and Jerusat that is recognized by Isabelle. If none
  6.2099 +is found, it falls back on SAT4J, which should always be available. If
  6.2100 +\textit{verbose} is enabled, Nitpick displays which SAT solver was chosen.
  6.2101 +
  6.2102 +\end{enum}
  6.2103 +\fussy
  6.2104 +
  6.2105 +\opt{batch\_size}{int\_or\_smart}{smart}
  6.2106 +Specifies the maximum number of Kodkod problems that should be lumped together
  6.2107 +when invoking Kodkodi. Each problem corresponds to one scope. Lumping problems
  6.2108 +together ensures that Kodkodi is launched less often, but it makes the verbose
  6.2109 +output less readable and is sometimes detrimental to performance. If
  6.2110 +\textit{batch\_size} is set to \textit{smart}, the actual value used is 1 if
  6.2111 +\textit{debug} (\S\ref{output-format}) is set and 64 otherwise.
  6.2112 +
  6.2113 +\optrue{destroy\_constrs}{dont\_destroy\_constrs}
  6.2114 +Specifies whether formulas involving (co)in\-duc\-tive datatype constructors should
  6.2115 +be rewritten to use (automatically generated) discriminators and destructors.
  6.2116 +This optimization can drastically reduce the size of the Boolean formulas given
  6.2117 +to the SAT solver.
  6.2118 +
  6.2119 +\nopagebreak
  6.2120 +{\small See also \textit{debug} (\S\ref{output-format}).}
  6.2121 +
  6.2122 +\optrue{specialize}{dont\_specialize}
  6.2123 +Specifies whether functions invoked with static arguments should be specialized.
  6.2124 +This optimization can drastically reduce the search space, especially for
  6.2125 +higher-order functions.
  6.2126 +
  6.2127 +\nopagebreak
  6.2128 +{\small See also \textit{debug} (\S\ref{output-format}) and
  6.2129 +\textit{show\_consts} (\S\ref{output-format}).}
  6.2130 +
  6.2131 +\optrue{skolemize}{dont\_skolemize}
  6.2132 +Specifies whether the formula should be skolemized. For performance reasons,
  6.2133 +(positive) $\forall$-quanti\-fiers that occur in the scope of a higher-order
  6.2134 +(positive) $\exists$-quanti\-fier are left unchanged.
  6.2135 +
  6.2136 +\nopagebreak
  6.2137 +{\small See also \textit{debug} (\S\ref{output-format}) and
  6.2138 +\textit{show\_skolems} (\S\ref{output-format}).}
  6.2139 +
  6.2140 +\optrue{star\_linear\_preds}{dont\_star\_linear\_preds}
  6.2141 +Specifies whether Nitpick should use Kodkod's transitive closure operator to
  6.2142 +encode non-well-founded ``linear inductive predicates,'' i.e., inductive
  6.2143 +predicates for which each the predicate occurs in at most one assumption of each
  6.2144 +introduction rule. Using the reflexive transitive closure is in principle
  6.2145 +equivalent to setting \textit{iter} to the cardinality of the predicate's
  6.2146 +domain, but it is usually more efficient.
  6.2147 +
  6.2148 +{\small See also \textit{wf} (\S\ref{scope-of-search}), \textit{debug}
  6.2149 +(\S\ref{output-format}), and \textit{iter} (\S\ref{scope-of-search}).}
  6.2150 +
  6.2151 +\optrue{uncurry}{dont\_uncurry}
  6.2152 +Specifies whether Nitpick should uncurry functions. Uncurrying has on its own no
  6.2153 +tangible effect on efficiency, but it creates opportunities for the boxing 
  6.2154 +optimization.
  6.2155 +
  6.2156 +\nopagebreak
  6.2157 +{\small See also \textit{box} (\S\ref{scope-of-search}), \textit{debug}
  6.2158 +(\S\ref{output-format}), and \textit{format} (\S\ref{output-format}).}
  6.2159 +
  6.2160 +\optrue{fast\_descrs}{full\_descrs}
  6.2161 +Specifies whether Nitpick should optimize the definite and indefinite
  6.2162 +description operators (THE and SOME). The optimized versions usually help
  6.2163 +Nitpick generate more counterexamples or at least find them faster, but only the
  6.2164 +unoptimized versions are complete when all types occurring in the formula are
  6.2165 +finite.
  6.2166 +
  6.2167 +{\small See also \textit{debug} (\S\ref{output-format}).}
  6.2168 +
  6.2169 +\optrue{peephole\_optim}{no\_peephole\_optim}
  6.2170 +Specifies whether Nitpick should simplify the generated Kodkod formulas using a
  6.2171 +peephole optimizer. These optimizations can make a significant difference.
  6.2172 +Unless you are tracking down a bug in Nitpick or distrust the peephole
  6.2173 +optimizer, you should leave this option enabled.
  6.2174 +
  6.2175 +\opt{sym\_break}{int}{20}
  6.2176 +Specifies an upper bound on the number of relations for which Kodkod generates
  6.2177 +symmetry breaking predicates. According to the Kodkod documentation
  6.2178 +\cite{kodkod-2009-options}, ``in general, the higher this value, the more
  6.2179 +symmetries will be broken, and the faster the formula will be solved. But,
  6.2180 +setting the value too high may have the opposite effect and slow down the
  6.2181 +solving.''
  6.2182 +
  6.2183 +\opt{sharing\_depth}{int}{3}
  6.2184 +Specifies the depth to which Kodkod should check circuits for equivalence during
  6.2185 +the translation to SAT. The default of 3 is the same as in Alloy. The minimum
  6.2186 +allowed depth is 1. Increasing the sharing may result in a smaller SAT problem,
  6.2187 +but can also slow down Kodkod.
  6.2188 +
  6.2189 +\opfalse{flatten\_props}{dont\_flatten\_props}
  6.2190 +Specifies whether Kodkod should try to eliminate intermediate Boolean variables.
  6.2191 +Although this might sound like a good idea, in practice it can drastically slow
  6.2192 +down Kodkod.
  6.2193 +
  6.2194 +\opt{max\_threads}{int}{0}
  6.2195 +Specifies the maximum number of threads to use in Kodkod. If this option is set
  6.2196 +to 0, Kodkod will compute an appropriate value based on the number of processor
  6.2197 +cores available.
  6.2198 +
  6.2199 +\nopagebreak
  6.2200 +{\small See also \textit{batch\_size} (\S\ref{optimizations}) and
  6.2201 +\textit{timeout} (\S\ref{timeouts}).}
  6.2202 +\end{enum}
  6.2203 +
  6.2204 +\subsection{Timeouts}
  6.2205 +\label{timeouts}
  6.2206 +
  6.2207 +\begin{enum}
  6.2208 +\opt{timeout}{time}{$\mathbf{30}$ s}
  6.2209 +Specifies the maximum amount of time that the \textbf{nitpick} command should
  6.2210 +spend looking for a counterexample. Nitpick tries to honor this constraint as
  6.2211 +well as it can but offers no guarantees. For automatic runs,
  6.2212 +\textit{auto\_timeout} is used instead.
  6.2213 +
  6.2214 +\nopagebreak
  6.2215 +{\small See also \textit{auto} (\S\ref{mode-of-operation})
  6.2216 +and \textit{max\_threads} (\S\ref{optimizations}).}
  6.2217 +
  6.2218 +\opt{auto\_timeout}{time}{$\mathbf{5}$ s}
  6.2219 +Specifies the maximum amount of time that Nitpick should use to find a
  6.2220 +counterexample when running automatically. Nitpick tries to honor this
  6.2221 +constraint as well as it can but offers no guarantees.
  6.2222 +
  6.2223 +\nopagebreak
  6.2224 +{\small See also \textit{auto} (\S\ref{mode-of-operation}).}
  6.2225 +
  6.2226 +\opt{tac\_timeout}{time}{$\mathbf{500}$ ms}
  6.2227 +Specifies the maximum amount of time that the \textit{auto} tactic should use
  6.2228 +when checking a counterexample, and similarly that \textit{lexicographic\_order}
  6.2229 +and \textit{sizechange} should use when checking whether a (co)in\-duc\-tive
  6.2230 +predicate is well-founded. Nitpick tries to honor this constraint as well as it
  6.2231 +can but offers no guarantees.
  6.2232 +
  6.2233 +\nopagebreak
  6.2234 +{\small See also \textit{wf} (\S\ref{scope-of-search}),
  6.2235 +\textit{check\_potential} (\S\ref{authentication}),
  6.2236 +and \textit{check\_genuine} (\S\ref{authentication}).}
  6.2237 +\end{enum}
  6.2238 +
  6.2239 +\section{Attribute Reference}
  6.2240 +\label{attribute-reference}
  6.2241 +
  6.2242 +Nitpick needs to consider the definitions of all constants occurring in a
  6.2243 +formula in order to falsify it. For constants introduced using the
  6.2244 +\textbf{definition} command, the definition is simply the associated
  6.2245 +\textit{\_def} axiom. In contrast, instead of using the internal representation
  6.2246 +of functions synthesized by Isabelle's \textbf{primrec}, \textbf{function}, and
  6.2247 +\textbf{nominal\_primrec} packages, Nitpick relies on the more natural
  6.2248 +equational specification entered by the user.
  6.2249 +
  6.2250 +Behind the scenes, Isabelle's built-in packages and theories rely on the
  6.2251 +following attributes to affect Nitpick's behavior:
  6.2252 +
  6.2253 +\begin{itemize}
  6.2254 +\flushitem{\textit{nitpick\_def}}
  6.2255 +
  6.2256 +\nopagebreak
  6.2257 +This attribute specifies an alternative definition of a constant. The
  6.2258 +alternative definition should be logically equivalent to the constant's actual
  6.2259 +axiomatic definition and should be of the form
  6.2260 +
  6.2261 +\qquad $c~{?}x_1~\ldots~{?}x_n \,\equiv\, t$,
  6.2262 +
  6.2263 +where ${?}x_1, \ldots, {?}x_n$ are distinct variables and $c$ does not occur in
  6.2264 +$t$.
  6.2265 +
  6.2266 +\flushitem{\textit{nitpick\_simp}}
  6.2267 +
  6.2268 +\nopagebreak
  6.2269 +This attribute specifies the equations that constitute the specification of a
  6.2270 +constant. For functions defined using the \textbf{primrec}, \textbf{function},
  6.2271 +and \textbf{nominal\_\allowbreak primrec} packages, this corresponds to the
  6.2272 +\textit{simps} rules. The equations must be of the form
  6.2273 +
  6.2274 +\qquad $c~t_1~\ldots\ t_n \,=\, u.$
  6.2275 +
  6.2276 +\flushitem{\textit{nitpick\_psimp}}
  6.2277 +
  6.2278 +\nopagebreak
  6.2279 +This attribute specifies the equations that constitute the partial specification
  6.2280 +of a constant. For functions defined using the \textbf{function} package, this
  6.2281 +corresponds to the \textit{psimps} rules. The conditional equations must be of
  6.2282 +the form
  6.2283 +
  6.2284 +\qquad $\lbrakk P_1;\> \ldots;\> P_m\rbrakk \,\Longrightarrow\, c\ t_1\ \ldots\ t_n \,=\, u$.
  6.2285 +
  6.2286 +\flushitem{\textit{nitpick\_intro}}
  6.2287 +
  6.2288 +\nopagebreak
  6.2289 +This attribute specifies the introduction rules of a (co)in\-duc\-tive predicate.
  6.2290 +For predicates defined using the \textbf{inductive} or \textbf{coinductive}
  6.2291 +command, this corresponds to the \textit{intros} rules. The introduction rules
  6.2292 +must be of the form
  6.2293 +
  6.2294 +\qquad $\lbrakk P_1;\> \ldots;\> P_m;\> M~(c\ t_{11}\ \ldots\ t_{1n});\>
  6.2295 +\ldots;\> M~(c\ t_{k1}\ \ldots\ t_{kn})\rbrakk \,\Longrightarrow\, c\ u_1\
  6.2296 +\ldots\ u_n$,
  6.2297 +
  6.2298 +where the $P_i$'s are side conditions that do not involve $c$ and $M$ is an
  6.2299 +optional monotonic operator. The order of the assumptions is irrelevant.
  6.2300 +
  6.2301 +\end{itemize}
  6.2302 +
  6.2303 +When faced with a constant, Nitpick proceeds as follows:
  6.2304 +
  6.2305 +\begin{enum}
  6.2306 +\item[1.] If the \textit{nitpick\_simp} set associated with the constant
  6.2307 +is not empty, Nitpick uses these rules as the specification of the constant.
  6.2308 +
  6.2309 +\item[2.] Otherwise, if the \textit{nitpick\_psimp} set associated with
  6.2310 +the constant is not empty, it uses these rules as the specification of the
  6.2311 +constant.
  6.2312 +
  6.2313 +\item[3.] Otherwise, it looks up the definition of the constant:
  6.2314 +
  6.2315 +\begin{enum}
  6.2316 +\item[1.] If the \textit{nitpick\_def} set associated with the constant
  6.2317 +is not empty, it uses the latest rule added to the set as the definition of the
  6.2318 +constant; otherwise it uses the actual definition axiom.
  6.2319 +\item[2.] If the definition is of the form
  6.2320 +
  6.2321 +\qquad $c~{?}x_1~\ldots~{?}x_m \,\equiv\, \lambda y_1~\ldots~y_n.\; \textit{lfp}~(\lambda f.\; t)$,
  6.2322 +
  6.2323 +then Nitpick assumes that the definition was made using an inductive package and
  6.2324 +based on the introduction rules marked with \textit{nitpick\_\allowbreak
  6.2325 +ind\_\allowbreak intros} tries to determine whether the definition is
  6.2326 +well-founded.
  6.2327 +\end{enum}
  6.2328 +\end{enum}
  6.2329 +
  6.2330 +As an illustration, consider the inductive definition
  6.2331 +
  6.2332 +\prew
  6.2333 +\textbf{inductive}~\textit{odd}~\textbf{where} \\
  6.2334 +``\textit{odd}~1'' $\,\mid$ \\
  6.2335 +``\textit{odd}~$n\,\Longrightarrow\, \textit{odd}~(\textit{Suc}~(\textit{Suc}~n))$''
  6.2336 +\postw
  6.2337 +
  6.2338 +Isabelle automatically attaches the \textit{nitpick\_intro} attribute to
  6.2339 +the above rules. Nitpick then uses the \textit{lfp}-based definition in
  6.2340 +conjunction with these rules. To override this, we can specify an alternative
  6.2341 +definition as follows:
  6.2342 +
  6.2343 +\prew
  6.2344 +\textbf{lemma} $\mathit{odd\_def}'$ [\textit{nitpick\_def}]: ``$\textit{odd}~n \,\equiv\, n~\textrm{mod}~2 = 1$''
  6.2345 +\postw
  6.2346 +
  6.2347 +Nitpick then expands all occurrences of $\mathit{odd}~n$ to $n~\textrm{mod}~2
  6.2348 += 1$. Alternatively, we can specify an equational specification of the constant:
  6.2349 +
  6.2350 +\prew
  6.2351 +\textbf{lemma} $\mathit{odd\_simp}'$ [\textit{nitpick\_simp}]: ``$\textit{odd}~n = (n~\textrm{mod}~2 = 1)$''
  6.2352 +\postw
  6.2353 +
  6.2354 +Such tweaks should be done with great care, because Nitpick will assume that the
  6.2355 +constant is completely defined by its equational specification. For example, if
  6.2356 +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
  6.2357 +$\textit{odd}~n$ arbitrarily for even values of $n$. The \textit{debug}
  6.2358 +(\S\ref{output-format}) option is extremely useful to understand what is going
  6.2359 +on when experimenting with \textit{nitpick\_} attributes.
  6.2360 +
  6.2361 +\section{Standard ML Interface}
  6.2362 +\label{standard-ml-interface}
  6.2363 +
  6.2364 +Nitpick provides a rich Standard ML interface used mainly for internal purposes
  6.2365 +and debugging. Among the most interesting functions exported by Nitpick are
  6.2366 +those that let you invoke the tool programmatically and those that let you
  6.2367 +register and unregister custom coinductive datatypes.
  6.2368 +
  6.2369 +\subsection{Invocation of Nitpick}
  6.2370 +\label{invocation-of-nitpick}
  6.2371 +
  6.2372 +The \textit{Nitpick} structure offers the following functions for invoking your
  6.2373 +favorite counterexample generator:
  6.2374 +
  6.2375 +\prew
  6.2376 +$\textbf{val}\,~\textit{pick\_nits\_in\_term} : \\
  6.2377 +\hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{term~list} \rightarrow \textit{term} \\
  6.2378 +\hbox{}\quad{\rightarrow}\; \textit{string} * \textit{Proof.state}$ \\
  6.2379 +$\textbf{val}\,~\textit{pick\_nits\_in\_subgoal} : \\
  6.2380 +\hbox{}\quad\textit{Proof.state} \rightarrow \textit{params} \rightarrow \textit{bool} \rightarrow \textit{int} \rightarrow \textit{string} * \textit{Proof.state}$
  6.2381 +\postw
  6.2382 +
  6.2383 +The return value is a new proof state paired with an outcome string
  6.2384 +(``genuine'', ``likely\_genuine'', ``potential'', ``none'', or ``unknown''). The
  6.2385 +\textit{params} type is a large record that lets you set Nitpick's options. The
  6.2386 +current default options can be retrieved by calling the following function
  6.2387 +defined in the \textit{NitpickIsar} structure:
  6.2388 +
  6.2389 +\prew
  6.2390 +$\textbf{val}\,~\textit{default\_params} :\,
  6.2391 +\textit{theory} \rightarrow (\textit{string} * \textit{string})~\textit{list} \rightarrow \textit{params}$
  6.2392 +\postw
  6.2393 +
  6.2394 +The second argument lets you override option values before they are parsed and
  6.2395 +put into a \textit{params} record. Here is an example:
  6.2396 +
  6.2397 +\prew
  6.2398 +$\textbf{val}\,~\textit{params} = \textit{NitpickIsar.default\_params}~\textit{thy}~[(\textrm{``}\textrm{timeout}\textrm{''},\, \textrm{``}\textrm{none}\textrm{''})]$ \\
  6.2399 +$\textbf{val}\,~(\textit{outcome},\, \textit{state}') = \textit{Nitpick.pick\_nits\_in\_subgoal}~\begin{aligned}[t]
  6.2400 +& \textit{state}~\textit{params}~\textit{false} \\[-2pt]
  6.2401 +& \textit{subgoal}\end{aligned}$
  6.2402 +\postw
  6.2403 +
  6.2404 +\subsection{Registration of Coinductive Datatypes}
  6.2405 +\label{registration-of-coinductive-datatypes}
  6.2406 +
  6.2407 +\let\antiq=\textrm
  6.2408 +
  6.2409 +If you have defined a custom coinductive datatype, you can tell Nitpick about
  6.2410 +it, so that it can use an efficient Kodkod axiomatization similar to the one it
  6.2411 +uses for lazy lists. The interface for registering and unregistering coinductive
  6.2412 +datatypes consists of the following pair of functions defined in the
  6.2413 +\textit{Nitpick} structure:
  6.2414 +
  6.2415 +\prew
  6.2416 +$\textbf{val}\,~\textit{register\_codatatype} :\,
  6.2417 +\textit{typ} \rightarrow \textit{string} \rightarrow \textit{styp~list} \rightarrow \textit{theory} \rightarrow \textit{theory}$ \\
  6.2418 +$\textbf{val}\,~\textit{unregister\_codatatype} :\,
  6.2419 +\textit{typ} \rightarrow \textit{theory} \rightarrow \textit{theory}$
  6.2420 +\postw
  6.2421 +
  6.2422 +The type $'a~\textit{llist}$ of lazy lists is already registered; had it
  6.2423 +not been, you could have told Nitpick about it by adding the following line
  6.2424 +to your theory file:
  6.2425 +
  6.2426 +\prew
  6.2427 +$\textbf{setup}~\,\{{*}\,~\!\begin{aligned}[t]
  6.2428 +& \textit{Nitpick.register\_codatatype} \\[-2pt]
  6.2429 +& \qquad @\{\antiq{typ}~``\kern1pt'a~\textit{llist}\textrm{''}\}~@\{\antiq{const\_name}~ \textit{llist\_case}\} \\[-2pt] %% TYPESETTING
  6.2430 +& \qquad (\textit{map}~\textit{dest\_Const}~[@\{\antiq{term}~\textit{LNil}\},\, @\{\antiq{term}~\textit{LCons}\}])\,\ {*}\}\end{aligned}$
  6.2431 +\postw
  6.2432 +
  6.2433 +The \textit{register\_codatatype} function takes a coinductive type, its case
  6.2434 +function, and the list of its constructors. The case function must take its
  6.2435 +arguments in the order that the constructors are listed. If no case function
  6.2436 +with the correct signature is available, simply pass the empty string.
  6.2437 +
  6.2438 +On the other hand, if your goal is to cripple Nitpick, add the following line to
  6.2439 +your theory file and try to check a few conjectures about lazy lists:
  6.2440 +
  6.2441 +\prew
  6.2442 +$\textbf{setup}~\,\{{*}\,~\textit{Nitpick.unregister\_codatatype}~@\{\antiq{typ}~``
  6.2443 +\kern1pt'a~\textit{list}\textrm{''}\}\ \,{*}\}$
  6.2444 +\postw
  6.2445 +
  6.2446 +\section{Known Bugs and Limitations}
  6.2447 +\label{known-bugs-and-limitations}
  6.2448 +
  6.2449 +Here are the known bugs and limitations in Nitpick at the time of writing:
  6.2450 +
  6.2451 +\begin{enum}
  6.2452 +\item[$\bullet$] Underspecified functions defined using the \textbf{primrec},
  6.2453 +\textbf{function}, or \textbf{nominal\_\allowbreak primrec} packages can lead
  6.2454 +Nitpick to generate spurious counterexamples for theorems that refer to values
  6.2455 +for which the function is not defined. For example:
  6.2456 +
  6.2457 +\prew
  6.2458 +\textbf{primrec} \textit{prec} \textbf{where} \\
  6.2459 +``$\textit{prec}~(\textit{Suc}~n) = n$'' \\[2\smallskipamount]
  6.2460 +\textbf{lemma} ``$\textit{prec}~0 = \undef$'' \\
  6.2461 +\textbf{nitpick} \\[2\smallskipamount]
  6.2462 +\quad{\slshape Nitpick found a counterexample for \textit{card nat}~= 2: 
  6.2463 +\nopagebreak
  6.2464 +\\[2\smallskipamount]
  6.2465 +\hbox{}\qquad Empty assignment} \nopagebreak\\[2\smallskipamount]
  6.2466 +\textbf{by}~(\textit{auto simp}: \textit{prec\_def})
  6.2467 +\postw
  6.2468 +
  6.2469 +Such theorems are considered bad style because they rely on the internal
  6.2470 +representation of functions synthesized by Isabelle, which is an implementation
  6.2471 +detail.
  6.2472 +
  6.2473 +\item[$\bullet$] Nitpick produces spurious counterexamples when invoked after a
  6.2474 +\textbf{guess} command in a structured proof.
  6.2475 +
  6.2476 +\item[$\bullet$] The \textit{nitpick\_} attributes and the
  6.2477 +\textit{Nitpick.register\_} functions can cause havoc if used improperly.
  6.2478 +
  6.2479 +\item[$\bullet$] Local definitions are not supported and result in an error.
  6.2480 +
  6.2481 +\item[$\bullet$] All constants and types whose names start with
  6.2482 +\textit{Nitpick}{.} are reserved for internal use.
  6.2483 +\end{enum}
  6.2484 +
  6.2485 +\let\em=\sl
  6.2486 +\bibliography{../manual}{}
  6.2487 +\bibliographystyle{abbrv}
  6.2488 +
  6.2489 +\end{document}
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  7.2217 +0.4 w
  7.2218 +S 
  7.2219 +n
  7.2220 +1975.2 2084.7 m
  7.2221 +1976.6 2083.4 1975.7 2081.1 1976 2079.4 C
  7.2222 +1979.3 2079.5 1980.9 2086.2 1984.8 2084 C
  7.2223 +1992.9 2078.9 2001.7 2075.6 2010 2071.2 C
  7.2224 +2011 2064.6 2010.2 2057.3 2010.8 2050.6 C
  7.2225 +2015.4 2046.9 2021.1 2045.9 2025.9 2042.4 C
  7.2226 +2026.5 2033.2 2026.8 2022.9 2025.6 2013.9 C
  7.2227 +2020.5 2008.1 2014.5 2003.1 2009.3 1997.6 C
  7.2228 +2004.1 1996.7 2000.7 2001.6 1995.9 2002.6 C
  7.2229 +1995.2 1996.7 1996.3 1990.2 1994.9 1984.6 C
  7.2230 +1989.8 1978.7 1983.6 1973.7 1978.4 1968 C
  7.2231 +1977.3 1969.3 1976 1967.6 1974.8 1968.5 C
  7.2232 +1967.7 1972.7 1960.4 1976.3 1952.9 1979.6 C
  7.2233 +1946.5 1976.9 1943.1 1970.5 1937.8 1966.1 C
  7.2234 +1928.3 1968.2 1920.6 1974.8 1911.6 1978.4 C
  7.2235 +1901.9 1979.7 1893.9 1986.6 1885 1990.6 C
  7.2236 +1884.3 1991 1884.3 1991.7 1884 1992.3 C
  7.2237 +1884.5 2001 1884.2 2011 1884.3 2019.9 C
  7.2238 +1890.9 2025.3 1895.9 2031.9 1902.3 2037.4 C
  7.2239 +1904.2 2037.9 1905.6 2034.2 1906.8 2035.7 C
  7.2240 +1907.4 2040.9 1905.7 2046.1 1907.3 2050.8 C
  7.2241 +1913.6 2056.2 1919.2 2062.6 1925.1 2067.9 C
  7.2242 +1926.9 2067.8 1928 2066.3 1929.6 2065.7 C
  7.2243 +1929.9 2070.5 1929.2 2076 1930.1 2080.8 C
  7.2244 +1936.5 2086.1 1941.6 2092.8 1948.4 2097.6 C
  7.2245 +1957.3 2093.3 1966.2 2088.8 1975.2 2084.7 C
  7.2246 +[0 0 0 0]  vc
  7.2247 +f 
  7.2248 +S 
  7.2249 +n
  7.2250 +1954.8 2093.8 m
  7.2251 +1961.6 2090.5 1968.2 2087 1975 2084 C
  7.2252 +1975 2082.8 1975.6 2080.9 1974.8 2080.6 C
  7.2253 +1974.3 2075.2 1974.6 2069.6 1974.5 2064 C
  7.2254 +1977.5 2059.7 1984.5 2060 1988.9 2056.4 C
  7.2255 +1989.5 2055.5 1990.5 2055.3 1990.8 2054.4 C
  7.2256 +1991.1 2045.7 1991.4 2036.1 1990.6 2027.8 C
  7.2257 +1990.7 2026.6 1992 2027.3 1992.8 2027.1 C
  7.2258 +1997 2032.4 2002.6 2037.8 2007.6 2042.2 C
  7.2259 +2008.7 2042.3 2007.8 2040.6 2007.4 2040 C
  7.2260 +2002.3 2035.6 1997.5 2030 1992.8 2025.2 C
  7.2261 +1991.6 2024.7 1990.8 2024.9 1990.1 2025.4 C
  7.2262 +1989.4 2024.9 1988.1 2025.2 1987.2 2024.4 C
  7.2263 +1987.1 2025.8 1988.3 2026.5 1989.4 2026.8 C
  7.2264 +1989.4 2026.6 1989.3 2026.2 1989.6 2026.1 C
  7.2265 +1989.9 2026.2 1989.9 2026.6 1989.9 2026.8 C
  7.2266 +1989.8 2026.6 1990 2026.5 1990.1 2026.4 C
  7.2267 +1990.2 2027 1991.1 2028.3 1990.1 2028 C
  7.2268 +1989.9 2037.9 1990.5 2044.1 1989.6 2054.2 C
  7.2269 +1985.9 2058 1979.7 2057.4 1976 2061.2 C
  7.2270 +1974.5 2061.6 1975.2 2059.9 1974.5 2059.5 C
  7.2271 +1973.9 2058 1975.6 2057.8 1975 2056.6 C
  7.2272 +1974.5 2057.1 1974.6 2055.3 1973.6 2055.9 C
  7.2273 +1971.9 2059.3 1974.7 2062.1 1973.1 2065.5 C
  7.2274 +1973.1 2071.2 1972.9 2077 1973.3 2082.5 C
  7.2275 +1967.7 2085.6 1962 2088 1956.3 2090.7 C
  7.2276 +1953.9 2092.4 1951 2093 1948.6 2094.8 C
  7.2277 +1943.7 2089.9 1937.9 2084.3 1933 2079.6 C
  7.2278 +1931.3 2076.1 1933.2 2071.3 1932.3 2067.2 C
  7.2279 +1931.3 2062.9 1933.3 2060.6 1932 2057.6 C
  7.2280 +1932.7 2056.5 1930.9 2053.3 1933.2 2051.8 C
  7.2281 +1936.8 2050.1 1940.1 2046.9 1944 2046.8 C
  7.2282 +1946.3 2049.7 1949.3 2051.9 1952 2054.4 C
  7.2283 +1954.5 2054.2 1956.4 2052.3 1958.7 2051.3 C
  7.2284 +1960.8 2050 1963.2 2049 1965.6 2048.4 C
  7.2285 +1968.3 2050.8 1970.7 2054.3 1973.6 2055.4 C
  7.2286 +1973 2052.2 1969.7 2050.4 1967.6 2048.2 C
  7.2287 +1967.1 2046.7 1968.8 2046.6 1969.5 2045.8 C
  7.2288 +1972.8 2043.3 1980.6 2043.4 1979.3 2038.4 C
  7.2289 +1979.4 2038.6 1979.2 2038.7 1979.1 2038.8 C
  7.2290 +1978.7 2038.6 1978.9 2038.1 1978.8 2037.6 C
  7.2291 +1978.9 2037.9 1978.7 2038 1978.6 2038.1 C
  7.2292 +1978.2 2032.7 1978.4 2027.1 1978.4 2021.6 C
  7.2293 +1979.3 2021.1 1980 2020.2 1981.5 2020.1 C
  7.2294 +1983.5 2020.5 1984 2021.8 1985.1 2023.5 C
  7.2295 +1985.7 2024 1987.4 2023.7 1986 2022.8 C
  7.2296 +1984.7 2021.7 1983.3 2020.8 1983.9 2018.7 C
  7.2297 +1987.2 2015.9 1993 2015.4 1994.9 2011.5 C
  7.2298 +1992.2 2004.9 1999.3 2005.2 2002.1 2002.4 C
  7.2299 +2005.9 2002.7 2004.8 1997.4 2009.1 1999 C
  7.2300 +2011 1999.3 2010 2002.9 2012.7 2002.4 C
  7.2301 +2010.2 2000.7 2009.4 1996.1 2005.5 1998.5 C
  7.2302 +2002.1 2000.3 1999 2002.5 1995.4 2003.8 C
  7.2303 +1995.2 2003.6 1994.9 2003.3 1994.7 2003.1 C
  7.2304 +1994.3 1997 1995.6 1991.1 1994.4 1985.3 C
  7.2305 +1994.3 1986 1993.8 1985 1994 1985.6 C
  7.2306 +1993.8 1995.4 1994.4 2001.6 1993.5 2011.7 C
  7.2307 +1989.7 2015.5 1983.6 2014.9 1979.8 2018.7 C
  7.2308 +1978.3 2019.1 1979.1 2017.4 1978.4 2017 C
  7.2309 +1977.8 2015.5 1979.4 2015.3 1978.8 2014.1 C
  7.2310 +1978.4 2014.6 1978.5 2012.8 1977.4 2013.4 C
  7.2311 +1975.8 2016.8 1978.5 2019.6 1976.9 2023 C
  7.2312 +1977 2028.7 1976.7 2034.5 1977.2 2040 C
  7.2313 +1971.6 2043.1 1965.8 2045.6 1960.1 2048.2 C
  7.2314 +1957.7 2049.9 1954.8 2050.5 1952.4 2052.3 C
  7.2315 +1947.6 2047.4 1941.8 2041.8 1936.8 2037.2 C
  7.2316 +1935.2 2033.6 1937.1 2028.8 1936.1 2024.7 C
  7.2317 +1935.1 2020.4 1937.1 2018.1 1935.9 2015.1 C
  7.2318 +1936.5 2014.1 1934.7 2010.8 1937.1 2009.3 C
  7.2319 +1944.4 2004.8 1952 2000.9 1959.9 1997.8 C
  7.2320 +1963.9 1997 1963.9 2001.9 1966.8 2003.3 C
  7.2321 +1970.3 2006.9 1973.7 2009.9 1976.9 2012.9 C
  7.2322 +1977.9 2013 1977.1 2011.4 1976.7 2010.8 C
  7.2323 +1971.6 2006.3 1966.8 2000.7 1962 1995.9 C
  7.2324 +1960 1995.2 1960.1 1996.6 1958.2 1995.6 C
  7.2325 +1957 1997 1955.1 1998.8 1953.2 1998 C
  7.2326 +1951.7 1994.5 1954.1 1993.4 1952.9 1991.1 C
  7.2327 +1952.1 1990.5 1953.3 1990.2 1953.2 1989.6 C
  7.2328 +1954.2 1986.8 1950.9 1981.4 1954.4 1981.2 C
  7.2329 +1954.7 1981.6 1954.7 1981.7 1955.1 1982 C
  7.2330 +1961.9 1979.1 1967.6 1975 1974.3 1971.6 C
  7.2331 +1974.7 1969.8 1976.7 1969.5 1978.4 1969.7 C
  7.2332 +1980.3 1970 1979.3 1973.6 1982 1973.1 C
  7.2333 +1975.8 1962.2 1968 1975.8 1960.8 1976.7 C
  7.2334 +1956.9 1977.4 1953.3 1982.4 1949.1 1978.8 C
  7.2335 +1946 1975.8 1941.2 1971 1939.5 1969.2 C
  7.2336 +1938.5 1968.6 1938.9 1967.4 1937.8 1966.8 C
  7.2337 +1928.7 1969.4 1920.6 1974.5 1912.4 1979.1 C
  7.2338 +1904 1980 1896.6 1985 1889.3 1989.4 C
  7.2339 +1887.9 1990.4 1885.1 1990.3 1885 1992.5 C
  7.2340 +1885.4 2000.6 1885.2 2012.9 1885.2 2019.9 C
  7.2341 +1886.1 2022 1889.7 2019.5 1888.4 2022.8 C
  7.2342 +1889 2023.3 1889.8 2024.4 1890.3 2024 C
  7.2343 +1891.2 2023.5 1891.8 2028.2 1893.4 2026.6 C
  7.2344 +1894.2 2026.3 1893.9 2027.3 1894.4 2027.6 C
  7.2345 +1893.4 2027.6 1894.7 2028.3 1894.1 2028.5 C
  7.2346 +1894.4 2029.6 1896 2030 1896 2029.2 C
  7.2347 +1896.2 2029 1896.3 2029 1896.5 2029.2 C
  7.2348 +1896.8 2029.8 1897.3 2030 1897 2030.7 C
  7.2349 +1896.5 2030.7 1896.9 2031.5 1897.2 2031.6 C
  7.2350 +1898.3 2034 1899.5 2030.6 1899.6 2033.3 C
  7.2351 +1898.5 2033 1899.6 2034.4 1900.1 2034.8 C
  7.2352 +1901.3 2035.8 1903.2 2034.6 1902.5 2036.7 C
  7.2353 +1904.4 2036.9 1906.1 2032.2 1907.6 2035.5 C
  7.2354 +1907.5 2040.1 1907.7 2044.9 1907.3 2049.4 C
  7.2355 +1908 2050.2 1908.3 2051.4 1909.5 2051.6 C
  7.2356 +1910.1 2051.1 1911.6 2051.1 1911.4 2052.3 C
  7.2357 +1909.7 2052.8 1912.4 2054 1912.6 2054.7 C
  7.2358 +1913.4 2055.2 1913 2053.7 1913.6 2054.4 C
  7.2359 +1913.6 2054.5 1913.6 2055.3 1913.6 2054.7 C
  7.2360 +1913.7 2054.4 1913.9 2054.4 1914 2054.7 C
  7.2361 +1914 2054.9 1914.1 2055.3 1913.8 2055.4 C
  7.2362 +1913.7 2056 1915.2 2057.6 1916 2057.6 C
  7.2363 +1915.9 2057.3 1916.1 2057.2 1916.2 2057.1 C
  7.2364 +1917 2056.8 1916.7 2057.7 1917.2 2058 C
  7.2365 +1917 2058.3 1916.7 2058.3 1916.4 2058.3 C
  7.2366 +1917.1 2059 1917.3 2060.1 1918.4 2060.4 C
  7.2367 +1918.1 2059.2 1919.1 2060.6 1919.1 2059.5 C
  7.2368 +1919 2060.6 1920.6 2060.1 1919.8 2061.2 C
  7.2369 +1919.6 2061.2 1919.3 2061.2 1919.1 2061.2 C
  7.2370 +1919.6 2061.9 1921.4 2064.2 1921.5 2062.6 C
  7.2371 +1922.4 2062.1 1921.6 2063.9 1922.2 2064.3 C
  7.2372 +1922.9 2067.3 1926.1 2064.3 1925.6 2067.2 C
  7.2373 +1927.2 2066.8 1928.4 2064.6 1930.1 2065.2 C
  7.2374 +1931.8 2067.8 1931 2071.8 1930.8 2074.8 C
  7.2375 +1930.6 2076.4 1930.1 2078.6 1930.6 2080.4 C
  7.2376 +1936.6 2085.4 1941.8 2091.6 1948.1 2096.9 C
  7.2377 +1950.7 2096.7 1952.6 2094.8 1954.8 2093.8 C
  7.2378 +[0 0.33 0.33 0.99]  vc
  7.2379 +f 
  7.2380 +S 
  7.2381 +n
  7.2382 +1989.4 2080.6 m
  7.2383 +1996.1 2077.3 2002.7 2073.8 2009.6 2070.8 C
  7.2384 +2009.6 2069.6 2010.2 2067.7 2009.3 2067.4 C
  7.2385 +2008.9 2062 2009.1 2056.4 2009.1 2050.8 C
  7.2386 +2012.3 2046.6 2019 2046.6 2023.5 2043.2 C
  7.2387 +2024 2042.3 2025.1 2042.1 2025.4 2041.2 C
  7.2388 +2025.3 2032.7 2025.6 2023.1 2025.2 2014.6 C
  7.2389 +2025 2015.3 2024.6 2014.2 2024.7 2014.8 C
  7.2390 +2024.5 2024.7 2025.1 2030.9 2024.2 2041 C
  7.2391 +2020.4 2044.8 2014.3 2044.2 2010.5 2048 C
  7.2392 +2009 2048.4 2009.8 2046.7 2009.1 2046.3 C
  7.2393 +2008.5 2044.8 2010.2 2044.6 2009.6 2043.4 C
  7.2394 +2009.1 2043.9 2009.2 2042.1 2008.1 2042.7 C
  7.2395 +2006.5 2046.1 2009.3 2048.9 2007.6 2052.3 C
  7.2396 +2007.7 2058 2007.5 2063.8 2007.9 2069.3 C
  7.2397 +2002.3 2072.4 1996.5 2074.8 1990.8 2077.5 C
  7.2398 +1988.4 2079.2 1985.6 2079.8 1983.2 2081.6 C
  7.2399 +1980.5 2079 1977.9 2076.5 1975.5 2074.1 C
  7.2400 +1975.5 2075.1 1975.5 2076.2 1975.5 2077.2 C
  7.2401 +1977.8 2079.3 1980.3 2081.6 1982.7 2083.7 C
  7.2402 +1985.3 2083.5 1987.1 2081.6 1989.4 2080.6 C
  7.2403 +f 
  7.2404 +S 
  7.2405 +n
  7.2406 +1930.1 2079.9 m
  7.2407 +1931.1 2075.6 1929.2 2071.1 1930.8 2067.2 C
  7.2408 +1930.3 2066.3 1930.1 2064.6 1928.7 2065.5 C
  7.2409 +1927.7 2066.4 1926.5 2067 1925.3 2067.4 C
  7.2410 +1924.5 2066.9 1925.6 2065.7 1924.4 2066 C
  7.2411 +1924.2 2067.2 1923.6 2065.5 1923.2 2065.7 C
  7.2412 +1922.3 2063.6 1917.8 2062.1 1919.6 2060.4 C
  7.2413 +1919.3 2060.5 1919.2 2060.3 1919.1 2060.2 C
  7.2414 +1919.7 2060.9 1918.2 2061 1917.6 2060.2 C
  7.2415 +1917 2059.6 1916.1 2058.8 1916.4 2058 C
  7.2416 +1915.5 2058 1917.4 2057.1 1915.7 2057.8 C
  7.2417 +1914.8 2057.1 1913.4 2056.2 1913.3 2054.9 C
  7.2418 +1913.1 2055.4 1911.3 2054.3 1910.9 2053.2 C
  7.2419 +1910.7 2052.9 1910.2 2052.5 1910.7 2052.3 C
  7.2420 +1911.1 2052.5 1910.9 2052 1910.9 2051.8 C
  7.2421 +1910.5 2051.2 1909.9 2052.6 1909.2 2051.8 C
  7.2422 +1908.2 2051.4 1907.8 2050.2 1907.1 2049.4 C
  7.2423 +1907.5 2044.8 1907.3 2040 1907.3 2035.2 C
  7.2424 +1905.3 2033 1902.8 2039.3 1902.3 2035.7 C
  7.2425 +1899.6 2036 1898.4 2032.5 1896.3 2030.7 C
  7.2426 +1895.7 2030.1 1897.5 2030 1896.3 2029.7 C
  7.2427 +1896.3 2030.6 1895 2029.7 1894.4 2029.2 C
  7.2428 +1892.9 2028.1 1894.2 2027.4 1893.6 2027.1 C
  7.2429 +1892.1 2027.9 1891.7 2025.6 1890.8 2024.9 C
  7.2430 +1891.1 2024.6 1889.1 2024.3 1888.4 2023 C
  7.2431 +1887.5 2022.6 1888.2 2021.9 1888.1 2021.3 C
  7.2432 +1886.7 2022 1885.2 2020.4 1884.8 2019.2 C
  7.2433 +1884.8 2010 1884.6 2000.2 1885 1991.8 C
  7.2434 +1886.9 1989.6 1889.9 1989.3 1892.2 1987.5 C
  7.2435 +1898.3 1982.7 1905.6 1980.1 1912.8 1978.6 C
  7.2436 +1921 1974.2 1928.8 1968.9 1937.8 1966.6 C
  7.2437 +1939.8 1968.3 1938.8 1968.3 1940.4 1970 C
  7.2438 +1945.4 1972.5 1947.6 1981.5 1954.6 1979.3 C
  7.2439 +1952.3 1981 1950.4 1978.4 1948.6 1977.9 C
  7.2440 +1945.1 1973.9 1941.1 1970.6 1938 1966.6 C
  7.2441 +1928.4 1968.5 1920.6 1974.8 1911.9 1978.8 C
  7.2442 +1907.1 1979.2 1902.6 1981.7 1898.2 1983.6 C
  7.2443 +1893.9 1986 1889.9 1989 1885.5 1990.8 C
  7.2444 +1884.9 1991.2 1884.8 1991.8 1884.5 1992.3 C
  7.2445 +1884.9 2001.3 1884.7 2011.1 1884.8 2019.6 C
  7.2446 +1890.6 2025 1896.5 2031.2 1902.3 2036.9 C
  7.2447 +1904.6 2037.6 1905 2033 1907.3 2035.5 C
  7.2448 +1907.2 2040.2 1907 2044.8 1907.1 2049.6 C
  7.2449 +1913.6 2055.3 1918.4 2061.5 1925.1 2067.4 C
  7.2450 +1927.3 2068.2 1929.6 2062.5 1930.6 2066.9 C
  7.2451 +1929.7 2070.7 1930.3 2076 1930.1 2080.1 C
  7.2452 +1935.6 2085.7 1941.9 2090.7 1947.2 2096.7 C
  7.2453 +1942.2 2091.1 1935.5 2085.2 1930.1 2079.9 C
  7.2454 +[0.18 0.18 0 0.78]  vc
  7.2455 +f 
  7.2456 +S 
  7.2457 +n
  7.2458 +1930.8 2061.9 m
  7.2459 +1930.3 2057.8 1931.8 2053.4 1931.1 2050.4 C
  7.2460 +1931.3 2050.3 1931.7 2050.5 1931.6 2050.1 C
  7.2461 +1933 2051.1 1934.4 2049.5 1935.9 2048.7 C
  7.2462 +1937 2046.5 1939.5 2047.1 1941.2 2045.1 C
  7.2463 +1939.7 2042.6 1937.3 2041.2 1935.4 2039.3 C
  7.2464 +1934 2039.7 1934.5 2038.1 1933.7 2037.6 C
  7.2465 +1934 2033.3 1933.1 2027.9 1934.4 2024.4 C
  7.2466 +1934.3 2023.8 1933.9 2022.8 1933 2022.8 C
  7.2467 +1931.6 2023.1 1930.5 2024.4 1929.2 2024.9 C
  7.2468 +1928.4 2024.5 1929.8 2023.5 1928.7 2023.5 C
  7.2469 +1927.7 2024.1 1926.2 2022.6 1925.6 2021.6 C
  7.2470 +1926.9 2021.6 1924.8 2020.6 1925.6 2020.4 C
  7.2471 +1924.7 2021.7 1923.9 2019.6 1923.2 2019.2 C
  7.2472 +1923.3 2018.3 1923.8 2018.1 1923.2 2018 C
  7.2473 +1922.9 2017.8 1922.9 2017.5 1922.9 2017.2 C
  7.2474 +1922.8 2018.3 1921.3 2017.3 1920.3 2018 C
  7.2475 +1916.6 2019.7 1913 2022.1 1910 2024.7 C
  7.2476 +1910 2032.9 1910 2041.2 1910 2049.4 C
  7.2477 +1915.4 2055.2 1920 2058.7 1925.3 2064.8 C
  7.2478 +1927.2 2064 1929 2061.4 1930.8 2061.9 C
  7.2479 +[0 0 0 0]  vc
  7.2480 +f 
  7.2481 +S 
  7.2482 +n
  7.2483 +1907.6 2030.4 m
  7.2484 +1907.5 2027.1 1906.4 2021.7 1908.5 2019.9 C
  7.2485 +1908.8 2020.1 1908.9 2019 1909.2 2019.6 C
  7.2486 +1910 2019.6 1912 2019.2 1913.1 2018.2 C
  7.2487 +1913.7 2016.5 1920.2 2015.7 1917.4 2012.7 C
  7.2488 +1918.2 2011.2 1917 2013.8 1917.2 2012 C
  7.2489 +1916.9 2012.3 1916 2012.4 1915.2 2012 C
  7.2490 +1912.5 2010.5 1916.6 2008.8 1913.6 2009.6 C
  7.2491 +1912.6 2009.2 1911.1 2009 1910.9 2007.6 C
  7.2492 +1911 1999.2 1911.8 1989.8 1911.2 1982.2 C
  7.2493 +1910.1 1981.1 1908.8 1982.2 1907.6 1982.2 C
  7.2494 +1900.8 1986.5 1893.2 1988.8 1887.2 1994.2 C
  7.2495 +1887.2 2002.4 1887.2 2010.7 1887.2 2018.9 C
  7.2496 +1892.6 2024.7 1897.2 2028.2 1902.5 2034.3 C
  7.2497 +1904.3 2033.3 1906.2 2032.1 1907.6 2030.4 C
  7.2498 +f 
  7.2499 +S 
  7.2500 +n
  7.2501 +1910.7 2025.4 m
  7.2502 +1912.7 2022.4 1916.7 2020.8 1919.8 2018.9 C
  7.2503 +1920.2 2018.7 1920.6 2018.6 1921 2018.4 C
  7.2504 +1925 2020 1927.4 2028.5 1932 2024.2 C
  7.2505 +1932.3 2025 1932.5 2023.7 1932.8 2024.4 C
  7.2506 +1932.8 2028 1932.8 2031.5 1932.8 2035 C
  7.2507 +1931.9 2033.9 1932.5 2036.3 1932.3 2036.9 C
  7.2508 +1933.2 2036.4 1932.5 2038.5 1933 2038.4 C
  7.2509 +1933.1 2040.5 1935.6 2042.2 1936.6 2043.2 C
  7.2510 +1936.2 2042.4 1935.1 2040.8 1933.7 2040.3 C
  7.2511 +1932.2 2034.4 1933.8 2029.8 1933 2023.2 C
  7.2512 +1931.1 2024.9 1928.4 2026.4 1926.5 2023.5 C
  7.2513 +1925.1 2021.6 1923 2019.8 1921.5 2018.2 C
  7.2514 +1917.8 2018.9 1915.2 2022.5 1911.6 2023.5 C
  7.2515 +1910.8 2023.8 1911.2 2024.7 1910.4 2025.2 C
  7.2516 +1910.9 2031.8 1910.6 2039.1 1910.7 2045.6 C
  7.2517 +1910.1 2048 1910.7 2045.9 1911.2 2044.8 C
  7.2518 +1910.6 2038.5 1911.2 2031.8 1910.7 2025.4 C
  7.2519 +[0.07 0.06 0 0.58]  vc
  7.2520 +f 
  7.2521 +S 
  7.2522 +n
  7.2523 +1910.7 2048.9 m
  7.2524 +1910.3 2047.4 1911.3 2046.5 1911.6 2045.3 C
  7.2525 +1912.9 2045.3 1913.9 2047.1 1915.2 2045.8 C
  7.2526 +1915.2 2044.9 1916.6 2043.3 1917.2 2042.9 C
  7.2527 +1918.7 2042.9 1919.4 2044.4 1920.5 2043.2 C
  7.2528 +1921.2 2042.2 1921.4 2040.9 1922.4 2040.3 C
  7.2529 +1924.5 2040.3 1925.7 2040.9 1926.8 2039.6 C
  7.2530 +1927.1 2037.9 1926.8 2038.1 1927.7 2037.6 C
  7.2531 +1929 2037.5 1930.4 2037 1931.6 2037.2 C
  7.2532 +1932.3 2038.2 1933.1 2038.7 1932.8 2040.3 C
  7.2533 +1935 2041.8 1935.9 2043.8 1938.5 2044.8 C
  7.2534 +1938.6 2045 1938.3 2045.5 1938.8 2045.3 C
  7.2535 +1939.1 2042.9 1935.4 2044.2 1935.4 2042.2 C
  7.2536 +1932.1 2040.8 1932.8 2037.2 1932 2034.8 C
  7.2537 +1932.3 2034 1932.7 2035.4 1932.5 2034.8 C
  7.2538 +1931.3 2031.8 1935.5 2020.1 1928.9 2025.9 C
  7.2539 +1924.6 2024.7 1922.6 2014.5 1917.4 2020.4 C
  7.2540 +1915.5 2022.8 1912 2022.6 1910.9 2025.4 C
  7.2541 +1911.5 2031.9 1910.9 2038.8 1911.4 2045.3 C
  7.2542 +1911.1 2046.5 1910 2047.4 1910.4 2048.9 C
  7.2543 +1915.1 2054.4 1920.4 2058.3 1925.1 2063.8 C
  7.2544 +1920.8 2058.6 1914.9 2054.3 1910.7 2048.9 C
  7.2545 +[0.4 0.4 0 0]  vc
  7.2546 +f 
  7.2547 +S 
  7.2548 +n
  7.2549 +1934.7 2031.9 m
  7.2550 +1934.6 2030.7 1934.9 2029.5 1934.4 2028.5 C
  7.2551 +1934 2029.5 1934.3 2031.2 1934.2 2032.6 C
  7.2552 +1933.8 2031.7 1934.9 2031.6 1934.7 2031.9 C
  7.2553 +[0.92 0.92 0 0.67]  vc
  7.2554 +f 
  7.2555 +S 
  7.2556 +n
  7.2557 +vmrs
  7.2558 +1934.7 2019.4 m
  7.2559 +1934.1 2015.3 1935.6 2010.9 1934.9 2007.9 C
  7.2560 +1935.1 2007.8 1935.6 2008.1 1935.4 2007.6 C
  7.2561 +1936.8 2008.6 1938.2 2007 1939.7 2006.2 C
  7.2562 +1940.1 2004.3 1942.7 2005 1943.6 2003.8 C
  7.2563 +1945.1 2000.3 1954 2000.8 1950 1996.6 C
  7.2564 +1952.1 1993.3 1948.2 1989.2 1951.2 1985.6 C
  7.2565 +1953 1981.4 1948.4 1982.3 1947.9 1979.8 C
  7.2566 +1945.4 1979.6 1945.1 1975.5 1942.4 1975 C
  7.2567 +1942.4 1972.3 1938 1973.6 1938.5 1970.4 C
  7.2568 +1937.4 1969 1935.6 1970.1 1934.2 1970.2 C
  7.2569 +1927.5 1974.5 1919.8 1976.8 1913.8 1982.2 C
  7.2570 +1913.8 1990.4 1913.8 1998.7 1913.8 2006.9 C
  7.2571 +1919.3 2012.7 1923.8 2016.2 1929.2 2022.3 C
  7.2572 +1931.1 2021.6 1932.8 2018.9 1934.7 2019.4 C
  7.2573 +[0 0 0 0]  vc
  7.2574 +f 
  7.2575 +0.4 w
  7.2576 +2 J
  7.2577 +2 M
  7.2578 +S 
  7.2579 +n
  7.2580 +2024.2 2038.1 m
  7.2581 +2024.1 2029.3 2024.4 2021.7 2024.7 2014.4 C
  7.2582 +2024.4 2013.6 2020.6 2013.4 2021.3 2011.2 C
  7.2583 +2020.5 2010.3 2018.4 2010.6 2018.9 2008.6 C
  7.2584 +2019 2008.8 2018.8 2009 2018.7 2009.1 C
  7.2585 +2018.2 2006.7 2015.2 2007.9 2015.3 2005.5 C
  7.2586 +2014.7 2004.8 2012.4 2005.1 2013.2 2003.6 C
  7.2587 +2012.3 2004.2 2012.8 2002.4 2012.7 2002.6 C
  7.2588 +2009.4 2003.3 2011.2 1998.6 2008.4 1999.2 C
  7.2589 +2007 1999.1 2006.1 1999.4 2005.7 2000.4 C
  7.2590 +2006.9 1998.5 2007.7 2000.5 2009.3 2000.2 C
  7.2591 +2009.2 2003.7 2012.4 2002.1 2012.9 2005.2 C
  7.2592 +2015.9 2005.6 2015.2 2008.6 2017.7 2008.8 C
  7.2593 +2018.4 2009.6 2018.3 2011.4 2019.6 2011 C
  7.2594 +2021.1 2011.7 2021.4 2014.8 2023.7 2015.1 C
  7.2595 +2023.7 2023.5 2023.9 2031.6 2023.5 2040.5 C
  7.2596 +2021.8 2041.7 2020.7 2043.6 2018.4 2043.9 C
  7.2597 +2020.8 2042.7 2025.5 2041.8 2024.2 2038.1 C
  7.2598 +[0 0.87 0.91 0.83]  vc
  7.2599 +f 
  7.2600 +S 
  7.2601 +n
  7.2602 +2023.5 2040 m
  7.2603 +2023.5 2031.1 2023.5 2023.4 2023.5 2015.1 C
  7.2604 +2020.2 2015 2021.8 2010.3 2018.4 2011 C
  7.2605 +2018.6 2007.5 2014.7 2009.3 2014.8 2006.4 C
  7.2606 +2011.8 2006.3 2012.2 2002.3 2009.8 2002.4 C
  7.2607 +2009.7 2001.5 2009.2 2000.1 2008.4 2000.2 C
  7.2608 +2008.7 2000.9 2009.7 2001.2 2009.3 2002.4 C
  7.2609 +2008.4 2004.2 2007.5 2003.1 2007.9 2005.5 C
  7.2610 +2007.9 2010.8 2007.7 2018.7 2008.1 2023.2 C
  7.2611 +2009 2024.3 2007.3 2023.4 2007.9 2024 C
  7.2612 +2007.7 2024.6 2007.3 2026.3 2008.6 2027.1 C
  7.2613 +2009.7 2026.8 2010 2027.6 2010.5 2028 C
  7.2614 +2010.5 2028.2 2010.5 2029.1 2010.5 2028.5 C
  7.2615 +2011.5 2028 2010.5 2030 2011.5 2030 C
  7.2616 +2014.2 2029.7 2012.9 2032.2 2014.8 2032.6 C
  7.2617 +2015.1 2033.6 2015.3 2033 2016 2033.3 C
  7.2618 +2017 2033.9 2016.6 2035.4 2017.2 2036.2 C
  7.2619 +2018.7 2036.4 2019.2 2039 2021.3 2038.4 C
  7.2620 +2021.6 2035.4 2019.7 2029.5 2021.1 2027.3 C
  7.2621 +2020.9 2023.5 2021.5 2018.5 2020.6 2016 C
  7.2622 +2020.9 2013.9 2021.5 2015.4 2022.3 2014.4 C
  7.2623 +2022.2 2015.1 2023.3 2014.8 2023.2 2015.6 C
  7.2624 +2022.7 2019.8 2023.3 2024.3 2022.8 2028.5 C
  7.2625 +2022.3 2028.2 2022.6 2027.6 2022.5 2027.1 C
  7.2626 +2022.5 2027.8 2022.5 2029.2 2022.5 2029.2 C
  7.2627 +2022.6 2029.2 2022.7 2029.1 2022.8 2029 C
  7.2628 +2023.9 2032.8 2022.6 2037 2023 2040.8 C
  7.2629 +2022.3 2041.2 2021.6 2041.5 2021.1 2042.2 C
  7.2630 +2022 2041.2 2022.9 2041.4 2023.5 2040 C
  7.2631 +[0 1 1 0.23]  vc
  7.2632 +f 
  7.2633 +S 
  7.2634 +n
  7.2635 +2009.1 1997.8 m
  7.2636 +2003.8 1997.7 2000.1 2002.4 1995.4 2003.1 C
  7.2637 +1995 1999.5 1995.2 1995 1995.2 1992 C
  7.2638 +1995.2 1995.8 1995 1999.7 1995.4 2003.3 C
  7.2639 +2000.3 2002.2 2003.8 1997.9 2009.1 1997.8 C
  7.2640 +2012.3 2001.2 2015.6 2004.8 2018.7 2008.1 C
  7.2641 +2021.6 2011.2 2027.5 2013.9 2025.9 2019.9 C
  7.2642 +2026.1 2017.9 2025.6 2016.2 2025.4 2014.4 C
  7.2643 +2020.2 2008.4 2014 2003.6 2009.1 1997.8 C
  7.2644 +[0.18 0.18 0 0.78]  vc
  7.2645 +f 
  7.2646 +S 
  7.2647 +n
  7.2648 +2009.3 1997.8 m
  7.2649 +2008.7 1997.4 2007.9 1997.6 2007.2 1997.6 C
  7.2650 +2007.9 1997.6 2008.9 1997.4 2009.6 1997.8 C
  7.2651 +2014.7 2003.6 2020.8 2008.8 2025.9 2014.8 C
  7.2652 +2025.8 2017.7 2026.1 2014.8 2025.6 2014.1 C
  7.2653 +2020.4 2008.8 2014.8 2003.3 2009.3 1997.8 C
  7.2654 +[0.07 0.06 0 0.58]  vc
  7.2655 +f 
  7.2656 +S 
  7.2657 +n
  7.2658 +2009.6 1997.6 m
  7.2659 +2009 1997.1 2008.1 1997.4 2007.4 1997.3 C
  7.2660 +2008.1 1997.4 2009 1997.1 2009.6 1997.6 C
  7.2661 +2014.8 2003.7 2021.1 2008.3 2025.9 2014.4 C
  7.2662 +2021.1 2008.3 2014.7 2003.5 2009.6 1997.6 C
  7.2663 +[0.4 0.4 0 0]  vc
  7.2664 +f 
  7.2665 +S 
  7.2666 +n
  7.2667 +2021.8 2011.5 m
  7.2668 +2021.9 2012.2 2022.3 2013.5 2023.7 2013.6 C
  7.2669 +2023.4 2012.7 2022.8 2011.8 2021.8 2011.5 C
  7.2670 +[0 0.33 0.33 0.99]  vc
  7.2671 +f 
  7.2672 +S 
  7.2673 +n
  7.2674 +2021.1 2042 m
  7.2675 +2022.1 2041.1 2020.9 2040.2 2020.6 2039.6 C
  7.2676 +2018.4 2039.5 2018.1 2036.9 2016.3 2036.4 C
  7.2677 +2015.8 2035.5 2015.3 2033.8 2014.8 2033.6 C
  7.2678 +2012.4 2033.8 2013 2030.4 2010.5 2030.2 C
  7.2679 +2009.6 2028.9 2009.6 2028.3 2008.4 2028 C
  7.2680 +2006.9 2026.7 2007.5 2024.3 2006 2023.2 C
  7.2681 +2006.6 2023.2 2005.7 2023.3 2005.7 2023 C
  7.2682 +2006.4 2022.5 2006.3 2021.1 2006.7 2020.6 C
  7.2683 +2006.6 2015 2006.9 2009 2006.4 2003.8 C
  7.2684 +2006.9 2002.5 2007.6 2001.1 2006.9 2000.7 C
  7.2685 +2004.6 2003.6 2003 2002.9 2000.2 2004.3 C
  7.2686 +1999.3 2005.8 1997.9 2006.3 1996.1 2006.7 C
  7.2687 +1995.7 2008.9 1996 2011.1 1995.9 2012.9 C
  7.2688 +1993.4 2015.1 1990.5 2016.2 1987.7 2017.7 C
  7.2689 +1987.1 2019.3 1991.1 2019.4 1990.4 2021.3 C
  7.2690 +1990.5 2021.5 1991.9 2022.3 1992 2023 C
  7.2691 +1994.8 2024.4 1996.2 2027.5 1998.5 2030 C
  7.2692 +2002.4 2033 2005.2 2037.2 2008.8 2041 C
  7.2693 +2010.2 2041.3 2011.6 2042 2011 2043.9 C
  7.2694 +2011.2 2044.8 2010.1 2045.3 2010.5 2046.3 C
  7.2695 +2013.8 2044.8 2017.5 2043.4 2021.1 2042 C
  7.2696 +[0 0.5 0.5 0.2]  vc
  7.2697 +f 
  7.2698 +S 
  7.2699 +n
  7.2700 +2019.4 2008.8 m
  7.2701 +2018.9 2009.2 2019.3 2009.9 2019.6 2010.3 C
  7.2702 +2022.2 2011.5 2020.3 2009.1 2019.4 2008.8 C
  7.2703 +[0 0.33 0.33 0.99]  vc
  7.2704 +f 
  7.2705 +S 
  7.2706 +n
  7.2707 +2018 2007.4 m
  7.2708 +2015.7 2006.7 2015.3 2003.6 2012.9 2002.8 C
  7.2709 +2013.5 2003.7 2013.5 2005.1 2015.6 2005.2 C
  7.2710 +2016.4 2006.1 2015.7 2007.7 2018 2007.4 C
  7.2711 +f 
  7.2712 +S 
  7.2713 +n
  7.2714 +vmrs
  7.2715 +1993.5 2008.8 m
  7.2716 +1993.4 2000 1993.7 1992.5 1994 1985.1 C
  7.2717 +1993.7 1984.3 1989.9 1984.1 1990.6 1982 C
  7.2718 +1989.8 1981.1 1987.7 1981.4 1988.2 1979.3 C
  7.2719 +1988.3 1979.6 1988.1 1979.7 1988 1979.8 C
  7.2720 +1987.5 1977.5 1984.5 1978.6 1984.6 1976.2 C
  7.2721 +1983.9 1975.5 1981.7 1975.8 1982.4 1974.3 C
  7.2722 +1981.6 1974.9 1982.1 1973.1 1982 1973.3 C
  7.2723 +1979 1973.7 1980 1968.8 1976.9 1969.7 C
  7.2724 +1975.9 1969.8 1975.3 1970.3 1975 1971.2 C
  7.2725 +1976.2 1969.2 1977 1971.2 1978.6 1970.9 C
  7.2726 +1978.5 1974.4 1981.7 1972.8 1982.2 1976 C
  7.2727 +1985.2 1976.3 1984.5 1979.3 1987 1979.6 C
  7.2728 +1987.7 1980.3 1987.5 1982.1 1988.9 1981.7 C
  7.2729 +1990.4 1982.4 1990.7 1985.5 1993 1985.8 C
  7.2730 +1992.9 1994.3 1993.2 2002.3 1992.8 2011.2 C
  7.2731 +1991.1 2012.4 1990 2014.4 1987.7 2014.6 C
  7.2732 +1990.1 2013.4 1994.7 2012.6 1993.5 2008.8 C
  7.2733 +[0 0.87 0.91 0.83]  vc
  7.2734 +f 
  7.2735 +0.4 w
  7.2736 +2 J
  7.2737 +2 M
  7.2738 +S 
  7.2739 +n
  7.2740 +1992.8 2010.8 m
  7.2741 +1992.8 2001.8 1992.8 1994.1 1992.8 1985.8 C
  7.2742 +1989.5 1985.7 1991.1 1981.1 1987.7 1981.7 C
  7.2743 +1987.9 1978.2 1983.9 1980 1984.1 1977.2 C
  7.2744 +1981.1 1977 1981.5 1973 1979.1 1973.1 C
  7.2745 +1979 1972.2 1978.5 1970.9 1977.6 1970.9 C
  7.2746 +1977.9 1971.6 1979 1971.9 1978.6 1973.1 C
  7.2747 +1977.6 1974.9 1976.8 1973.9 1977.2 1976.2 C
  7.2748 +1977.2 1981.5 1977 1989.4 1977.4 1994 C
  7.2749 +1978.3 1995 1976.6 1994.1 1977.2 1994.7 C
  7.2750 +1977 1995.3 1976.6 1997 1977.9 1997.8 C
  7.2751 +1979 1997.5 1979.3 1998.3 1979.8 1998.8 C
  7.2752 +1979.8 1998.9 1979.8 1999.8 1979.8 1999.2 C
  7.2753 +1980.8 1998.7 1979.7 2000.7 1980.8 2000.7 C
  7.2754 +1983.5 2000.4 1982.1 2003 1984.1 2003.3 C
  7.2755 +1984.4 2004.3 1984.5 2003.7 1985.3 2004 C
  7.2756 +1986.3 2004.6 1985.9 2006.1 1986.5 2006.9 C
  7.2757 +1988 2007.1 1988.4 2009.7 1990.6 2009.1 C
  7.2758 +1990.9 2006.1 1989 2000.2 1990.4 1998 C
  7.2759 +1990.2 1994.3 1990.8 1989.2 1989.9 1986.8 C
  7.2760 +1990.2 1984.7 1990.8 1986.2 1991.6 1985.1 C
  7.2761 +1991.5 1985.9 1992.6 1985.5 1992.5 1986.3 C
  7.2762 +1992 1990.5 1992.6 1995 1992 1999.2 C
  7.2763 +1991.6 1998.9 1991.9 1998.3 1991.8 1997.8 C
  7.2764 +1991.8 1998.5 1991.8 2000 1991.8 2000 C
  7.2765 +1991.9 1999.9 1992 1999.8 1992 1999.7 C
  7.2766 +1993.2 2003.5 1991.9 2007.7 1992.3 2011.5 C
  7.2767 +1991.6 2012 1990.9 2012.2 1990.4 2012.9 C
  7.2768 +1991.3 2011.9 1992.2 2012.1 1992.8 2010.8 C
  7.2769 +[0 1 1 0.23]  vc
  7.2770 +f 
  7.2771 +S 
  7.2772 +n
  7.2773 +1978.4 1968.5 m
  7.2774 +1977 1969.2 1975.8 1968.2 1974.5 1969 C
  7.2775 +1968.3 1973 1961.6 1976 1955.1 1979.1 C
  7.2776 +1962 1975.9 1968.8 1972.5 1975.5 1968.8 C
  7.2777 +1976.5 1968.8 1977.6 1968.8 1978.6 1968.8 C
  7.2778 +1981.7 1972.1 1984.8 1975.7 1988 1978.8 C
  7.2779 +1990.9 1981.9 1996.8 1984.6 1995.2 1990.6 C
  7.2780 +1995.3 1988.6 1994.9 1986.9 1994.7 1985.1 C
  7.2781 +1989.5 1979.1 1983.3 1974.3 1978.4 1968.5 C
  7.2782 +[0.18 0.18 0 0.78]  vc
  7.2783 +f 
  7.2784 +S 
  7.2785 +n
  7.2786 +1978.4 1968.3 m
  7.2787 +1977.9 1968.7 1977.1 1968.5 1976.4 1968.5 C
  7.2788 +1977.3 1968.8 1978.1 1967.9 1978.8 1968.5 C
  7.2789 +1984 1974.3 1990.1 1979.5 1995.2 1985.6 C
  7.2790 +1995.1 1988.4 1995.3 1985.6 1994.9 1984.8 C
  7.2791 +1989.5 1979.4 1983.9 1973.8 1978.4 1968.3 C
  7.2792 +[0.07 0.06 0 0.58]  vc
  7.2793 +f 
  7.2794 +S 
  7.2795 +n
  7.2796 +1978.6 1968 m
  7.2797 +1977.9 1968 1977.4 1968.6 1978.4 1968 C
  7.2798 +1983.9 1973.9 1990.1 1979.1 1995.2 1985.1 C
  7.2799 +1990.2 1979 1983.8 1974.1 1978.6 1968 C
  7.2800 +[0.4 0.4 0 0]  vc
  7.2801 +f 
  7.2802 +S 
  7.2803 +n
  7.2804 +1991.1 1982.2 m
  7.2805 +1991.2 1982.9 1991.6 1984.2 1993 1984.4 C
  7.2806 +1992.6 1983.5 1992.1 1982.5 1991.1 1982.2 C
  7.2807 +[0 0.33 0.33 0.99]  vc
  7.2808 +f 
  7.2809 +S 
  7.2810 +n
  7.2811 +1990.4 2012.7 m
  7.2812 +1991.4 2011.8 1990.2 2010.9 1989.9 2010.3 C
  7.2813 +1987.7 2010.2 1987.4 2007.6 1985.6 2007.2 C
  7.2814 +1985.1 2006.2 1984.6 2004.5 1984.1 2004.3 C
  7.2815 +1981.7 2004.5 1982.3 2001.2 1979.8 2000.9 C
  7.2816 +1978.8 1999.6 1978.8 1999.1 1977.6 1998.8 C
  7.2817 +1976.1 1997.4 1976.7 1995 1975.2 1994 C
  7.2818 +1975.8 1994 1975 1994 1975 1993.7 C
  7.2819 +1975.7 1993.2 1975.6 1991.8 1976 1991.3 C
  7.2820 +1975.9 1985.7 1976.1 1979.7 1975.7 1974.5 C
  7.2821 +1976.2 1973.3 1976.9 1971.8 1976.2 1971.4 C
  7.2822 +1973.9 1974.3 1972.2 1973.6 1969.5 1975 C
  7.2823 +1967.9 1977.5 1963.8 1977.1 1961.8 1980 C
  7.2824 +1959 1980 1957.6 1983 1954.8 1982.9 C
  7.2825 +1953.8 1984.2 1954.8 1985.7 1955.1 1987.2 C
  7.2826 +1956.2 1989.5 1959.7 1990.1 1959.9 1991.8 C
  7.2827 +1965.9 1998 1971.8 2005.2 1978.1 2011.7 C
  7.2828 +1979.5 2012 1980.9 2012.7 1980.3 2014.6 C
  7.2829 +1980.5 2015.6 1979.4 2016 1979.8 2017 C
  7.2830 +1983 2015.6 1986.8 2014.1 1990.4 2012.7 C
  7.2831 +[0 0.5 0.5 0.2]  vc
  7.2832 +f 
  7.2833 +S 
  7.2834 +n
  7.2835 +1988.7 1979.6 m
  7.2836 +1988.2 1979.9 1988.6 1980.6 1988.9 1981 C
  7.2837 +1991.4 1982.2 1989.6 1979.9 1988.7 1979.6 C
  7.2838 +[0 0.33 0.33 0.99]  vc
  7.2839 +f 
  7.2840 +S 
  7.2841 +n
  7.2842 +1987.2 1978.1 m
  7.2843 +1985 1977.5 1984.6 1974.3 1982.2 1973.6 C
  7.2844 +1982.7 1974.5 1982.8 1975.8 1984.8 1976 C
  7.2845 +1985.7 1976.9 1985 1978.4 1987.2 1978.1 C
  7.2846 +f 
  7.2847 +S 
  7.2848 +n
  7.2849 +1975.5 2084 m
  7.2850 +1975.5 2082 1975.3 2080 1975.7 2078.2 C
  7.2851 +1978.8 2079 1980.9 2085.5 1984.8 2083.5 C
  7.2852 +1993 2078.7 2001.6 2075 2010 2070.8 C
  7.2853 +2010.1 2064 2009.9 2057.2 2010.3 2050.6 C
  7.2854 +2014.8 2046.2 2020.9 2045.7 2025.6 2042 C
  7.2855 +2026.1 2035.1 2025.8 2028 2025.9 2021.1 C
  7.2856 +2025.8 2027.8 2026.1 2034.6 2025.6 2041.2 C
  7.2857 +2022.2 2044.9 2017.6 2046.8 2012.9 2048 C
  7.2858 +2012.5 2049.5 2010.4 2049.4 2009.8 2051.1 C
  7.2859 +2009.9 2057.6 2009.6 2064.2 2010 2070.5 C
  7.2860 +2001.2 2075.4 1992 2079.1 1983.2 2084 C
  7.2861 +1980.3 2082.3 1977.8 2079.2 1975.2 2077.5 C
  7.2862 +1974.9 2079.9 1977.2 2084.6 1973.3 2085.2 C
  7.2863 +1964.7 2088.6 1956.8 2093.7 1948.1 2097.2 C
  7.2864 +1949 2097.3 1949.6 2096.9 1950.3 2096.7 C
  7.2865 +1958.4 2091.9 1967.1 2088.2 1975.5 2084 C
  7.2866 +[0.18 0.18 0 0.78]  vc
  7.2867 +f 
  7.2868 +S 
  7.2869 +n
  7.2870 +vmrs
  7.2871 +1948.6 2094.5 m
  7.2872 +1950.2 2093.7 1951.8 2092.9 1953.4 2092.1 C
  7.2873 +1951.8 2092.9 1950.2 2093.7 1948.6 2094.5 C
  7.2874 +[0 0.87 0.91 0.83]  vc
  7.2875 +f 
  7.2876 +0.4 w
  7.2877 +2 J
  7.2878 +2 M
  7.2879 +S 
  7.2880 +n
  7.2881 +1971.6 2082.3 m
  7.2882 +1971.6 2081.9 1970.7 2081.1 1970.9 2081.3 C
  7.2883 +1970.7 2081.6 1970.6 2081.6 1970.4 2081.3 C
  7.2884 +1970.8 2080.1 1968.7 2081.7 1968.3 2080.8 C
  7.2885 +1966.6 2080.9 1966.7 2078 1964.2 2078.2 C
  7.2886 +1964.8 2075 1960.1 2075.8 1960.1 2072.9 C
  7.2887 +1958 2072.3 1957.5 2069.3 1955.3 2069.3 C
  7.2888 +1953.9 2070.9 1948.8 2067.8 1950 2072 C
  7.2889 +1949 2074 1943.2 2070.6 1944 2074.8 C
  7.2890 +1942.2 2076.6 1937.6 2073.9 1938 2078.2 C
  7.2891 +1936.7 2078.6 1935 2078.6 1933.7 2078.2 C
  7.2892 +1933.5 2080 1936.8 2080.7 1937.3 2082.8 C
  7.2893 +1939.9 2083.5 1940.6 2086.4 1942.6 2088 C
  7.2894 +1945.2 2089.2 1946 2091.3 1948.4 2093.6 C
  7.2895 +1956 2089.5 1963.9 2086.1 1971.6 2082.3 C
  7.2896 +[0 0.01 1 0]  vc
  7.2897 +f 
  7.2898 +S 
  7.2899 +n
  7.2900 +1958.2 2089.7 m
  7.2901 +1956.4 2090 1955.6 2091.3 1953.9 2091.9 C
  7.2902 +1955.6 2091.9 1956.5 2089.7 1958.2 2089.7 C
  7.2903 +[0 0.87 0.91 0.83]  vc
  7.2904 +f 
  7.2905 +S 
  7.2906 +n
  7.2907 +1929.9 2080.4 m
  7.2908 +1929.5 2077.3 1929.7 2073.9 1929.6 2070.8 C
  7.2909 +1929.8 2074.1 1929.2 2077.8 1930.1 2080.8 C
  7.2910 +1935.8 2085.9 1941.4 2091.3 1946.9 2096.9 C
  7.2911 +1941.2 2091 1935.7 2086 1929.9 2080.4 C
  7.2912 +[0.4 0.4 0 0]  vc
  7.2913 +f 
  7.2914 +S 
  7.2915 +n
  7.2916 +1930.1 2080.4 m
  7.2917 +1935.8 2086 1941.5 2090.7 1946.9 2096.7 C
  7.2918 +1941.5 2090.9 1935.7 2085.8 1930.1 2080.4 C
  7.2919 +[0.07 0.06 0 0.58]  vc
  7.2920 +f 
  7.2921 +S 
  7.2922 +n
  7.2923 +1940.9 2087.1 m
  7.2924 +1941.7 2088 1944.8 2090.6 1943.6 2089.2 C
  7.2925 +1942.5 2089 1941.6 2087.7 1940.9 2087.1 C
  7.2926 +[0 0.87 0.91 0.83]  vc
  7.2927 +f 
  7.2928 +S 
  7.2929 +n
  7.2930 +1972.8 2082.8 m
  7.2931 +1973 2075.3 1972.4 2066.9 1973.3 2059.5 C
  7.2932 +1972.5 2058.9 1972.8 2057.3 1973.1 2056.4 C
  7.2933 +1974.8 2055.2 1973.4 2055.5 1972.4 2055.4 C
  7.2934 +1970.1 2053.2 1967.9 2050.9 1965.6 2048.7 C
  7.2935 +1960.9 2049.9 1956.9 2052.7 1952.4 2054.7 C
  7.2936 +1949.3 2052.5 1946.3 2049.5 1943.6 2046.8 C
  7.2937 +1939.9 2047.7 1936.8 2050.1 1933.5 2051.8 C
  7.2938 +1930.9 2054.9 1933.5 2056.2 1932.3 2059.7 C
  7.2939 +1933.2 2059.7 1932.2 2060.5 1932.5 2060.2 C
  7.2940 +1933.2 2062.5 1931.6 2064.6 1932.5 2067.4 C
  7.2941 +1932.9 2069.7 1932.7 2072.2 1932.8 2074.6 C
  7.2942 +1933.6 2070.6 1932.2 2066.3 1933 2062.6 C
  7.2943 +1934.4 2058.2 1929.8 2053.5 1935.2 2051.1 C
  7.2944 +1937.7 2049.7 1940.2 2048 1942.8 2046.8 C
  7.2945 +1945.9 2049.2 1948.8 2052 1951.7 2054.7 C
  7.2946 +1952.7 2054.7 1953.6 2054.6 1954.4 2054.2 C
  7.2947 +1958.1 2052.5 1961.7 2049.3 1965.9 2049.2 C
  7.2948 +1968.2 2052.8 1975.2 2055 1972.6 2060.9 C
  7.2949 +1973.3 2062.4 1972.2 2065.2 1972.6 2067.6 C
  7.2950 +1972.7 2072.6 1972.4 2077.7 1972.8 2082.5 C
  7.2951 +1968.1 2084.9 1963.5 2087.5 1958.7 2089.5 C
  7.2952 +1963.5 2087.4 1968.2 2085 1972.8 2082.8 C
  7.2953 +f 
  7.2954 +S 
  7.2955 +n
  7.2956 +1935.2 2081.1 m
  7.2957 +1936.8 2083.4 1938.6 2084.6 1940.4 2086.6 C
  7.2958 +1938.8 2084.4 1936.7 2083.4 1935.2 2081.1 C
  7.2959 +f 
  7.2960 +S 
  7.2961 +n
  7.2962 +1983.2 2081.3 m
  7.2963 +1984.8 2080.5 1986.3 2079.7 1988 2078.9 C
  7.2964 +1986.3 2079.7 1984.8 2080.5 1983.2 2081.3 C
  7.2965 +f 
  7.2966 +S 
  7.2967 +n
  7.2968 +2006.2 2069.1 m
  7.2969 +2006.2 2068.7 2005.2 2067.9 2005.5 2068.1 C
  7.2970 +2005.3 2068.4 2005.2 2068.4 2005 2068.1 C
  7.2971 +2005.4 2066.9 2003.3 2068.5 2002.8 2067.6 C
  7.2972 +2001.2 2067.7 2001.2 2064.8 1998.8 2065 C
  7.2973 +1999.4 2061.8 1994.7 2062.6 1994.7 2059.7 C
  7.2974 +1992.4 2059.5 1992.4 2055.8 1990.1 2056.8 C
  7.2975 +1985.9 2059.5 1981.1 2061 1976.9 2063.8 C
  7.2976 +1977.2 2067.6 1974.9 2074.2 1978.8 2075.8 C
  7.2977 +1979.6 2077.8 1981.7 2078.4 1982.9 2080.4 C
  7.2978 +1990.6 2076.3 1998.5 2072.9 2006.2 2069.1 C
  7.2979 +[0 0.01 1 0]  vc
  7.2980 +f 
  7.2981 +S 
  7.2982 +n
  7.2983 +vmrs
  7.2984 +1992.8 2076.5 m
  7.2985 +1991 2076.8 1990.2 2078.1 1988.4 2078.7 C
  7.2986 +1990.2 2078.7 1991 2076.5 1992.8 2076.5 C
  7.2987 +[0 0.87 0.91 0.83]  vc
  7.2988 +f 
  7.2989 +0.4 w
  7.2990 +2 J
  7.2991 +2 M
  7.2992 +S 
  7.2993 +n
  7.2994 +1975.5 2073.4 m
  7.2995 +1976.1 2069.7 1973.9 2064.6 1977.4 2062.4 C
  7.2996 +1973.9 2064.5 1976.1 2069.9 1975.5 2073.6 C
  7.2997 +1976 2074.8 1979.3 2077.4 1978.1 2076 C
  7.2998 +1977 2075.7 1975.8 2074.5 1975.5 2073.4 C
  7.2999 +f 
  7.3000 +S 
  7.3001 +n
  7.3002 +2007.4 2069.6 m
  7.3003 +2007.6 2062.1 2007 2053.7 2007.9 2046.3 C
  7.3004 +2007.1 2045.7 2007.3 2044.1 2007.6 2043.2 C
  7.3005 +2009.4 2042 2007.9 2042.3 2006.9 2042.2 C
  7.3006 +2002.2 2037.4 1996.7 2032.4 1992.5 2027.3 C
  7.3007 +1992 2027.3 1991.6 2027.3 1991.1 2027.3 C
  7.3008 +1991.4 2035.6 1991.4 2045.6 1991.1 2054.4 C
  7.3009 +1990.5 2055.5 1988.4 2056.6 1990.6 2055.4 C
  7.3010 +1991.6 2055.4 1991.6 2054.1 1991.6 2053.2 C
  7.3011 +1990.8 2044.7 1991.9 2035.4 1991.6 2027.6 C
  7.3012 +1991.8 2027.6 1992 2027.6 1992.3 2027.6 C
  7.3013 +1997 2032.8 2002.5 2037.7 2007.2 2042.9 C
  7.3014 +2007.3 2044.8 2006.7 2047.4 2007.6 2048.4 C
  7.3015 +2006.9 2055.1 2007.1 2062.5 2007.4 2069.3 C
  7.3016 +2002.7 2071.7 1998.1 2074.3 1993.2 2076.3 C
  7.3017 +1998 2074.2 2002.7 2071.8 2007.4 2069.6 C
  7.3018 +f 
  7.3019 +S 
  7.3020 +n
  7.3021 +2006.7 2069.1 m
  7.3022 +2006.3 2068.6 2005.9 2067.7 2005.7 2066.9 C
  7.3023 +2005.7 2059.7 2005.9 2051.4 2005.5 2045.1 C
  7.3024 +2004.9 2045.3 2004.7 2044.5 2004.3 2045.3 C
  7.3025 +2005.1 2045.3 2004.2 2045.8 2004.8 2046 C
  7.3026 +2004.8 2052.2 2004.8 2059.2 2004.8 2064.5 C
  7.3027 +2005.7 2065.7 2005.1 2065.7 2005 2066.7 C
  7.3028 +2003.8 2067 2002.7 2067.2 2001.9 2066.4 C
  7.3029 +2001.3 2064.6 1998 2063.1 1998 2061.9 C
  7.3030 +1996.1 2062.3 1996.6 2058.3 1994.2 2058.8 C
  7.3031 +1992.6 2057.7 1992.7 2054.8 1989.9 2056.6 C
  7.3032 +1985.6 2059.3 1980.9 2060.8 1976.7 2063.6 C
  7.3033 +1976 2066.9 1976 2071.2 1976.7 2074.6 C
  7.3034 +1977.6 2070.8 1973.1 2062.1 1980.5 2061.2 C
  7.3035 +1984.3 2060.3 1987.5 2058.2 1990.8 2056.4 C
  7.3036 +1991.7 2056.8 1992.9 2057.2 1993.5 2059.2 C
  7.3037 +1994.3 2058.6 1994.4 2060.6 1994.7 2059.2 C
  7.3038 +1995.3 2062.7 1999.2 2061.4 1998.8 2064.8 C
  7.3039 +2001.8 2065.4 2002.5 2068.4 2005.2 2067.4 C
  7.3040 +2004.9 2067.9 2006 2068 2006.4 2069.1 C
  7.3041 +2001.8 2071.1 1997.4 2073.9 1992.8 2075.8 C
  7.3042 +1997.5 2073.8 2002 2071.2 2006.7 2069.1 C
  7.3043 +[0 0.2 1 0]  vc
  7.3044 +f 
  7.3045 +S 
  7.3046 +n
  7.3047 +1988.7 2056.6 m
  7.3048 +1985.1 2058.7 1981.1 2060.1 1977.6 2061.9 C
  7.3049 +1981.3 2060.5 1985.6 2058.1 1988.7 2056.6 C
  7.3050 +[0 0.87 0.91 0.83]  vc
  7.3051 +f 
  7.3052 +S 
  7.3053 +n
  7.3054 +1977.9 2059.5 m
  7.3055 +1975.7 2064.5 1973.7 2054.7 1975.2 2060.9 C
  7.3056 +1976 2060.6 1977.6 2059.7 1977.9 2059.5 C
  7.3057 +f 
  7.3058 +S 
  7.3059 +n
  7.3060 +1989.6 2051.3 m
  7.3061 +1990.1 2042.3 1989.8 2036.6 1989.9 2028 C
  7.3062 +1989.8 2027 1990.8 2028.3 1990.1 2027.3 C
  7.3063 +1988.9 2026.7 1986.7 2026.9 1986.8 2024.7 C
  7.3064 +1987.4 2023 1985.9 2024.6 1985.1 2023.7 C
  7.3065 +1984.1 2021.4 1982.5 2020.5 1980.3 2020.6 C
  7.3066 +1979.9 2020.8 1979.5 2021.1 1979.3 2021.6 C
  7.3067 +1979.7 2025.8 1978.4 2033 1979.6 2038.1 C
  7.3068 +1983.7 2042.9 1968.8 2044.6 1978.8 2042.7 C
  7.3069 +1979.3 2042.3 1979.6 2041.9 1980 2041.5 C
  7.3070 +1980 2034.8 1980 2027 1980 2021.6 C
  7.3071 +1981.3 2020.5 1981.7 2021.5 1982.9 2021.8 C
  7.3072 +1983.6 2024.7 1986.1 2023.8 1986.8 2026.4 C
  7.3073 +1987.1 2027.7 1988.6 2027.1 1989.2 2028.3 C
  7.3074 +1989.1 2036.7 1989.3 2044.8 1988.9 2053.7 C
  7.3075 +1987.2 2054.9 1986.2 2056.8 1983.9 2057.1 C
  7.3076 +1986.3 2055.9 1990.9 2055 1989.6 2051.3 C
  7.3077 +f 
  7.3078 +S 
  7.3079 +n
  7.3080 +1971.6 2078.9 m
  7.3081 +1971.4 2070.5 1972.1 2062.2 1971.6 2055.9 C
  7.3082 +1969.9 2053.7 1967.6 2051.7 1965.6 2049.6 C
  7.3083 +1961.4 2050.4 1957.6 2053.6 1953.4 2055.2 C
  7.3084 +1949.8 2055.6 1948.2 2051.2 1945.5 2049.6 C
  7.3085 +1945.1 2048.8 1944.5 2047.9 1943.6 2047.5 C
  7.3086 +1940.1 2047.8 1937.3 2051 1934 2052.3 C
  7.3087 +1933.7 2052.6 1933.7 2053 1933.2 2053.2 C
  7.3088 +1933.7 2060.8 1933.4 2067.2 1933.5 2074.6 C
  7.3089 +1933.8 2068.1 1934 2060.9 1933.2 2054 C
  7.3090 +1935.3 2050.9 1939.3 2049.6 1942.4 2047.5 C
  7.3091 +1942.8 2047.5 1943.4 2047.4 1943.8 2047.7 C
  7.3092 +1947.1 2050.2 1950.3 2057.9 1955.3 2054.4 C
  7.3093 +1955.4 2054.4 1955.5 2054.3 1955.6 2054.2 C
  7.3094 +1955.9 2057.6 1956.1 2061.8 1955.3 2064.8 C
  7.3095 +1955.4 2064.3 1955.1 2063.8 1955.6 2063.6 C
  7.3096 +1956 2066.6 1955.3 2068.7 1958.7 2069.8 C
  7.3097 +1959.2 2071.7 1961.4 2071.7 1962 2074.1 C
  7.3098 +1964.4 2074.2 1964 2077.7 1967.3 2078.4 C
  7.3099 +1967 2079.7 1968.1 2079.9 1969 2080.1 C
  7.3100 +1971.1 2079.9 1970 2079.2 1970.4 2078 C
  7.3101 +1969.5 2077.2 1970.3 2075.9 1969.7 2075.1 C
  7.3102 +1970.1 2069.8 1970.1 2063.6 1969.7 2058.8 C
  7.3103 +1969.2 2058.5 1970 2058.1 1970.2 2057.8 C
  7.3104 +1970.4 2058.3 1971.2 2057.7 1971.4 2058.3 C
  7.3105 +1971.5 2065.3 1971.2 2073.6 1971.6 2081.1 C
  7.3106 +1974.1 2081.4 1969.8 2084.3 1972.4 2082.5 C
  7.3107 +1971.9 2081.4 1971.6 2080.2 1971.6 2078.9 C
  7.3108 +[0 0.4 1 0]  vc
  7.3109 +f 
  7.3110 +S 
  7.3111 +n
  7.3112 +1952.4 2052 m
  7.3113 +1954.1 2051.3 1955.6 2050.4 1957.2 2049.6 C
  7.3114 +1955.6 2050.4 1954.1 2051.3 1952.4 2052 C
  7.3115 +[0 0.87 0.91 0.83]  vc
  7.3116 +f 
  7.3117 +S 
  7.3118 +n
  7.3119 +1975.5 2039.8 m
  7.3120 +1975.5 2039.4 1974.5 2038.7 1974.8 2038.8 C
  7.3121 +1974.6 2039.1 1974.5 2039.1 1974.3 2038.8 C
  7.3122 +1974.6 2037.6 1972.5 2039.3 1972.1 2038.4 C
  7.3123 +1970.4 2038.4 1970.5 2035.5 1968 2035.7 C
  7.3124 +1968.6 2032.5 1964 2033.3 1964 2030.4 C
  7.3125 +1961.9 2029.8 1961.4 2026.8 1959.2 2026.8 C
  7.3126 +1957.7 2028.5 1952.6 2025.3 1953.9 2029.5 C
  7.3127 +1952.9 2031.5 1947 2028.2 1947.9 2032.4 C
  7.3128 +1946 2034.2 1941.5 2031.5 1941.9 2035.7 C
  7.3129 +1940.6 2036.1 1938.9 2036.1 1937.6 2035.7 C
  7.3130 +1937.3 2037.5 1940.7 2038.2 1941.2 2040.3 C
  7.3131 +1943.7 2041.1 1944.4 2043.9 1946.4 2045.6 C
  7.3132 +1949.1 2046.7 1949.9 2048.8 1952.2 2051.1 C
  7.3133 +1959.9 2047.1 1967.7 2043.6 1975.5 2039.8 C
  7.3134 +[0 0.01 1 0]  vc
  7.3135 +f 
  7.3136 +S 
  7.3137 +n
  7.3138 +vmrs
  7.3139 +1962 2047.2 m
  7.3140 +1960.2 2047.5 1959.5 2048.9 1957.7 2049.4 C
  7.3141 +1959.5 2049.5 1960.3 2047.2 1962 2047.2 C
  7.3142 +[0 0.87 0.91 0.83]  vc
  7.3143 +f 
  7.3144 +0.4 w
  7.3145 +2 J
  7.3146 +2 M
  7.3147 +S 
  7.3148 +n
  7.3149 +2012.4 2046.3 m
  7.3150 +2010.3 2051.3 2008.3 2041.5 2009.8 2047.7 C
  7.3151 +2010.5 2047.4 2012.2 2046.5 2012.4 2046.3 C
  7.3152 +f 
  7.3153 +S 
  7.3154 +n
  7.3155 +1944.8 2044.6 m
  7.3156 +1945.5 2045.6 1948.6 2048.1 1947.4 2046.8 C
  7.3157 +1946.3 2046.5 1945.5 2045.2 1944.8 2044.6 C
  7.3158 +f 
  7.3159 +S 
  7.3160 +n
  7.3161 +1987.2 2054.9 m
  7.3162 +1983.7 2057.3 1979.6 2058 1976 2060.2 C
  7.3163 +1974.7 2058.2 1977.2 2055.8 1974.3 2054.9 C
  7.3164 +1973.1 2052 1970.4 2050.2 1968 2048 C
  7.3165 +1968 2047.7 1968 2047.4 1968.3 2047.2 C
  7.3166 +1969.5 2046.1 1983 2040.8 1972.4 2044.8 C
  7.3167 +1971.2 2046.6 1967.9 2046 1968 2048.2 C
  7.3168 +1970.5 2050.7 1973.8 2052.6 1974.3 2055.6 C
  7.3169 +1975.1 2055 1975.7 2056.7 1975.7 2057.1 C
  7.3170 +1975.7 2058.2 1974.8 2059.3 1975.5 2060.4 C
  7.3171 +1979.3 2058.2 1983.9 2057.7 1987.2 2054.9 C
  7.3172 +[0.18 0.18 0 0.78]  vc
  7.3173 +f 
  7.3174 +S 
  7.3175 +n
  7.3176 +1967.8 2047.5 m
  7.3177 +1968.5 2047 1969.1 2046.5 1969.7 2046 C
  7.3178 +1969.1 2046.5 1968.5 2047 1967.8 2047.5 C
  7.3179 +[0 0.87 0.91 0.83]  vc
  7.3180 +f 
  7.3181 +S 
  7.3182 +n
  7.3183 +1976.7 2040.3 m
  7.3184 +1976.9 2032.8 1976.3 2024.4 1977.2 2017 C
  7.3185 +1976.4 2016.5 1976.6 2014.8 1976.9 2013.9 C
  7.3186 +1978.7 2012.7 1977.2 2013 1976.2 2012.9 C
  7.3187 +1971.5 2008.1 1965.9 2003.1 1961.8 1998 C
  7.3188 +1960.9 1998 1960.1 1998 1959.2 1998 C
  7.3189 +1951.5 2001.1 1944.3 2005.5 1937.1 2009.6 C
  7.3190 +1935 2012.9 1937 2013.6 1936.1 2017.2 C
  7.3191 +1937.1 2017.2 1936 2018 1936.4 2017.7 C
  7.3192 +1937 2020.1 1935.5 2022.1 1936.4 2024.9 C
  7.3193 +1936.8 2027.2 1936.5 2029.7 1936.6 2032.1 C
  7.3194 +1937.4 2028.2 1936 2023.8 1936.8 2020.1 C
  7.3195 +1938.3 2015.7 1933.6 2011 1939 2008.6 C
  7.3196 +1945.9 2004.5 1953.1 2000.3 1960.6 1998.3 C
  7.3197 +1960.9 1998.3 1961.3 1998.3 1961.6 1998.3 C
  7.3198 +1966.2 2003.5 1971.8 2008.4 1976.4 2013.6 C
  7.3199 +1976.6 2015.5 1976 2018.1 1976.9 2019.2 C
  7.3200 +1976.1 2025.8 1976.4 2033.2 1976.7 2040 C
  7.3201 +1971.9 2042.4 1967.4 2045 1962.5 2047 C
  7.3202 +1967.3 2044.9 1972 2042.6 1976.7 2040.3 C
  7.3203 +f 
  7.3204 +S 
  7.3205 +n
  7.3206 +1939 2038.6 m
  7.3207 +1940.6 2040.9 1942.5 2042.1 1944.3 2044.1 C
  7.3208 +1942.7 2041.9 1940.6 2040.9 1939 2038.6 C
  7.3209 +f 
  7.3210 +S 
  7.3211 +n
  7.3212 +2006.2 2065.7 m
  7.3213 +2006 2057.3 2006.7 2049 2006.2 2042.7 C
  7.3214 +2002.1 2038.4 1997.7 2033.4 1993 2030 C
  7.3215 +1992.9 2029.3 1992.5 2028.6 1992 2028.3 C
  7.3216 +1992.1 2036.6 1991.9 2046.2 1992.3 2054.9 C
  7.3217 +1990.8 2056.2 1989 2056.7 1987.5 2058 C
  7.3218 +1988.7 2057.7 1990.7 2054.4 1993 2056.4 C
  7.3219 +1993.4 2058.8 1996 2058.2 1996.6 2060.9 C
  7.3220 +1999 2061 1998.5 2064.5 2001.9 2065.2 C
  7.3221 +2001.5 2066.5 2002.7 2066.7 2003.6 2066.9 C
  7.3222 +2005.7 2066.7 2004.6 2066 2005 2064.8 C
  7.3223 +2004 2064 2004.8 2062.7 2004.3 2061.9 C
  7.3224 +2004.6 2056.6 2004.6 2050.4 2004.3 2045.6 C
  7.3225 +2003.7 2045.3 2004.6 2044.9 2004.8 2044.6 C
  7.3226 +2005 2045.1 2005.7 2044.5 2006 2045.1 C
  7.3227 +2006 2052.1 2005.8 2060.4 2006.2 2067.9 C
  7.3228 +2008.7 2068.2 2004.4 2071.1 2006.9 2069.3 C
  7.3229 +2006.4 2068.2 2006.2 2067 2006.2 2065.7 C
  7.3230 +[0 0.4 1 0]  vc
  7.3231 +f 
  7.3232 +S 
  7.3233 +n
  7.3234 +2021.8 2041.7 m
  7.3235 +2018.3 2044.1 2014.1 2044.8 2010.5 2047 C
  7.3236 +2009.3 2045 2011.7 2042.6 2008.8 2041.7 C
  7.3237 +2004.3 2035.1 1997.6 2030.9 1993 2024.4 C
  7.3238 +1992.1 2024 1991.5 2024.3 1990.8 2024 C
  7.3239 +1993.2 2023.9 1995.3 2027.1 1996.8 2029 C
  7.3240 +2000.4 2032.6 2004.9 2036.9 2008.4 2040.8 C
  7.3241 +2008.2 2043.1 2011.4 2042.8 2009.8 2045.8 C
  7.3242 +2009.8 2046.3 2009.7 2046.9 2010 2047.2 C
  7.3243 +2013.8 2045 2018.5 2044.5 2021.8 2041.7 C
  7.3244 +[0.18 0.18 0 0.78]  vc
  7.3245 +f 
  7.3246 +S 
  7.3247 +n
  7.3248 +2001.6 2034 m
  7.3249 +2000.7 2033.1 1999.9 2032.3 1999 2031.4 C
  7.3250 +1999.9 2032.3 2000.7 2033.1 2001.6 2034 C
  7.3251 +[0 0.87 0.91 0.83]  vc
  7.3252 +f 
  7.3253 +S 
  7.3254 +n
  7.3255 +vmrs
  7.3256 +1989.4 2024.4 m
  7.3257 +1989.5 2025.4 1988.6 2024.3 1988.9 2024.7 C
  7.3258 +1990.5 2025.8 1990.7 2024.2 1992.8 2024.9 C
  7.3259 +1993.8 2025.9 1995 2027.1 1995.9 2028 C
  7.3260 +1994.3 2026 1991.9 2023.4 1989.4 2024.4 C
  7.3261 +[0 0.87 0.91 0.83]  vc
  7.3262 +f 
  7.3263 +0.4 w
  7.3264 +2 J
  7.3265 +2 M
  7.3266 +S 
  7.3267 +n
  7.3268 +1984.8 2019.9 m
  7.3269 +1984.6 2018.6 1986.3 2017.2 1987.7 2016.8 C
  7.3270 +1987.2 2017.5 1982.9 2017.9 1984.4 2020.6 C
  7.3271 +1984.1 2019.9 1984.9 2020 1984.8 2019.9 C
  7.3272 +f 
  7.3273 +S 
  7.3274 +n
  7.3275 +1981.7 2017 m
  7.3276 +1979.6 2022 1977.6 2012.3 1979.1 2018.4 C
  7.3277 +1979.8 2018.1 1981.5 2017.2 1981.7 2017 C
  7.3278 +f 
  7.3279 +S 
  7.3280 +n
  7.3281 +1884.3 2019.2 m
  7.3282 +1884.7 2010.5 1884.5 2000.6 1884.5 1991.8 C
  7.3283 +1886.6 1989.3 1889.9 1988.9 1892.4 1987 C
  7.3284 +1890.8 1988.7 1886 1989.1 1884.3 1992.3 C
  7.3285 +1884.7 2001 1884.5 2011.3 1884.5 2019.9 C
  7.3286 +1891 2025.1 1895.7 2031.5 1902 2036.9 C
  7.3287 +1896.1 2031 1890 2024.9 1884.3 2019.2 C
  7.3288 +[0.07 0.06 0 0.58]  vc
  7.3289 +f 
  7.3290 +S 
  7.3291 +n
  7.3292 +1884 2019.4 m
  7.3293 +1884.5 2010.6 1884.2 2000.4 1884.3 1991.8 C
  7.3294 +1884.8 1990.4 1887.8 1989 1884.8 1990.8 C
  7.3295 +1884.3 1991.3 1884.3 1992 1884 1992.5 C
  7.3296 +1884.5 2001.2 1884.2 2011.1 1884.3 2019.9 C
  7.3297 +1887.9 2023.1 1891.1 2026.4 1894.4 2030 C
  7.3298 +1891.7 2026.1 1887.1 2022.9 1884 2019.4 C
  7.3299 +[0.4 0.4 0 0]  vc
  7.3300 +f 
  7.3301 +S 
  7.3302 +n
  7.3303 +1885 2011.7 m
  7.3304 +1885 2006.9 1885 2001.9 1885 1997.1 C
  7.3305 +1885 2001.9 1885 2006.9 1885 2011.7 C
  7.3306 +[0 0.87 0.91 0.83]  vc
  7.3307 +f 
  7.3308 +S 
  7.3309 +n
  7.3310 +1975.5 2036.4 m
  7.3311 +1975.2 2028 1976 2019.7 1975.5 2013.4 C
  7.3312 +1971.1 2008.5 1965.6 2003.6 1961.6 1999 C
  7.3313 +1958.8 1998 1956 2000 1953.6 2001.2 C
  7.3314 +1948.2 2004.7 1941.9 2006.5 1937.1 2010.8 C
  7.3315 +1937.5 2018.3 1937.3 2024.7 1937.3 2032.1 C
  7.3316 +1937.6 2025.6 1937.9 2018.4 1937.1 2011.5 C
  7.3317 +1937.3 2011 1937.6 2010.5 1937.8 2010 C
  7.3318 +1944.6 2005.7 1951.9 2002.3 1959.2 1999 C
  7.3319 +1960.1 1998.5 1960.1 1999.8 1960.4 2000.4 C
  7.3320 +1959.7 2006.9 1959.7 2014.2 1959.4 2021.1 C
  7.3321 +1959 2021.1 1959.2 2021.9 1959.2 2022.3 C
  7.3322 +1959.2 2021.9 1959 2021.3 1959.4 2021.1 C
  7.3323 +1959.8 2024.1 1959.2 2026.2 1962.5 2027.3 C
  7.3324 +1963 2029.2 1965.3 2029.2 1965.9 2031.6 C
  7.3325 +1968.3 2031.8 1967.8 2035.2 1971.2 2036 C
  7.3326 +1970.8 2037.2 1971.9 2037.5 1972.8 2037.6 C
  7.3327 +1974.9 2037.4 1973.9 2036.7 1974.3 2035.5 C
  7.3328 +1973.3 2034.7 1974.1 2033.4 1973.6 2032.6 C
  7.3329 +1973.9 2027.3 1973.9 2021.1 1973.6 2016.3 C
  7.3330 +1973 2016 1973.9 2015.6 1974 2015.3 C
  7.3331 +1974.3 2015.9 1975 2015.3 1975.2 2015.8 C
  7.3332 +1975.3 2022.8 1975.1 2031.2 1975.5 2038.6 C
  7.3333 +1977.9 2039 1973.7 2041.8 1976.2 2040 C
  7.3334 +1975.7 2039 1975.5 2037.8 1975.5 2036.4 C
  7.3335 +[0 0.4 1 0]  vc
  7.3336 +f 
  7.3337 +S 
  7.3338 +n
  7.3339 +1991.1 2012.4 m
  7.3340 +1987.5 2014.8 1983.4 2015.6 1979.8 2017.7 C
  7.3341 +1978.5 2015.7 1981 2013.3 1978.1 2012.4 C
  7.3342 +1973.6 2005.8 1966.8 2001.6 1962.3 1995.2 C
  7.3343 +1961.4 1994.7 1960.8 1995 1960.1 1994.7 C
  7.3344 +1962.5 1994.6 1964.6 1997.8 1966.1 1999.7 C
  7.3345 +1969.7 2003.3 1974.2 2007.6 1977.6 2011.5 C
  7.3346 +1977.5 2013.8 1980.6 2013.5 1979.1 2016.5 C
  7.3347 +1979.1 2017 1979 2017.6 1979.3 2018 C
  7.3348 +1983.1 2015.7 1987.8 2015.2 1991.1 2012.4 C
  7.3349 +[0.18 0.18 0 0.78]  vc
  7.3350 +f 
  7.3351 +S 
  7.3352 +n
  7.3353 +1970.9 2004.8 m
  7.3354 +1970 2003.9 1969.2 2003 1968.3 2002.1 C
  7.3355 +1969.2 2003 1970 2003.9 1970.9 2004.8 C
  7.3356 +[0 0.87 0.91 0.83]  vc
  7.3357 +f 
  7.3358 +S 
  7.3359 +n
  7.3360 +1887.9 1994.9 m
  7.3361 +1888.5 1992.3 1891.4 1992.2 1893.2 1990.8 C
  7.3362 +1898.4 1987.5 1904 1984.8 1909.5 1982.2 C
  7.3363 +1909.7 1982.7 1910.3 1982.1 1910.4 1982.7 C
  7.3364 +1909.5 1990.5 1910.1 1996.4 1910 2004.5 C
  7.3365 +1909.1 2003.4 1909.7 2005.8 1909.5 2006.4 C
  7.3366 +1910.4 2006 1909.7 2008 1910.2 2007.9 C
  7.3367 +1911.3 2010.6 1912.5 2012.6 1915.7 2013.4 C
  7.3368 +1915.8 2013.7 1915.5 2014.4 1916 2014.4 C
  7.3369 +1916.3 2015 1915.4 2016 1915.2 2016 C
  7.3370 +1916.1 2015.5 1916.5 2014.5 1916 2013.6 C
  7.3371 +1913.4 2013.3 1913.1 2010.5 1910.9 2009.8 C
  7.3372 +1910.7 2008.8 1910.4 2007.9 1910.2 2006.9 C
  7.3373 +1911.1 1998.8 1909.4 1990.7 1910.7 1982.4 C
  7.3374 +1910 1982.1 1908.9 1982.1 1908.3 1982.4 C
  7.3375 +1901.9 1986.1 1895 1988.7 1888.8 1993 C
  7.3376 +1888 1993.4 1888.4 1994.3 1887.6 1994.7 C
  7.3377 +1888.1 2001.3 1887.8 2008.6 1887.9 2015.1 C
  7.3378 +1887.3 2017.5 1887.9 2015.4 1888.4 2014.4 C
  7.3379 +1887.8 2008 1888.4 2001.3 1887.9 1994.9 C
  7.3380 +[0.07 0.06 0 0.58]  vc
  7.3381 +f 
  7.3382 +S 
  7.3383 +n
  7.3384 +vmrs
  7.3385 +1887.9 2018.4 m
  7.3386 +1887.5 2016.9 1888.5 2016 1888.8 2014.8 C
  7.3387 +1890.1 2014.8 1891.1 2016.6 1892.4 2015.3 C
  7.3388 +1892.4 2014.4 1893.8 2012.9 1894.4 2012.4 C
  7.3389 +1895.9 2012.4 1896.6 2013.9 1897.7 2012.7 C
  7.3390 +1898.4 2011.7 1898.6 2010.4 1899.6 2009.8 C
  7.3391 +1901.7 2009.9 1902.9 2010.4 1904 2009.1 C
  7.3392 +1904.3 2007.4 1904 2007.6 1904.9 2007.2 C
  7.3393 +1906.2 2007 1907.6 2006.5 1908.8 2006.7 C
  7.3394 +1910.6 2008.2 1909.8 2011.5 1912.6 2012 C
  7.3395 +1912.4 2013 1913.8 2012.7 1914 2013.2 C
  7.3396 +1911.5 2011.1 1909.1 2007.9 1909.2 2004.3 C
  7.3397 +1909.5 2003.5 1909.9 2004.9 1909.7 2004.3 C
  7.3398 +1909.9 1996.2 1909.3 1990.5 1910.2 1982.7 C
  7.3399 +1909.5 1982.6 1909.5 1982.6 1908.8 1982.7 C
  7.3400 +1903.1 1985.7 1897 1987.9 1891.7 1992 C
  7.3401 +1890.5 1993 1888.2 1992.9 1888.1 1994.9 C
  7.3402 +1888.7 2001.4 1888.1 2008.4 1888.6 2014.8 C
  7.3403 +1888.3 2016 1887.2 2016.9 1887.6 2018.4 C
  7.3404 +1892.3 2023.9 1897.6 2027.9 1902.3 2033.3 C
  7.3405 +1898 2028.2 1892.1 2023.8 1887.9 2018.4 C
  7.3406 +[0.4 0.4 0 0]  vc
  7.3407 +f 
  7.3408 +0.4 w
  7.3409 +2 J
  7.3410 +2 M
  7.3411 +S 
  7.3412 +n
  7.3413 +1910.9 1995.2 m
  7.3414 +1910.4 1999.8 1911 2003.3 1910.9 2008.1 C
  7.3415 +1910.9 2003.8 1910.9 1999.2 1910.9 1995.2 C
  7.3416 +[0.18 0.18 0 0.78]  vc
  7.3417 +f 
  7.3418 +S 
  7.3419 +n
  7.3420 +1911.2 2004.3 m
  7.3421 +1911.2 2001.9 1911.2 1999.7 1911.2 1997.3 C
  7.3422 +1911.2 1999.7 1911.2 2001.9 1911.2 2004.3 C
  7.3423 +[0 0.87 0.91 0.83]  vc
  7.3424 +f 
  7.3425 +S 
  7.3426 +n
  7.3427 +1958.7 1995.2 m
  7.3428 +1959 1995.6 1956.2 1995 1956.5 1996.8 C
  7.3429 +1955.8 1997.6 1954.2 1998.5 1953.6 1997.3 C
  7.3430 +1953.6 1990.8 1954.9 1989.6 1953.4 1983.9 C
  7.3431 +1953.4 1983.3 1953.3 1982.1 1954.4 1982 C
  7.3432 +1955.5 1982.6 1956.5 1981.3 1957.5 1981 C
  7.3433 +1956.3 1981.8 1954.7 1982.6 1953.9 1981.5 C
  7.3434 +1951.4 1983 1954.7 1988.8 1952.9 1990.6 C
  7.3435 +1953.8 1990.6 1953.2 1992.7 1953.4 1993.7 C
  7.3436 +1953.8 1994.5 1952.3 1996.1 1953.2 1997.8 C
  7.3437 +1956.3 1999.4 1957.5 1994 1959.9 1995.6 C
  7.3438 +1962 1994.4 1963.7 1997.7 1965.2 1998.8 C
  7.3439 +1963.5 1996.7 1961.2 1994.1 1958.7 1995.2 C
  7.3440 +f 
  7.3441 +S 
  7.3442 +n
  7.3443 +1945 2000.7 m
  7.3444 +1945.4 1998.7 1945.4 1997.9 1945 1995.9 C
  7.3445 +1944.5 1995.3 1944.2 1992.6 1945.7 1993.2 C
  7.3446 +1946 1992.2 1948.7 1992.5 1948.4 1990.6 C
  7.3447 +1947.5 1990.3 1948.1 1988.7 1947.9 1988.2 C
  7.3448 +1948.9 1987.8 1950.5 1986.8 1950.5 1984.6 C
  7.3449 +1951.5 1980.9 1946.7 1983 1947.2 1979.8 C
  7.3450 +1944.5 1979.9 1945.2 1976.6 1943.1 1976.7 C
  7.3451 +1941.8 1975.7 1942.1 1972.7 1939.2 1973.8 C
  7.3452 +1938.2 1974.6 1939.3 1971.6 1938.3 1970.9 C
  7.3453 +1938.8 1969.2 1933.4 1970.3 1937.3 1970 C
  7.3454 +1939.4 1971.2 1937.2 1973 1937.6 1974.3 C
  7.3455 +1937.2 1976.3 1937.1 1981.2 1937.8 1984.1 C
  7.3456 +1938.8 1982.3 1937.9 1976.6 1938.5 1973.1 C
  7.3457 +1938.9 1975 1938.5 1976.4 1939.7 1977.2 C
  7.3458 +1939.5 1983.5 1938.9 1991.3 1940.2 1997.3 C
  7.3459 +1939.4 1999.1 1938.6 1997.1 1937.8 1997.1 C
  7.3460 +1937.4 1996.7 1937.6 1996.1 1937.6 1995.6 C
  7.3461 +1936.5 1998.5 1940.1 1998.4 1940.9 2000.7 C
  7.3462 +1942.1 2000.4 1943.2 2001.3 1943.1 2002.4 C
  7.3463 +1943.6 2003.1 1941.1 2004.6 1942.8 2003.8 C
  7.3464 +1943.9 2002.5 1942.6 2000.6 1945 2000.7 C
  7.3465 +[0.65 0.65 0 0.42]  vc
  7.3466 +f 
  7.3467 +S 
  7.3468 +n
  7.3469 +1914.5 2006.4 m
  7.3470 +1914.1 2004.9 1915.2 2004 1915.5 2002.8 C
  7.3471 +1916.7 2002.8 1917.8 2004.6 1919.1 2003.3 C
  7.3472 +1919 2002.4 1920.4 2000.9 1921 2000.4 C
  7.3473 +1922.5 2000.4 1923.2 2001.9 1924.4 2000.7 C
  7.3474 +1925 1999.7 1925.3 1998.4 1926.3 1997.8 C
  7.3475 +1928.4 1997.9 1929.5 1998.4 1930.6 1997.1 C
  7.3476 +1930.9 1995.4 1930.7 1995.6 1931.6 1995.2 C
  7.3477 +1932.8 1995 1934.3 1994.5 1935.4 1994.7 C
  7.3478 +1936.1 1995.8 1936.9 1996.2 1936.6 1997.8 C
  7.3479 +1938.9 1999.4 1939.7 2001.3 1942.4 2002.4 C
  7.3480 +1942.4 2002.5 1942.2 2003 1942.6 2002.8 C
  7.3481 +1942.9 2000.4 1939.2 2001.8 1939.2 1999.7 C
  7.3482 +1936.2 1998.6 1937 1995.3 1935.9 1993.5 C
  7.3483 +1937.1 1986.5 1935.2 1977.9 1937.6 1971.2 C
  7.3484 +1937.6 1970.3 1936.6 1971 1936.4 1970.4 C
  7.3485 +1930.2 1973.4 1924 1976 1918.4 1980 C
  7.3486 +1917.2 1981 1914.9 1980.9 1914.8 1982.9 C
  7.3487 +1915.3 1989.4 1914.7 1996.4 1915.2 2002.8 C
  7.3488 +1914.9 2004 1913.9 2004.9 1914.3 2006.4 C
  7.3489 +1919 2011.9 1924.2 2015.9 1928.9 2021.3 C
  7.3490 +1924.6 2016.2 1918.7 2011.8 1914.5 2006.4 C
  7.3491 +[0.4 0.4 0 0]  vc
  7.3492 +f 
  7.3493 +S 
  7.3494 +n
  7.3495 +1914.5 1982.9 m
  7.3496 +1915.1 1980.3 1918 1980.2 1919.8 1978.8 C
  7.3497 +1925 1975.5 1930.6 1972.8 1936.1 1970.2 C
  7.3498 +1939.4 1970.6 1936.1 1974.2 1936.6 1976.4 C
  7.3499 +1936.5 1981.9 1936.8 1987.5 1936.4 1992.8 C
  7.3500 +1935.9 1992.8 1936.2 1993.5 1936.1 1994 C
  7.3501 +1937.1 1993.6 1936.2 1995.9 1936.8 1995.9 C
  7.3502 +1937 1998 1939.5 1999.7 1940.4 2000.7 C
  7.3503 +1940.1 1998.6 1935 1997.2 1937.6 1993.7 C
  7.3504 +1938.3 1985.7 1935.9 1976.8 1937.8 1970.7 C
  7.3505 +1936.9 1969.8 1935.4 1970.3 1934.4 1970.7 C
  7.3506 +1928.3 1974.4 1921.4 1976.7 1915.5 1981 C
  7.3507 +1914.6 1981.4 1915.1 1982.3 1914.3 1982.7 C
  7.3508 +1914.7 1989.3 1914.5 1996.6 1914.5 2003.1 C
  7.3509 +1913.9 2005.5 1914.5 2003.4 1915 2002.4 C
  7.3510 +1914.5 1996 1915.1 1989.3 1914.5 1982.9 C
  7.3511 +[0.07 0.06 0 0.58]  vc
  7.3512 +f 
  7.3513 +S 
  7.3514 +n
  7.3515 +1939.2 1994.9 m
  7.3516 +1939.3 1995 1939.4 1995.1 1939.5 1995.2 C
  7.3517 +1939.1 1989 1939.3 1981.6 1939 1976.7 C
  7.3518 +1938.6 1976.3 1938.6 1974.6 1938.5 1973.3 C
  7.3519 +1938.7 1976.1 1938.1 1979.4 1939 1981.7 C
  7.3520 +1937.3 1986 1937.7 1991.6 1938 1996.4 C
  7.3521 +1937.3 1994.3 1939.6 1996.2 1939.2 1994.9 C
  7.3522 +[0.18 0.18 0 0.78]  vc
  7.3523 +f 
  7.3524 +S 
  7.3525 +n
  7.3526 +1938.3 1988.4 m
  7.3527 +1938.5 1990.5 1937.9 1994.1 1938.8 1994.7 C
  7.3528 +1937.9 1992.6 1939 1990.6 1938.3 1988.4 C
  7.3529 +[0 0.87 0.91 0.83]  vc
  7.3530 +f 
  7.3531 +S 
  7.3532 +n
  7.3533 +1938.8 1985.8 m
  7.3534 +1938.5 1985.9 1938.4 1985.7 1938.3 1985.6 C
  7.3535 +1938.4 1986.2 1938 1989.5 1938.8 1987.2 C
  7.3536 +1938.8 1986.8 1938.8 1986.3 1938.8 1985.8 C
  7.3537 +f 
  7.3538 +S 
  7.3539 +n
  7.3540 +vmrs
  7.3541 +1972.8 2062.1 m
  7.3542 +1971.9 2061 1972.5 2059.4 1972.4 2058 C
  7.3543 +1972.2 2063.8 1971.9 2073.7 1972.4 2081.3 C
  7.3544 +1972.5 2074.9 1971.9 2067.9 1972.8 2062.1 C
  7.3545 +[0 1 1 0.36]  vc
  7.3546 +f 
  7.3547 +0.4 w
  7.3548 +2 J
  7.3549 +2 M
  7.3550 +S 
  7.3551 +n
  7.3552 +1940.2 2071.7 m
  7.3553 +1941.3 2072 1943.1 2072.3 1944 2071.5 C
  7.3554 +1943.6 2069.9 1945.2 2069.1 1946 2068.8 C
  7.3555 +1950 2071.1 1948.7 2065.9 1951.7 2066.2 C
  7.3556 +1953.5 2063.9 1956.9 2069.4 1955.6 2063.8 C
  7.3557 +1955.5 2064.2 1955.7 2064.8 1955.3 2065 C
  7.3558 +1954.3 2063.7 1956.2 2063.6 1955.6 2062.1 C
  7.3559 +1954.5 2060 1958.3 2050.3 1952.2 2055.6 C
  7.3560 +1949.1 2053.8 1946 2051 1943.8 2048 C
  7.3561 +1940.3 2048 1937.5 2051.3 1934.2 2052.5 C
  7.3562 +1933.1 2054.6 1934.4 2057.3 1934 2060 C
  7.3563 +1934 2065.1 1934 2069.7 1934 2074.6 C
  7.3564 +1934.4 2069 1934.1 2061.5 1934.2 2054.9 C
  7.3565 +1934.6 2054.5 1935.3 2054.7 1935.9 2054.7 C
  7.3566 +1937 2055.3 1935.9 2056.1 1935.9 2056.8 C
  7.3567 +1936.5 2063 1935.6 2070.5 1935.9 2074.6 C
  7.3568 +1936.7 2074.4 1937.3 2075.2 1938 2074.6 C
  7.3569 +1937.9 2073.6 1939.1 2072.1 1940.2 2071.7 C
  7.3570 +[0 0.2 1 0]  vc
  7.3571 +f 
  7.3572 +S 
  7.3573 +n
  7.3574 +1933.2 2074.1 m
  7.3575 +1933.2 2071.5 1933.2 2069 1933.2 2066.4 C
  7.3576 +1933.2 2069 1933.2 2071.5 1933.2 2074.1 C
  7.3577 +[0 1 1 0.36]  vc
  7.3578 +f 
  7.3579 +S 
  7.3580 +n
  7.3581 +2007.4 2048.9 m
  7.3582 +2006.5 2047.8 2007.1 2046.2 2006.9 2044.8 C
  7.3583 +2006.7 2050.6 2006.5 2060.5 2006.9 2068.1 C
  7.3584 +2007.1 2061.7 2006.5 2054.7 2007.4 2048.9 C
  7.3585 +f 
  7.3586 +S 
  7.3587 +n
  7.3588 +1927.2 2062.4 m
  7.3589 +1925.8 2060.1 1928.1 2058.2 1927 2056.4 C
  7.3590 +1927.3 2055.5 1926.5 2053.5 1926.8 2051.8 C
  7.3591 +1926.8 2052.8 1926 2052.5 1925.3 2052.5 C
  7.3592 +1924.1 2052.8 1925 2050.5 1924.4 2050.1 C
  7.3593 +1925.3 2050.2 1925.4 2048.8 1926.3 2049.4 C
  7.3594 +1926.5 2052.3 1928.4 2047.2 1928.4 2051.1 C
  7.3595 +1928.9 2050.5 1929 2051.4 1928.9 2051.8 C
  7.3596 +1928.9 2052 1928.9 2052.3 1928.9 2052.5 C
  7.3597 +1929.4 2051.4 1928.9 2049 1930.1 2048.2 C
  7.3598 +1928.9 2047.1 1930.5 2047.1 1930.4 2046.5 C
  7.3599 +1931.9 2046.2 1933.1 2046.1 1934.7 2046.5 C
  7.3600 +1934.6 2046.9 1935.2 2047.9 1934.4 2048.4 C
  7.3601 +1936.9 2048.1 1933.6 2043.8 1935.9 2043.9 C
  7.3602 +1935.7 2043.9 1934.8 2041.3 1933.2 2041.7 C
  7.3603 +1932.5 2041.6 1932.4 2039.6 1932.3 2041 C
  7.3604 +1930.8 2042.6 1929 2040.6 1927.7 2042 C
  7.3605 +1927.5 2041.4 1927.1 2040.9 1927.2 2040.3 C
  7.3606 +1927.8 2040.6 1927.4 2039.1 1928.2 2038.6 C
  7.3607 +1929.4 2038 1930.5 2038.8 1931.3 2037.9 C
  7.3608 +1931.7 2039 1932.5 2038.6 1931.8 2037.6 C
  7.3609 +1930.9 2037 1928.7 2037.8 1928.2 2037.9 C
  7.3610 +1926.7 2037.8 1928 2039 1927 2038.8 C
  7.3611 +1927.4 2040.4 1925.6 2040.8 1925.1 2041 C
  7.3612 +1924.3 2040.4 1923.2 2040.5 1922.2 2040.5 C
  7.3613 +1921.4 2041.7 1921 2043.9 1919.3 2043.9 C
  7.3614 +1918.8 2043.4 1917.2 2043.3 1916.4 2043.4 C
  7.3615 +1915.9 2044.4 1915.7 2046 1914.3 2046.5 C
  7.3616 +1913.1 2046.6 1912 2044.5 1911.4 2046.3 C
  7.3617 +1912.8 2046.5 1913.8 2047.4 1915.7 2047 C
  7.3618 +1916.9 2047.7 1915.6 2048.8 1916 2049.4 C
  7.3619 +1915.4 2049.3 1913.9 2050.3 1913.3 2051.1 C
  7.3620 +1913.9 2054.1 1916 2050.2 1916.7 2053 C
  7.3621 +1916.9 2053.8 1915.5 2054.1 1916.7 2054.4 C
  7.3622 +1917 2054.7 1920.2 2054.3 1919.3 2056.6 C
  7.3623 +1918.8 2056.1 1920.2 2058.6 1920.3 2057.6 C
  7.3624 +1921.2 2057.9 1922.1 2057.5 1922.4 2059 C
  7.3625 +1922.3 2059.1 1922.2 2059.3 1922 2059.2 C
  7.3626 +1922.1 2059.7 1922.4 2060.3 1922.9 2060.7 C
  7.3627 +1923.2 2060.1 1923.8 2060.4 1924.6 2060.7 C
  7.3628 +1925.9 2062.6 1923.2 2062 1925.6 2063.6 C
  7.3629 +1926.1 2063.1 1927.3 2062.5 1927.2 2062.4 C
  7.3630 +[0.21 0.21 0 0]  vc
  7.3631 +f 
  7.3632 +S 
  7.3633 +n
  7.3634 +1933.2 2063.3 m
  7.3635 +1933.2 2060.7 1933.2 2058.2 1933.2 2055.6 C
  7.3636 +1933.2 2058.2 1933.2 2060.7 1933.2 2063.3 C
  7.3637 +[0 1 1 0.36]  vc
  7.3638 +f 
  7.3639 +S 
  7.3640 +n
  7.3641 +1965.2 2049.2 m
  7.3642 +1967.1 2050.1 1969.9 2053.7 1972.1 2056.4 C
  7.3643 +1970.5 2054 1967.6 2051.3 1965.2 2049.2 C
  7.3644 +f 
  7.3645 +S 
  7.3646 +n
  7.3647 +1991.8 2034.8 m
  7.3648 +1991.7 2041.5 1992 2048.5 1991.6 2055.2 C
  7.3649 +1990.5 2056.4 1991.9 2054.9 1991.8 2054.4 C
  7.3650 +1991.8 2047.9 1991.8 2041.3 1991.8 2034.8 C
  7.3651 +f 
  7.3652 +S 
  7.3653 +n
  7.3654 +1988.9 2053.2 m
  7.3655 +1988.9 2044.3 1988.9 2036.6 1988.9 2028.3 C
  7.3656 +1985.7 2028.2 1987.2 2023.5 1983.9 2024.2 C
  7.3657 +1983.9 2022.4 1982 2021.6 1981 2021.3 C
  7.3658 +1980.6 2021.1 1980.6 2021.7 1980.3 2021.6 C
  7.3659 +1980.3 2027 1980.3 2034.8 1980.3 2041.5 C
  7.3660 +1979.3 2043.2 1977.6 2043 1976.2 2043.6 C
  7.3661 +1977.1 2043.8 1978.5 2043.2 1978.8 2044.1 C
  7.3662 +1978.5 2045.3 1979.9 2045.3 1980.3 2045.8 C
  7.3663 +1980.5 2046.8 1980.7 2046.2 1981.5 2046.5 C
  7.3664 +1982.4 2047.1 1982 2048.6 1982.7 2049.4 C
  7.3665 +1984.2 2049.6 1984.6 2052.2 1986.8 2051.6 C
  7.3666 +1987.1 2048.6 1985.1 2042.7 1986.5 2040.5 C
  7.3667 +1986.3 2036.7 1986.9 2031.7 1986 2029.2 C
  7.3668 +1986.3 2027.1 1986.9 2028.6 1987.7 2027.6 C
  7.3669 +1987.7 2028.3 1988.7 2028 1988.7 2028.8 C
  7.3670 +1988.1 2033 1988.7 2037.5 1988.2 2041.7 C
  7.3671 +1987.8 2041.4 1988 2040.8 1988 2040.3 C
  7.3672 +1988 2041 1988 2042.4 1988 2042.4 C
  7.3673 +1988 2042.4 1988.1 2042.3 1988.2 2042.2 C
  7.3674 +1989.3 2046 1988 2050.2 1988.4 2054 C
  7.3675 +1987.8 2054.4 1987.1 2054.7 1986.5 2055.4 C
  7.3676 +1987.4 2054.4 1988.4 2054.6 1988.9 2053.2 C
  7.3677 +[0 1 1 0.23]  vc
  7.3678 +f 
  7.3679 +S 
  7.3680 +n
  7.3681 +1950.8 2054.4 m
  7.3682 +1949.7 2053.4 1948.7 2052.3 1947.6 2051.3 C
  7.3683 +1948.7 2052.3 1949.7 2053.4 1950.8 2054.4 C
  7.3684 +[0 1 1 0.36]  vc
  7.3685 +f 
  7.3686 +S 
  7.3687 +n
  7.3688 +vmrs
  7.3689 +2006.7 2043.2 m
  7.3690 +2004.5 2040.8 2002.4 2038.4 2000.2 2036 C
  7.3691 +2002.4 2038.4 2004.5 2040.8 2006.7 2043.2 C
  7.3692 +[0 1 1 0.36]  vc
  7.3693 +f 
  7.3694 +0.4 w
  7.3695 +2 J
  7.3696 +2 M
  7.3697 +S 
  7.3698 +n
  7.3699 +1976.7 2019.6 m
  7.3700 +1975.8 2018.6 1976.4 2016.9 1976.2 2015.6 C
  7.3701 +1976 2021.3 1975.8 2031.2 1976.2 2038.8 C
  7.3702 +1976.4 2032.4 1975.8 2025.5 1976.7 2019.6 C
  7.3703 +f 
  7.3704 +S 
  7.3705 +n
  7.3706 +1988.4 2053.5 m
  7.3707 +1988.6 2049.2 1988.1 2042.8 1988 2040 C
  7.3708 +1988.4 2040.4 1988.1 2041 1988.2 2041.5 C
  7.3709 +1988.3 2037.2 1988 2032.7 1988.4 2028.5 C
  7.3710 +1987.6 2027.1 1987.2 2028.6 1986.8 2028 C
  7.3711 +1985.9 2028.5 1986.5 2029.7 1986.3 2030.4 C
  7.3712 +1986.9 2029.8 1986.6 2031 1987 2031.2 C
  7.3713 +1987.4 2039.6 1985 2043 1987.2 2050.4 C
  7.3714 +1987.2 2051.6 1985.9 2052.3 1984.6 2051.3 C
  7.3715 +1981.9 2049.7 1982.9 2047 1980.3 2046.5 C
  7.3716 +1980.3 2045.2 1978.1 2046.2 1978.6 2043.9 C
  7.3717 +1975.6 2043.3 1979.3 2045.6 1979.6 2046.5 C
  7.3718 +1980.8 2046.6 1981.5 2048.5 1982.2 2049.9 C
  7.3719 +1983.7 2050.8 1984.8 2052.8 1986.5 2053 C
  7.3720 +1986.7 2053.5 1987.5 2054.1 1987 2054.7 C
  7.3721 +1987.4 2053.9 1988.3 2054.3 1988.4 2053.5 C
  7.3722 +[0 1 1 0.23]  vc
  7.3723 +f 
  7.3724 +S 
  7.3725 +n
  7.3726 +1988 2038.1 m
  7.3727 +1988 2036.7 1988 2035.4 1988 2034 C
  7.3728 +1988 2035.4 1988 2036.7 1988 2038.1 C
  7.3729 +[0 1 1 0.36]  vc
  7.3730 +f 
  7.3731 +S 
  7.3732 +n
  7.3733 +1999.7 2035.7 m
  7.3734 +1997.6 2033.5 1995.4 2031.2 1993.2 2029 C
  7.3735 +1995.4 2031.2 1997.6 2033.5 1999.7 2035.7 C
  7.3736 +f 
  7.3737 +S 
  7.3738 +n
  7.3739 +1944 2029.2 m
  7.3740 +1945.2 2029.5 1946.9 2029.8 1947.9 2029 C
  7.3741 +1947.4 2027.4 1949 2026.7 1949.8 2026.4 C
  7.3742 +1953.9 2028.6 1952.6 2023.4 1955.6 2023.7 C
  7.3743 +1957.4 2021.4 1960.7 2027 1959.4 2021.3 C
  7.3744 +1959.3 2021.7 1959.6 2022.3 1959.2 2022.5 C
  7.3745 +1958.1 2021.2 1960.1 2021.1 1959.4 2019.6 C
  7.3746 +1959.1 2012.7 1959.9 2005.1 1959.6 1999.2 C
  7.3747 +1955.3 2000.1 1951.3 2003.1 1947.2 2005 C
  7.3748 +1943.9 2006 1941.2 2008.7 1938 2010 C
  7.3749 +1936.9 2012.1 1938.2 2014.8 1937.8 2017.5 C
  7.3750 +1937.8 2022.6 1937.8 2027.3 1937.8 2032.1 C
  7.3751 +1938.2 2026.5 1938 2019 1938 2012.4 C
  7.3752 +1938.5 2012 1939.2 2012.3 1939.7 2012.2 C
  7.3753 +1940.8 2012.8 1939.7 2013.6 1939.7 2014.4 C
  7.3754 +1940.4 2020.5 1939.4 2028 1939.7 2032.1 C
  7.3755 +1940.6 2031.9 1941.2 2032.7 1941.9 2032.1 C
  7.3756 +1941.7 2031.2 1943 2029.7 1944 2029.2 C
  7.3757 +[0 0.2 1 0]  vc
  7.3758 +f 
  7.3759 +S 
  7.3760 +n
  7.3761 +1937.1 2031.6 m
  7.3762 +1937.1 2029.1 1937.1 2026.5 1937.1 2024 C
  7.3763 +1937.1 2026.5 1937.1 2029.1 1937.1 2031.6 C
  7.3764 +[0 1 1 0.36]  vc
  7.3765 +f 
  7.3766 +S 
  7.3767 +n
  7.3768 +1991.8 2028 m
  7.3769 +1992.5 2027.8 1993.2 2029.9 1994 2030.2 C
  7.3770 +1992.9 2029.6 1993.1 2028.1 1991.8 2028 C
  7.3771 +[0 1 1 0.23]  vc
  7.3772 +f 
  7.3773 +S 
  7.3774 +n
  7.3775 +1991.8 2027.8 m
  7.3776 +1992.4 2027.6 1992.6 2028.3 1993 2028.5 C
  7.3777 +1992.6 2028.2 1992.2 2027.6 1991.6 2027.8 C
  7.3778 +1991.6 2028.5 1991.6 2029.1 1991.6 2029.7 C
  7.3779 +1991.6 2029.1 1991.4 2028.3 1991.8 2027.8 C
  7.3780 +[0 1 1 0.36]  vc
  7.3781 +f 
  7.3782 +S 
  7.3783 +n
  7.3784 +1985.8 2025.4 m
  7.3785 +1985.3 2025.2 1984.8 2024.7 1984.1 2024.9 C
  7.3786 +1983.3 2025.3 1983.6 2027.3 1983.9 2027.6 C
  7.3787 +1985 2028 1986.9 2026.9 1985.8 2025.4 C
  7.3788 +[0 1 1 0.23]  vc
  7.3789 +f 
  7.3790 +S 
  7.3791 +n
  7.3792 +vmrs
  7.3793 +1993.5 2024.4 m
  7.3794 +1992.4 2023.7 1991.3 2022.9 1990.1 2023.2 C
  7.3795 +1990.7 2023.7 1989.8 2023.8 1989.4 2023.7 C
  7.3796 +1989.1 2023.7 1988.6 2023.9 1988.4 2023.5 C
  7.3797 +1988.5 2023.2 1988.3 2022.7 1988.7 2022.5 C
  7.3798 +1989 2022.6 1988.9 2023 1988.9 2023.2 C
  7.3799 +1989.1 2022.8 1990.4 2022.3 1990.6 2021.3 C
  7.3800 +1990.4 2021.8 1990 2021.3 1990.1 2021.1 C
  7.3801 +1990.1 2020.9 1990.1 2020.1 1990.1 2020.6 C
  7.3802 +1989.9 2021.1 1989.5 2020.6 1989.6 2020.4 C
  7.3803 +1989.6 2019.8 1988.7 2019.6 1988.2 2019.2 C
  7.3804 +1987.5 2018.7 1987.7 2020.2 1987 2019.4 C
  7.3805 +1987.5 2020.4 1986 2021.1 1987.5 2021.8 C
  7.3806 +1986.8 2023.1 1986.6 2021.1 1986 2021.1 C
  7.3807 +1986.1 2020.1 1985.9 2019 1986.3 2018.2 C
  7.3808 +1986.7 2018.4 1986.5 2019 1986.5 2019.4 C
  7.3809 +1986.5 2018.7 1986.4 2017.8 1987.2 2017.7 C
  7.3810 +1986.5 2017.2 1985.5 2019.3 1985.3 2020.4 C
  7.3811 +1986.2 2022 1987.3 2023.5 1989.2 2024.2 C
  7.3812 +1990.8 2024.3 1991.6 2022.9 1993.2 2024.4 C
  7.3813 +1993.8 2025.4 1995 2026.6 1995.9 2027.1 C
  7.3814 +1995 2026.5 1994.1 2025.5 1993.5 2024.4 C
  7.3815 +[0 1 1 0.36]  vc
  7.3816 +f 
  7.3817 +0.4 w
  7.3818 +2 J
  7.3819 +2 M
  7.3820 +[0 0.5 0.5 0.2]  vc
  7.3821 +S 
  7.3822 +n
  7.3823 +2023 2040.3 m
  7.3824 +2023.2 2036 2022.7 2029.6 2022.5 2026.8 C
  7.3825 +2022.9 2027.2 2022.7 2027.8 2022.8 2028.3 C
  7.3826 +2022.8 2024 2022.6 2019.5 2023 2015.3 C
  7.3827 +2022.2 2013.9 2021.7 2015.4 2021.3 2014.8 C
  7.3828 +2020.4 2015.3 2021 2016.5 2020.8 2017.2 C
  7.3829 +2021.4 2016.6 2021.1 2017.8 2021.6 2018 C
  7.3830 +2022 2026.4 2019.6 2029.8 2021.8 2037.2 C
  7.3831 +2021.7 2038.4 2020.5 2039.1 2019.2 2038.1 C
  7.3832 +2016.5 2036.5 2017.5 2033.8 2014.8 2033.3 C
  7.3833 +2014.9 2032 2012.6 2033 2013.2 2030.7 C
  7.3834 +2011.9 2030.8 2011.2 2030.1 2010.8 2029.2 C
  7.3835 +2010.8 2029.1 2010.8 2028.2 2010.8 2028.8 C
  7.3836 +2010 2028.8 2010.4 2026.5 2008.6 2027.3 C
  7.3837 +2007.9 2026.6 2007.3 2025.9 2007.9 2027.1 C
  7.3838 +2009.7 2028 2010 2030.1 2012.2 2030.9 C
  7.3839 +2012.9 2032.1 2013.7 2033.6 2015.1 2033.6 C
  7.3840 +2015.7 2035.1 2016.9 2036.7 2018.4 2038.4 C
  7.3841 +2019.8 2039.3 2022 2039.4 2021.6 2041.5 C
  7.3842 +2021.9 2040.7 2022.9 2041.1 2023 2040.3 C
  7.3843 +[0 1 1 0.23]  vc
  7.3844 +f 
  7.3845 +S 
  7.3846 +n
  7.3847 +2022.5 2024.9 m
  7.3848 +2022.5 2023.5 2022.5 2022.2 2022.5 2020.8 C
  7.3849 +2022.5 2022.2 2022.5 2023.5 2022.5 2024.9 C
  7.3850 +[0 1 1 0.36]  vc
  7.3851 +f 
  7.3852 +S 
  7.3853 +n
  7.3854 +1983.2 2022.8 m
  7.3855 +1982.4 2022.5 1982.1 2021.6 1981.2 2022.3 C
  7.3856 +1981.1 2022.9 1980.5 2024 1981 2024.2 C
  7.3857 +1981.8 2024.6 1982.9 2024.4 1983.2 2022.8 C
  7.3858 +[0 1 1 0.23]  vc
  7.3859 +f 
  7.3860 +S 
  7.3861 +n
  7.3862 +1931.1 2019.9 m
  7.3863 +1929.6 2017.7 1932 2015.7 1930.8 2013.9 C
  7.3864 +1931.1 2013 1930.3 2011 1930.6 2009.3 C
  7.3865 +1930.6 2010.3 1929.8 2010 1929.2 2010 C
  7.3866 +1928 2010.3 1928.8 2008.1 1928.2 2007.6 C
  7.3867 +1929.1 2007.8 1929.3 2006.3 1930.1 2006.9 C
  7.3868 +1930.3 2009.8 1932.2 2004.8 1932.3 2008.6 C
  7.3869 +1932.7 2008 1932.8 2009 1932.8 2009.3 C
  7.3870 +1932.8 2009.6 1932.8 2009.8 1932.8 2010 C
  7.3871 +1933.2 2009 1932.7 2006.6 1934 2005.7 C
  7.3872 +1932.7 2004.6 1934.3 2004.6 1934.2 2004 C
  7.3873 +1935.8 2003.7 1937 2003.6 1938.5 2004 C
  7.3874 +1938.5 2004.5 1939.1 2005.4 1938.3 2006 C
  7.3875 +1940.7 2005.7 1937.4 2001.3 1939.7 2001.4 C
  7.3876 +1939.5 2001.4 1938.6 1998.8 1937.1 1999.2 C
  7.3877 +1936.3 1999.1 1936.2 1997.1 1936.1 1998.5 C
  7.3878 +1934.7 2000.1 1932.9 1998.2 1931.6 1999.5 C
  7.3879 +1931.3 1998.9 1930.9 1998.5 1931.1 1997.8 C
  7.3880 +1931.6 1998.2 1931.3 1996.6 1932 1996.1 C
  7.3881 +1933.2 1995.5 1934.3 1996.4 1935.2 1995.4 C
  7.3882 +1935.5 1996.5 1936.3 1996.1 1935.6 1995.2 C
  7.3883 +1934.7 1994.5 1932.5 1995.3 1932 1995.4 C
  7.3884 +1930.5 1995.3 1931.9 1996.5 1930.8 1996.4 C
  7.3885 +1931.2 1997.9 1929.5 1998.3 1928.9 1998.5 C
  7.3886 +1928.1 1997.9 1927.1 1998 1926 1998 C
  7.3887 +1925.3 1999.2 1924.8 2001.4 1923.2 2001.4 C
  7.3888 +1922.6 2000.9 1921 2000.9 1920.3 2000.9 C
  7.3889 +1919.7 2001.9 1919.6 2003.5 1918.1 2004 C
  7.3890 +1916.9 2004.1 1915.8 2002 1915.2 2003.8 C
  7.3891 +1916.7 2004 1917.6 2004.9 1919.6 2004.5 C
  7.3892 +1920.7 2005.2 1919.4 2006.3 1919.8 2006.9 C
  7.3893 +1919.2 2006.9 1917.7 2007.8 1917.2 2008.6 C
  7.3894 +1917.8 2011.6 1919.8 2007.8 1920.5 2010.5 C
  7.3895 +1920.8 2011.3 1919.3 2011.6 1920.5 2012 C
  7.3896 +1920.8 2012.3 1924 2011.8 1923.2 2014.1 C
  7.3897 +1922.6 2013.6 1924.1 2016.1 1924.1 2015.1 C
  7.3898 +1925.1 2015.4 1925.9 2015 1926.3 2016.5 C
  7.3899 +1926.2 2016.6 1926 2016.8 1925.8 2016.8 C
  7.3900 +1925.9 2017.2 1926.2 2017.8 1926.8 2018.2 C
  7.3901 +1927.1 2017.6 1927.7 2018 1928.4 2018.2 C
  7.3902 +1929.7 2020.1 1927.1 2019.5 1929.4 2021.1 C
  7.3903 +1929.9 2020.7 1931.1 2020 1931.1 2019.9 C
  7.3904 +[0.21 0.21 0 0]  vc
  7.3905 +f 
  7.3906 +S 
  7.3907 +n
  7.3908 +1937.1 2020.8 m
  7.3909 +1937.1 2018.3 1937.1 2015.7 1937.1 2013.2 C
  7.3910 +1937.1 2015.7 1937.1 2018.3 1937.1 2020.8 C
  7.3911 +[0 1 1 0.36]  vc
  7.3912 +f 
  7.3913 +S 
  7.3914 +n
  7.3915 +2020.4 2012.2 m
  7.3916 +2019.8 2012 2019.3 2011.5 2018.7 2011.7 C
  7.3917 +2017.9 2012.1 2018.1 2014.1 2018.4 2014.4 C
  7.3918 +2019.6 2014.8 2021.4 2013.7 2020.4 2012.2 C
  7.3919 +[0 1 1 0.23]  vc
  7.3920 +f 
  7.3921 +S 
  7.3922 +n
  7.3923 +1976 2013.9 m
  7.3924 +1973.8 2011.5 1971.6 2009.1 1969.5 2006.7 C
  7.3925 +1971.6 2009.1 1973.8 2011.5 1976 2013.9 C
  7.3926 +[0 1 1 0.36]  vc
  7.3927 +f 
  7.3928 +S 
  7.3929 +n
  7.3930 +1995.4 2012.7 m
  7.3931 +1996.1 2010.3 1993.8 2006.2 1997.3 2005.7 C
  7.3932 +1998.9 2005.4 2000 2003.7 2001.4 2003.1 C
  7.3933 +2003.9 2003.1 2005.3 2001.3 2006.9 1999.7 C
  7.3934 +2004.5 2003.5 2000 2002.2 1997.6 2005.7 C
  7.3935 +1996.5 2005.9 1994.8 2006.1 1995.2 2007.6 C
  7.3936 +1995.7 2009.4 1995.2 2011.6 1994.7 2012.9 C
  7.3937 +1992 2015.8 1987.8 2015.7 1985.3 2018.7 C
  7.3938 +1988.3 2016.3 1992.3 2015.3 1995.4 2012.7 C
  7.3939 +[0.18 0.18 0 0.78]  vc
  7.3940 +f 
  7.3941 +S 
  7.3942 +n
  7.3943 +1995.6 2012.4 m
  7.3944 +1995.6 2011.2 1995.6 2010 1995.6 2008.8 C
  7.3945 +1995.6 2010 1995.6 2011.2 1995.6 2012.4 C
  7.3946 +[0 1 1 0.36]  vc
  7.3947 +f 
  7.3948 +S 
  7.3949 +n
  7.3950 +vmrs
  7.3951 +2017.7 2009.6 m
  7.3952 +2016.9 2009.3 2016.7 2008.4 2015.8 2009.1 C
  7.3953 +2014.2 2010.6 2016 2010.6 2016.5 2011.5 C
  7.3954 +2017.2 2010.9 2018.1 2010.8 2017.7 2009.6 C
  7.3955 +[0 1 1 0.23]  vc
  7.3956 +f 
  7.3957 +0.4 w
  7.3958 +2 J
  7.3959 +2 M
  7.3960 +S 
  7.3961 +n
  7.3962 +2014.4 2006.4 m
  7.3963 +2013.5 2006.8 2012.1 2005.6 2012 2006.7 C
  7.3964 +2013 2007.3 2011.9 2009.2 2012.9 2008.4 C
  7.3965 +2014.2 2008.3 2014.6 2007.8 2014.4 2006.4 C
  7.3966 +f 
  7.3967 +S 
  7.3968 +n
  7.3969 +1969 2006.4 m
  7.3970 +1966.5 2003.8 1964 2001.2 1961.6 1998.5 C
  7.3971 +1964 2001.2 1966.5 2003.8 1969 2006.4 C
  7.3972 +[0 1 1 0.36]  vc
  7.3973 +f 
  7.3974 +S 
  7.3975 +n
  7.3976 +2012 2005.2 m
  7.3977 +2012.2 2004.2 2011.4 2003.3 2010.3 2003.3 C
  7.3978 +2009 2003.6 2010 2004.7 2009.6 2004.8 C
  7.3979 +2009.3 2005.7 2011.4 2006.7 2012 2005.2 C
  7.3980 +[0 1 1 0.23]  vc
  7.3981 +f 
  7.3982 +S 
  7.3983 +n
  7.3984 +1962.8 1995.2 m
  7.3985 +1961.7 1994.4 1960.6 1993.7 1959.4 1994 C
  7.3986 +1959.5 1994.9 1957.5 1994.1 1956.8 1994.7 C
  7.3987 +1955.9 1995.5 1956.7 1997 1955.1 1997.3 C
  7.3988 +1956.9 1996.7 1956.8 1994 1959.2 1994.7 C
  7.3989 +1961.1 1991 1968.9 2003.2 1962.8 1995.2 C
  7.3990 +[0 1 1 0.36]  vc
  7.3991 +f 
  7.3992 +S 
  7.3993 +n
  7.3994 +1954.6 1995.6 m
  7.3995 +1955.9 1994.7 1955.1 1989.8 1955.3 1988 C
  7.3996 +1954.5 1988.3 1954.9 1986.6 1954.4 1986 C
  7.3997 +1955.7 1989.2 1953.9 1991.1 1954.8 1994.2 C
  7.3998 +1954.5 1995.9 1953.5 1995.3 1953.9 1997.3 C
  7.3999 +1955.3 1998.3 1953.2 1995.5 1954.6 1995.6 C
  7.4000 +f 
  7.4001 +S 
  7.4002 +n
  7.4003 +1992.3 2011 m
  7.4004 +1992.5 2006.7 1992 2000.3 1991.8 1997.6 C
  7.4005 +1992.2 1997.9 1992 1998.5 1992 1999 C
  7.4006 +1992.1 1994.7 1991.9 1990.2 1992.3 1986 C
  7.4007 +1991.4 1984.6 1991 1986.1 1990.6 1985.6 C
  7.4008 +1989.7 1986 1990.3 1987.2 1990.1 1988 C
  7.4009 +1990.7 1987.4 1990.4 1988.5 1990.8 1988.7 C
  7.4010 +1991.3 1997.1 1988.9 2000.6 1991.1 2007.9 C
  7.4011 +1991 2009.1 1989.8 2009.9 1988.4 2008.8 C
  7.4012 +1985.7 2007.2 1986.8 2004.5 1984.1 2004 C
  7.4013 +1984.2 2002.7 1981.9 2003.7 1982.4 2001.4 C
  7.4014 +1981.2 2001.5 1980.5 2000.8 1980 2000 C
  7.4015 +1980 1999.8 1980 1998.9 1980 1999.5 C
  7.4016 +1979.3 1999.5 1979.7 1997.2 1977.9 1998 C
  7.4017 +1977.2 1997.3 1976.6 1996.7 1977.2 1997.8 C
  7.4018 +1979 1998.7 1979.3 2000.8 1981.5 2001.6 C
  7.4019 +1982.2 2002.8 1983 2004.3 1984.4 2004.3 C
  7.4020 +1985 2005.8 1986.2 2007.5 1987.7 2009.1 C
  7.4021 +1989 2010 1991.3 2010.2 1990.8 2012.2 C
  7.4022 +1991.2 2011.4 1992.2 2011.8 1992.3 2011 C
  7.4023 +[0 1 1 0.23]  vc
  7.4024 +f 
  7.4025 +S 
  7.4026 +n
  7.4027 +1991.8 1995.6 m
  7.4028 +1991.8 1994.3 1991.8 1992.9 1991.8 1991.6 C
  7.4029 +1991.8 1992.9 1991.8 1994.3 1991.8 1995.6 C
  7.4030 +[0 1 1 0.36]  vc
  7.4031 +f 
  7.4032 +S 
  7.4033 +n
  7.4034 +1959.2 1994.2 m
  7.4035 +1958.8 1993.3 1960.7 1993.9 1961.1 1993.7 C
  7.4036 +1961.5 1993.9 1961.2 1994.4 1961.8 1994.2 C
  7.4037 +1960.9 1994 1960.8 1992.9 1959.9 1992.5 C
  7.4038 +1959.6 1993.5 1958.3 1993.5 1958.2 1994.2 C
  7.4039 +1958.1 1994.1 1958 1994 1958 1994 C
  7.4040 +1957.2 1994.9 1958 1993.4 1956.8 1993 C
  7.4041 +1955.6 1992.5 1956 1991 1956.3 1989.9 C
  7.4042 +1956.5 1989.8 1956.6 1990 1956.8 1990.1 C
  7.4043 +1957.1 1989 1956 1989.1 1955.8 1988.2 C
  7.4044 +1955.1 1990.4 1956.2 1995 1954.8 1995.9 C
  7.4045 +1954.1 1995.5 1954.5 1996.5 1954.4 1997.1 C
  7.4046 +1955 1996.8 1954.8 1997.4 1955.6 1996.8 C
  7.4047 +1956 1996 1956.3 1993.2 1958.7 1994.2 C
  7.4048 +1958.9 1994.2 1959.7 1994.2 1959.2 1994.2 C
  7.4049 +[0 1 1 0.23]  vc
  7.4050 +f 
  7.4051 +S 
  7.4052 +n
  7.4053 +1958.2 1994 m
  7.4054 +1958.4 1993.5 1959.7 1993.1 1959.9 1992 C
  7.4055 +1959.7 1992.5 1959.3 1992 1959.4 1991.8 C
  7.4056 +1959.4 1991.6 1959.4 1990.8 1959.4 1991.3 C
  7.4057 +1959.2 1991.8 1958.8 1991.3 1958.9 1991.1 C
  7.4058 +1958.9 1990.5 1958 1990.3 1957.5 1989.9 C
  7.4059 +1956.8 1989.5 1956.9 1991 1956.3 1990.1 C
  7.4060 +1956.7 1991 1955.4 1992.1 1956.5 1992.3 C
  7.4061 +1956.8 1993.5 1958.3 1992.9 1957.2 1994 C
  7.4062 +1957.8 1994.3 1958.1 1992.4 1958.2 1994 C
  7.4063 +[0 0.5 0.5 0.2]  vc
  7.4064 +f 
  7.4065 +S 
  7.4066 +n
  7.4067 +vmrs
  7.4068 +1954.4 1982.7 m
  7.4069 +1956.1 1982.7 1954.1 1982.5 1953.9 1982.9 C
  7.4070 +1953.9 1983.7 1953.7 1984.7 1954.1 1985.3 C
  7.4071 +1954.4 1984.2 1953.6 1983.6 1954.4 1982.7 C
  7.4072 +[0 1 1 0.36]  vc
  7.4073 +f 
  7.4074 +0.4 w
  7.4075 +2 J
  7.4076 +2 M
  7.4077 +S 
  7.4078 +n
  7.4079 +1989.6 1982.9 m
  7.4080 +1989.1 1982.7 1988.6 1982.3 1988 1982.4 C
  7.4081 +1987.2 1982.8 1987.4 1984.8 1987.7 1985.1 C
  7.4082 +1988.9 1985.6 1990.7 1984.4 1989.6 1982.9 C
  7.4083 +[0 1 1 0.23]  vc
  7.4084 +f 
  7.4085 +S 
  7.4086 +n
  7.4087 +1987 1980.3 m
  7.4088 +1986.2 1980 1986 1979.1 1985.1 1979.8 C
  7.4089 +1983.5 1981.4 1985.3 1981.4 1985.8 1982.2 C
  7.4090 +1986.5 1981.7 1987.4 1981.5 1987 1980.3 C
  7.4091 +f 
  7.4092 +S 
  7.4093 +n
  7.4094 +1983.6 1977.2 m
  7.4095 +1982.7 1977.5 1981.4 1976.3 1981.2 1977.4 C
  7.4096 +1982.3 1978 1981.2 1979.9 1982.2 1979.1 C
  7.4097 +1983.5 1979 1983.9 1978.5 1983.6 1977.2 C
  7.4098 +f 
  7.4099 +S 
  7.4100 +n
  7.4101 +1981.2 1976 m
  7.4102 +1981.5 1974.9 1980.6 1974 1979.6 1974 C
  7.4103 +1978.3 1974.3 1979.3 1975.4 1978.8 1975.5 C
  7.4104 +1978.6 1976.4 1980.7 1977.4 1981.2 1976 C
  7.4105 +f 
  7.4106 +S 
  7.4107 +n
  7.4108 +1972.1 2082.3 m
  7.4109 +1971.8 2081.8 1971.3 2080.9 1971.2 2080.1 C
  7.4110 +1971.1 2072.9 1971.3 2064.6 1970.9 2058.3 C
  7.4111 +1970.3 2058.5 1970.1 2057.7 1969.7 2058.5 C
  7.4112 +1970.6 2058.5 1969.7 2059 1970.2 2059.2 C
  7.4113 +1970.2 2065.4 1970.2 2072.4 1970.2 2077.7 C
  7.4114 +1971.1 2078.9 1970.6 2078.9 1970.4 2079.9 C
  7.4115 +1969.2 2080.2 1968.2 2080.4 1967.3 2079.6 C
  7.4116 +1966.8 2077.8 1963.4 2076.3 1963.5 2075.1 C
  7.4117 +1961.5 2075.5 1962 2071.5 1959.6 2072 C
  7.4118 +1959.2 2070 1956.5 2069.3 1955.8 2067.6 C
  7.4119 +1956 2068.4 1955.3 2069.7 1956.5 2069.8 C
  7.4120 +1958.6 2068.9 1958.1 2073.5 1960.1 2072.4 C
  7.4121 +1960.7 2075.9 1964.7 2074.6 1964.2 2078 C
  7.4122 +1967.2 2078.6 1967.9 2081.6 1970.7 2080.6 C
  7.4123 +1970.3 2081.1 1971.5 2081.2 1971.9 2082.3 C
  7.4124 +1967.2 2084.3 1962.9 2087.1 1958.2 2089 C
  7.4125 +1962.9 2087 1967.4 2084.4 1972.1 2082.3 C
  7.4126 +[0 0.2 1 0]  vc
  7.4127 +f 
  7.4128 +S 
  7.4129 +n
  7.4130 +1971.9 2080.1 m
  7.4131 +1971.9 2075.1 1971.9 2070 1971.9 2065 C
  7.4132 +1971.9 2070 1971.9 2075.1 1971.9 2080.1 C
  7.4133 +[0 1 1 0.23]  vc
  7.4134 +f 
  7.4135 +S 
  7.4136 +n
  7.4137 +2010.8 2050.6 m
  7.4138 +2013.2 2049 2010.5 2050.1 2010.5 2051.3 C
  7.4139 +2010.5 2057.7 2010.5 2064.1 2010.5 2070.5 C
  7.4140 +2008.7 2072.4 2006 2073.3 2003.6 2074.4 C
  7.4141 +2016.4 2073.7 2008 2058.4 2010.8 2050.6 C
  7.4142 +[0.4 0.4 0 0]  vc
  7.4143 +f 
  7.4144 +S 
  7.4145 +n
  7.4146 +2006.4 2066.9 m
  7.4147 +2006.4 2061.9 2006.4 2056.8 2006.4 2051.8 C
  7.4148 +2006.4 2056.8 2006.4 2061.9 2006.4 2066.9 C
  7.4149 +[0 1 1 0.23]  vc
  7.4150 +f 
  7.4151 +S 
  7.4152 +n
  7.4153 +1971.9 2060.7 m
  7.4154 +1972.2 2060.3 1971.4 2068.2 1972.4 2061.9 C
  7.4155 +1971.8 2061.6 1972.4 2060.9 1971.9 2060.7 C
  7.4156 +f 
  7.4157 +S 
  7.4158 +n
  7.4159 +vmrs
  7.4160 +1986.5 2055.2 m
  7.4161 +1987.5 2054.3 1986.3 2053.4 1986 2052.8 C
  7.4162 +1983.8 2052.7 1983.6 2050.1 1981.7 2049.6 C
  7.4163 +1981.2 2048.7 1980.8 2047 1980.3 2046.8 C
  7.4164 +1978.5 2047 1978 2044.6 1976.7 2043.9 C
  7.4165 +1974 2044.4 1972 2046.6 1969.2 2047 C
  7.4166 +1969 2047.2 1968.8 2047.5 1968.5 2047.7 C
  7.4167 +1970.6 2049.6 1973.1 2051.3 1974.3 2054.2 C
  7.4168 +1975.7 2054.5 1977 2055.2 1976.4 2057.1 C
  7.4169 +1976.7 2058 1975.5 2058.5 1976 2059.5 C
  7.4170 +1979.2 2058 1983 2056.6 1986.5 2055.2 C
  7.4171 +[0 0.5 0.5 0.2]  vc
  7.4172 +f 
  7.4173 +0.4 w
  7.4174 +2 J
  7.4175 +2 M
  7.4176 +S 
  7.4177 +n
  7.4178 +1970.2 2054.2 m
  7.4179 +1971.5 2055.3 1972.5 2056.8 1972.1 2058.3 C
  7.4180 +1972.8 2056.5 1971.6 2055.6 1970.2 2054.2 C
  7.4181 +[0 1 1 0.23]  vc
  7.4182 +f 
  7.4183 +S 
  7.4184 +n
  7.4185 +1992 2052.5 m
  7.4186 +1992 2053.4 1992.2 2054.4 1991.8 2055.2 C
  7.4187 +1992.2 2054.4 1992 2053.4 1992 2052.5 C
  7.4188 +f 
  7.4189 +S 
  7.4190 +n
  7.4191 +1957.2 2053 m
  7.4192 +1958.1 2052.6 1959 2052.2 1959.9 2051.8 C
  7.4193 +1959 2052.2 1958.1 2052.6 1957.2 2053 C
  7.4194 +f 
  7.4195 +S 
  7.4196 +n
  7.4197 +2006.4 2047.5 m
  7.4198 +2006.8 2047.1 2006 2055 2006.9 2048.7 C
  7.4199 +2006.4 2048.4 2007 2047.7 2006.4 2047.5 C
  7.4200 +f 
  7.4201 +S 
  7.4202 +n
  7.4203 +2004.8 2041 m
  7.4204 +2006.1 2042.1 2007.1 2043.6 2006.7 2045.1 C
  7.4205 +2007.3 2043.3 2006.2 2042.4 2004.8 2041 C
  7.4206 +f 
  7.4207 +S 
  7.4208 +n
  7.4209 +1976 2039.8 m
  7.4210 +1975.6 2039.3 1975.2 2038.4 1975 2037.6 C
  7.4211 +1974.9 2030.4 1975.2 2022.1 1974.8 2015.8 C
  7.4212 +1974.2 2016 1974 2015.3 1973.6 2016 C
  7.4213 +1974.4 2016 1973.5 2016.5 1974 2016.8 C
  7.4214 +1974 2022.9 1974 2030 1974 2035.2 C
  7.4215 +1974.9 2036.4 1974.4 2036.4 1974.3 2037.4 C
  7.4216 +1973.1 2037.7 1972 2037.9 1971.2 2037.2 C
  7.4217 +1970.6 2035.3 1967.3 2033.9 1967.3 2032.6 C
  7.4218 +1965.3 2033 1965.9 2029.1 1963.5 2029.5 C
  7.4219 +1963 2027.6 1960.4 2026.8 1959.6 2025.2 C
  7.4220 +1959.8 2025.9 1959.2 2027.2 1960.4 2027.3 C
  7.4221 +1962.5 2026.4 1961.9 2031 1964 2030 C
  7.4222 +1964.6 2033.4 1968.5 2032.1 1968 2035.5 C
  7.4223 +1971 2036.1 1971.8 2039.1 1974.5 2038.1 C
  7.4224 +1974.2 2038.7 1975.3 2038.7 1975.7 2039.8 C
  7.4225 +1971 2041.8 1966.7 2044.6 1962 2046.5 C
  7.4226 +1966.8 2044.5 1971.3 2041.9 1976 2039.8 C
  7.4227 +[0 0.2 1 0]  vc
  7.4228 +f 
  7.4229 +S 
  7.4230 +n
  7.4231 +1975.7 2037.6 m
  7.4232 +1975.7 2032.6 1975.7 2027.6 1975.7 2022.5 C
  7.4233 +1975.7 2027.6 1975.7 2032.6 1975.7 2037.6 C
  7.4234 +[0 1 1 0.23]  vc
  7.4235 +f 
  7.4236 +S 
  7.4237 +n
  7.4238 +1992 2035.5 m
  7.4239 +1992 2034.2 1992 2032.9 1992 2031.6 C
  7.4240 +1992 2032.9 1992 2034.2 1992 2035.5 C
  7.4241 +f 
  7.4242 +S 
  7.4243 +n
  7.4244 +2015.3 2036 m
  7.4245 +2015.4 2034.1 2013.3 2034 2012.9 2033.3 C
  7.4246 +2011.5 2031 2009.3 2029.4 2007.4 2028 C
  7.4247 +2006.9 2027.1 2006.6 2023.8 2005 2024.9 C
  7.4248 +2004 2024.9 2002.9 2024.9 2001.9 2024.9 C
  7.4249 +2001.4 2026.5 2001 2028.4 2003.8 2028.3 C
  7.4250 +2006.6 2030.4 2008.9 2033.7 2011.2 2036.2 C
  7.4251 +2011.8 2036.4 2012.9 2035.8 2012.9 2036.7 C
  7.4252 +2013 2035.5 2015.3 2037.4 2015.3 2036 C
  7.4253 +[0 0 0 0]  vc
  7.4254 +f 
  7.4255 +S 
  7.4256 +n
  7.4257 +vmrs
  7.4258 +2009.1 2030.4 m
  7.4259 +2009.1 2029 2007.5 2029.4 2006.9 2028.3 C
  7.4260 +2007.2 2027.1 2006.5 2025.5 2005.7 2024.7 C
  7.4261 +2004.6 2025.1 2003.1 2024.9 2001.9 2024.9 C
  7.4262 +2001.8 2026.2 2000.9 2027 2002.4 2028 C
  7.4263 +2004.5 2027.3 2004.9 2029.4 2006.9 2029 C
  7.4264 +2007 2030.2 2007.6 2030.7 2008.4 2031.4 C
  7.4265 +2008.8 2031.5 2009.1 2031.1 2009.1 2030.4 C
  7.4266 +[0 0 0 0.18]  vc
  7.4267 +f 
  7.4268 +0.4 w
  7.4269 +2 J
  7.4270 +2 M
  7.4271 +S 
  7.4272 +n
  7.4273 +2003.8 2029.5 m
  7.4274 +2003 2029.4 2001.9 2029.1 2002.4 2030.4 C
  7.4275 +2003.1 2031.3 2005.2 2030.3 2003.8 2029.5 C
  7.4276 +[0 1 1 0.23]  vc
  7.4277 +f 
  7.4278 +S 
  7.4279 +n
  7.4280 +1999.2 2025.2 m
  7.4281 +1999.1 2025.6 1998 2025.7 1998.8 2026.6 C
  7.4282 +2000.9 2028.5 1999.5 2023.4 1999.2 2025.2 C
  7.4283 +f 
  7.4284 +S 
  7.4285 +n
  7.4286 +2007.6 2024.2 m
  7.4287 +2007.6 2022.9 2008.4 2024.2 2007.6 2022.8 C
  7.4288 +2007.6 2017.5 2007.8 2009.1 2007.4 2003.8 C
  7.4289 +2007.9 2003.7 2008.7 2002.8 2009.1 2002.1 C
  7.4290 +2009.6 2000.8 2008.3 2000.8 2007.9 2000.2 C
  7.4291 +2004.9 2000 2008.9 2001.3 2007.2 2002.1 C
  7.4292 +2006.7 2007.7 2007 2015.1 2006.9 2021.1 C
  7.4293 +2006.7 2022.1 2005.4 2022.8 2006.2 2023.5 C
  7.4294 +2006.6 2023.1 2008 2025.9 2007.6 2024.2 C
  7.4295 +f 
  7.4296 +S 
  7.4297 +n
  7.4298 +1989.9 2023.5 m
  7.4299 +1989.5 2022.6 1991.4 2023.2 1991.8 2023 C
  7.4300 +1992.2 2023.2 1991.9 2023.7 1992.5 2023.5 C
  7.4301 +1991.6 2023.2 1991.6 2022.2 1990.6 2021.8 C
  7.4302 +1990.4 2022.8 1989 2022.8 1988.9 2023.5 C
  7.4303 +1988.5 2023 1988.7 2022.6 1988.7 2023.5 C
  7.4304 +1989.1 2023.5 1990.2 2023.5 1989.9 2023.5 C
  7.4305 +f 
  7.4306 +[0 0.5 0.5 0.2]  vc
  7.4307 +S 
  7.4308 +n
  7.4309 +2003.3 2023.5 m
  7.4310 +2003.1 2023.3 2003.1 2023.2 2003.3 2023 C
  7.4311 +2003.7 2023.1 2003.9 2022.9 2003.8 2022.5 C
  7.4312 +2003.4 2022.2 2001.2 2022.3 2002.4 2023 C
  7.4313 +2002.6 2022.9 2002.7 2023.1 2002.8 2023.2 C
  7.4314 +2000.7 2023.7 2003.9 2023.4 2003.3 2023.5 C
  7.4315 +[0 1 1 0.23]  vc
  7.4316 +f 
  7.4317 +S 
  7.4318 +n
  7.4319 +1986.8 2019.4 m
  7.4320 +1987.8 2019.8 1987.5 2018.6 1987.2 2018 C
  7.4321 +1986.2 2017.8 1987.3 2020.5 1986.3 2019.2 C
  7.4322 +1986.3 2017.7 1986.3 2020.6 1986.3 2021.3 C
  7.4323 +1988.5 2023.1 1985.6 2020.3 1986.8 2019.4 C
  7.4324 +f 
  7.4325 +S 
  7.4326 +n
  7.4327 +1975.7 2018.2 m
  7.4328 +1976.1 2017.8 1975.2 2025.7 1976.2 2019.4 C
  7.4329 +1975.7 2019.2 1976.3 2018.4 1975.7 2018.2 C
  7.4330 +f 
  7.4331 +S 
  7.4332 +n
  7.4333 +1974 2011.7 m
  7.4334 +1975.4 2012.8 1976.4 2014.3 1976 2015.8 C
  7.4335 +1976.6 2014 1975.5 2013.1 1974 2011.7 C
  7.4336 +f 
  7.4337 +S 
  7.4338 +n
  7.4339 +1984.6 2006.7 m
  7.4340 +1984.7 2004.8 1982.6 2004.8 1982.2 2004 C
  7.4341 +1980.8 2001.7 1978.6 2000.1 1976.7 1998.8 C
  7.4342 +1976.1 1997.8 1975.8 1994.5 1974.3 1995.6 C
  7.4343 +1973.3 1995.6 1972.2 1995.6 1971.2 1995.6 C
  7.4344 +1970.7 1997.2 1970.3 1999.1 1973.1 1999 C
  7.4345 +1975.8 2001.2 1978.2 2004.4 1980.5 2006.9 C
  7.4346 +1981.1 2007.1 1982.1 2006.5 1982.2 2007.4 C
  7.4347 +1982.3 2006.2 1984.5 2008.1 1984.6 2006.7 C
  7.4348 +[0 0 0 0]  vc
  7.4349 +f 
  7.4350 +S 
  7.4351 +n
  7.4352 +vmrs
  7.4353 +1978.4 2001.2 m
  7.4354 +1978.4 1999.7 1976.8 2000.1 1976.2 1999 C
  7.4355 +1976.5 1997.8 1975.8 1996.2 1975 1995.4 C
  7.4356 +1973.9 1995.8 1972.4 1995.6 1971.2 1995.6 C
  7.4357 +1971 1997 1970.2 1997.7 1971.6 1998.8 C
  7.4358 +1973.8 1998 1974.2 2000.1 1976.2 1999.7 C
  7.4359 +1976.3 2000.9 1976.9 2001.4 1977.6 2002.1 C
  7.4360 +1978.1 2002.2 1978.4 2001.8 1978.4 2001.2 C
  7.4361 +[0 0 0 0.18]  vc
  7.4362 +f 
  7.4363 +0.4 w
  7.4364 +2 J
  7.4365 +2 M
  7.4366 +S 
  7.4367 +n
  7.4368 +1973.1 2000.2 m
  7.4369 +1972.3 2000.1 1971.2 1999.8 1971.6 2001.2 C
  7.4370 +1972.4 2002 1974.5 2001 1973.1 2000.2 C
  7.4371 +[0 1 1 0.23]  vc
  7.4372 +f 
  7.4373 +S 
  7.4374 +n
  7.4375 +1960.8 1998.5 m
  7.4376 +1961.6 1998.2 1962.6 2000.3 1963.2 2000.9 C
  7.4377 +1962.3 2000.1 1962.2 1998.7 1960.8 1998.5 C
  7.4378 +f 
  7.4379 +S 
  7.4380 +n
  7.4381 +1968.5 1995.9 m
  7.4382 +1968.4 1996.4 1967.3 1996.4 1968 1997.3 C
  7.4383 +1970.1 1999.2 1968.8 1994.1 1968.5 1995.9 C
  7.4384 +f 
  7.4385 +S 
  7.4386 +n
  7.4387 +1976.9 1994.9 m
  7.4388 +1976.9 1993.7 1977.6 1994.9 1976.9 1993.5 C
  7.4389 +1976.9 1988.2 1977.1 1979.8 1976.7 1974.5 C
  7.4390 +1977.2 1974.5 1978 1973.5 1978.4 1972.8 C
  7.4391 +1978.8 1971.5 1977.6 1971.5 1977.2 1970.9 C
  7.4392 +1974.2 1970.7 1978.2 1972 1976.4 1972.8 C
  7.4393 +1976 1978.4 1976.3 1985.8 1976.2 1991.8 C
  7.4394 +1976 1992.8 1974.6 1993.5 1975.5 1994.2 C
  7.4395 +1975.9 1993.8 1977.3 1996.6 1976.9 1994.9 C
  7.4396 +f 
  7.4397 +S 
  7.4398 +n
  7.4399 +1972.6 1994.2 m
  7.4400 +1972.4 1994 1972.4 1993.9 1972.6 1993.7 C
  7.4401 +1973 1993.8 1973.1 1993.7 1973.1 1993.2 C
  7.4402 +1972.7 1992.9 1970.5 1993.1 1971.6 1993.7 C
  7.4403 +1971.9 1993.7 1972 1993.8 1972.1 1994 C
  7.4404 +1970 1994.4 1973.1 1994.1 1972.6 1994.2 C
  7.4405 +f 
  7.4406 +S 
  7.4407 +n
  7.4408 +1948.1 2093.8 m
  7.4409 +1947 2092.7 1945.9 2091.6 1944.8 2090.4 C
  7.4410 +1945.9 2091.6 1947 2092.7 1948.1 2093.8 C
  7.4411 +[0 0.4 1 0]  vc
  7.4412 +f 
  7.4413 +S 
  7.4414 +n
  7.4415 +1953.4 2091.4 m
  7.4416 +1954.8 2090.7 1956.3 2090 1957.7 2089.2 C
  7.4417 +1956.3 2090 1954.8 2090.7 1953.4 2091.4 C
  7.4418 +[0 0.2 1 0]  vc
  7.4419 +f 
  7.4420 +S 
  7.4421 +n
  7.4422 +1954.1 2091.4 m
  7.4423 +1956.6 2089.6 1957.2 2089.6 1954.1 2091.4 C
  7.4424 +[0 0.4 1 0]  vc
  7.4425 +f 
  7.4426 +S 
  7.4427 +n
  7.4428 +1962.3 2087.3 m
  7.4429 +1963.7 2086.6 1965.2 2085.9 1966.6 2085.2 C
  7.4430 +1965.2 2085.9 1963.7 2086.6 1962.3 2087.3 C
  7.4431 +f 
  7.4432 +S 
  7.4433 +n
  7.4434 +vmrs
  7.4435 +1967.1 2084.9 m
  7.4436 +1968.3 2084.4 1969.7 2083.8 1970.9 2083.2 C
  7.4437 +1969.7 2083.8 1968.3 2084.4 1967.1 2084.9 C
  7.4438 +[0 0.4 1 0]  vc
  7.4439 +f 
  7.4440 +0.4 w
  7.4441 +2 J
  7.4442 +2 M
  7.4443 +S 
  7.4444 +n
  7.4445 +1982.7 2080.6 m
  7.4446 +1981.5 2079.5 1980.5 2078.4 1979.3 2077.2 C
  7.4447 +1980.5 2078.4 1981.5 2079.5 1982.7 2080.6 C
  7.4448 +f 
  7.4449 +S 
  7.4450 +n
  7.4451 +1988 2078.2 m
  7.4452 +1989.4 2077.5 1990.8 2076.8 1992.3 2076 C
  7.4453 +1990.8 2076.8 1989.4 2077.5 1988 2078.2 C
  7.4454 +[0 0.2 1 0]  vc
  7.4455 +f 
  7.4456 +S 
  7.4457 +n
  7.4458 +1988.7 2078.2 m
  7.4459 +1991.1 2076.4 1991.8 2076.4 1988.7 2078.2 C
  7.4460 +[0 0.4 1 0]  vc
  7.4461 +f 
  7.4462 +S 
  7.4463 +n
  7.4464 +1976.2 2063.8 m
  7.4465 +1978.6 2062.2 1976 2063.3 1976 2064.5 C
  7.4466 +1976.1 2067.8 1975.5 2071.4 1976.4 2074.4 C
  7.4467 +1975.7 2071.1 1975.9 2067.2 1976.2 2063.8 C
  7.4468 +f 
  7.4469 +S 
  7.4470 +n
  7.4471 +1996.8 2074.1 m
  7.4472 +1998.3 2073.4 1999.7 2072.7 2001.2 2072 C
  7.4473 +1999.7 2072.7 1998.3 2073.4 1996.8 2074.1 C
  7.4474 +f 
  7.4475 +S 
  7.4476 +n
  7.4477 +2001.6 2071.7 m
  7.4478 +2002.9 2071.2 2004.2 2070.6 2005.5 2070 C
  7.4479 +2004.2 2070.6 2002.9 2071.2 2001.6 2071.7 C
  7.4480 +f 
  7.4481 +S 
  7.4482 +n
  7.4483 +1981.5 2060.7 m
  7.4484 +1980.2 2061.2 1978.9 2061.5 1977.9 2062.6 C
  7.4485 +1978.9 2061.5 1980.2 2061.2 1981.5 2060.7 C
  7.4486 +f 
  7.4487 +S 
  7.4488 +n
  7.4489 +1982 2060.4 m
  7.4490 +1982.7 2060.1 1983.6 2059.8 1984.4 2059.5 C
  7.4491 +1983.6 2059.8 1982.7 2060.1 1982 2060.4 C
  7.4492 +f 
  7.4493 +S 
  7.4494 +n
  7.4495 +1952 2051.3 m
  7.4496 +1950.8 2050.2 1949.7 2049.1 1948.6 2048 C
  7.4497 +1949.7 2049.1 1950.8 2050.2 1952 2051.3 C
  7.4498 +f 
  7.4499 +S 
  7.4500 +n
  7.4501 +vmrs
  7.4502 +1977.4 2047.7 m
  7.4503 +1975.8 2047.8 1974.8 2046.1 1974.5 2045.3 C
  7.4504 +1974.9 2044.4 1976 2044.5 1976.7 2044.8 C
  7.4505 +1977.9 2045 1977 2048.4 1979.3 2047.5 C
  7.4506 +1979.9 2047.5 1980.8 2048.6 1979.8 2049.2 C
  7.4507 +1978.2 2050.4 1980.8 2049.5 1980.3 2049.4 C
  7.4508 +1981.4 2049.8 1980.3 2048.4 1980.3 2048 C
  7.4509 +1979.8 2047.5 1979 2046.6 1978.4 2046.5 C
  7.4510 +1977.3 2045.9 1977.2 2043.3 1975.2 2044.6 C
  7.4511 +1974.7 2045.3 1973.6 2045 1973.3 2045.8 C
  7.4512 +1975 2046.3 1975.8 2049.8 1978.1 2049.4 C
  7.4513 +1978.4 2050.9 1978.7 2048.5 1977.9 2049.2 C
  7.4514 +1977.7 2048.7 1977.2 2047.8 1977.4 2047.7 C
  7.4515 +[0 0.5 0.5 0.2]  vc
  7.4516 +f 
  7.4517 +0.4 w
  7.4518 +2 J
  7.4519 +2 M
  7.4520 +S 
  7.4521 +n
  7.4522 +1957.2 2048.9 m
  7.4523 +1958.7 2048.2 1960.1 2047.5 1961.6 2046.8 C
  7.4524 +1960.1 2047.5 1958.7 2048.2 1957.2 2048.9 C
  7.4525 +[0 0.2 1 0]  vc
  7.4526 +f 
  7.4527 +S 
  7.4528 +n
  7.4529 +1958 2048.9 m
  7.4530 +1960.4 2047.1 1961.1 2047.1 1958 2048.9 C
  7.4531 +[0 0.4 1 0]  vc
  7.4532 +f 
  7.4533 +S 
  7.4534 +n
  7.4535 +1966.1 2044.8 m
  7.4536 +1967.6 2044.1 1969 2043.4 1970.4 2042.7 C
  7.4537 +1969 2043.4 1967.6 2044.1 1966.1 2044.8 C
  7.4538 +f 
  7.4539 +S 
  7.4540 +n
  7.4541 +1970.9 2042.4 m
  7.4542 +1972.2 2041.9 1973.5 2041.3 1974.8 2040.8 C
  7.4543 +1973.5 2041.3 1972.2 2041.9 1970.9 2042.4 C
  7.4544 +f 
  7.4545 +S 
  7.4546 +n
  7.4547 +2012 2034.5 m
  7.4548 +2010.4 2034.6 2009.3 2032.9 2009.1 2032.1 C
  7.4549 +2009.4 2031 2010.3 2031.3 2011.2 2031.6 C
  7.4550 +2012.5 2031.8 2011.6 2035.2 2013.9 2034.3 C
  7.4551 +2014.4 2034.3 2015.4 2035.4 2014.4 2036 C
  7.4552 +2012.7 2037.2 2015.3 2036.3 2014.8 2036.2 C
  7.4553 +2015.9 2036.6 2014.8 2035.2 2014.8 2034.8 C
  7.4554 +2014.4 2034.3 2013.6 2033.4 2012.9 2033.3 C
  7.4555 +2011.5 2031 2009.3 2029.4 2007.4 2028 C
  7.4556 +2007.5 2026.5 2007.3 2027.9 2007.2 2028.3 C
  7.4557 +2007.9 2028.8 2008.7 2029.1 2009.3 2030 C
  7.4558 +2009.6 2030.7 2009 2031.9 2008.4 2031.6 C
  7.4559 +2006.7 2031 2007.7 2028 2005 2028.8 C
  7.4560 +2004.8 2028.6 2004.3 2028.2 2003.8 2028.3 C
  7.4561 +2006.6 2030.4 2008.9 2033.7 2011.2 2036.2 C
  7.4562 +2011.8 2036.4 2012.9 2035.8 2012.9 2036.7 C
  7.4563 +2012.7 2036.1 2011.8 2035 2012 2034.5 C
  7.4564 +[0 0.5 0.5 0.2]  vc
  7.4565 +f 
  7.4566 +S 
  7.4567 +n
  7.4568 +1981.2 2005.2 m
  7.4569 +1979.7 2005.3 1978.6 2003.6 1978.4 2002.8 C
  7.4570 +1978.7 2001.8 1979.6 2002.1 1980.5 2002.4 C
  7.4571 +1981.8 2002.5 1980.9 2005.9 1983.2 2005 C
  7.4572 +1983.7 2005.1 1984.7 2006.1 1983.6 2006.7 C
  7.4573 +1982 2007.9 1984.6 2007 1984.1 2006.9 C
  7.4574 +1985.2 2007.3 1984.1 2006 1984.1 2005.5 C
  7.4575 +1983.6 2005 1982.9 2004.1 1982.2 2004 C
  7.4576 +1980.8 2001.7 1978.6 2000.1 1976.7 1998.8 C
  7.4577 +1976.7 1997.2 1976.6 1998.6 1976.4 1999 C
  7.4578 +1977.2 1999.5 1978 1999.8 1978.6 2000.7 C
  7.4579 +1978.8 2001.5 1978.3 2002.7 1977.6 2002.4 C
  7.4580 +1976 2001.8 1977 1998.7 1974.3 1999.5 C
  7.4581 +1974.1 1999.3 1973.6 1998.9 1973.1 1999 C
  7.4582 +1975.8 2001.2 1978.2 2004.4 1980.5 2006.9 C
  7.4583 +1981.1 2007.1 1982.1 2006.5 1982.2 2007.4 C
  7.4584 +1982 2006.8 1981.1 2005.7 1981.2 2005.2 C
  7.4585 +f 
  7.4586 +S 
  7.4587 +n
  7.4588 +1966.8 1976.4 m
  7.4589 +1969.4 1973 1974.4 1974.6 1976.2 1970.4 C
  7.4590 +1972.7 1974 1968 1975.1 1964 1977.4 C
  7.4591 +1960.9 1979.9 1957.1 1981.8 1953.9 1982.7 C
  7.4592 +1958.4 1981.1 1962.6 1978.8 1966.8 1976.4 C
  7.4593 +[0.18 0.18 0 0.78]  vc
  7.4594 +f 
  7.4595 +S 
  7.4596 +n
  7.4597 +1948.4 2093.8 m
  7.4598 +1949.8 2093.1 1951.2 2092.5 1952.7 2091.9 C
  7.4599 +1951.2 2092.5 1949.8 2093.1 1948.4 2093.8 C
  7.4600 +[0 0.2 1 0]  vc
  7.4601 +f 
  7.4602 +S 
  7.4603 +n
  7.4604 +1948.1 2093.6 m
  7.4605 +1947.3 2092.8 1946.5 2091.9 1945.7 2091.2 C
  7.4606 +1946.5 2091.9 1947.3 2092.8 1948.1 2093.6 C
  7.4607 +f 
  7.4608 +S 
  7.4609 +n
  7.4610 +vmrs
  7.4611 +1942.1 2087.8 m
  7.4612 +1943.5 2088.4 1944.3 2089.5 1945.2 2090.7 C
  7.4613 +1944.8 2089.3 1943.3 2088.3 1942.1 2087.8 C
  7.4614 +[0 0.2 1 0]  vc
  7.4615 +f 
  7.4616 +0.4 w
  7.4617 +2 J
  7.4618 +2 M
  7.4619 +S 
  7.4620 +n
  7.4621 +1933.5 2078.4 m
  7.4622 +1933.5 2078 1933.2 2079 1933.7 2079.4 C
  7.4623 +1935 2080.4 1936.2 2081.3 1937.1 2082.8 C
  7.4624 +1936.7 2080.7 1933.7 2080.7 1933.5 2078.4 C
  7.4625 +f 
  7.4626 +S 
  7.4627 +n
  7.4628 +1982.9 2080.6 m
  7.4629 +1984.4 2079.9 1985.8 2079.3 1987.2 2078.7 C
  7.4630 +1985.8 2079.3 1984.4 2079.9 1982.9 2080.6 C
  7.4631 +f 
  7.4632 +S 
  7.4633 +n
  7.4634 +1982.7 2080.4 m
  7.4635 +1981.9 2079.6 1981.1 2078.7 1980.3 2078 C
  7.4636 +1981.1 2078.7 1981.9 2079.6 1982.7 2080.4 C
  7.4637 +f 
  7.4638 +S 
  7.4639 +n
  7.4640 +1977.4 2075.1 m
  7.4641 +1977.9 2075.3 1979.1 2076.4 1979.8 2077.5 C
  7.4642 +1979 2076.8 1978.7 2075.1 1977.4 2075.1 C
  7.4643 +f 
  7.4644 +S 
  7.4645 +n
  7.4646 +1952.2 2051.3 m
  7.4647 +1953.6 2050.7 1955.1 2050.1 1956.5 2049.4 C
  7.4648 +1955.1 2050.1 1953.6 2050.7 1952.2 2051.3 C
  7.4649 +f 
  7.4650 +S 
  7.4651 +n
  7.4652 +1952 2051.1 m
  7.4653 +1951.2 2050.3 1950.3 2049.5 1949.6 2048.7 C
  7.4654 +1950.3 2049.5 1951.2 2050.3 1952 2051.1 C
  7.4655 +f 
  7.4656 +S 
  7.4657 +n
  7.4658 +1946 2045.3 m
  7.4659 +1947.3 2045.9 1948.1 2047 1949.1 2048.2 C
  7.4660 +1948.6 2046.8 1947.1 2045.8 1946 2045.3 C
  7.4661 +f 
  7.4662 +S 
  7.4663 +n
  7.4664 +1937.3 2036 m
  7.4665 +1937.4 2035.5 1937 2036.5 1937.6 2036.9 C
  7.4666 +1938.8 2037.9 1940.1 2038.8 1940.9 2040.3 C
  7.4667 +1940.6 2038.2 1937.6 2038.2 1937.3 2036 C
  7.4668 +f 
  7.4669 +S 
  7.4670 +n
  7.4671 +1935.2 2073.2 m
  7.4672 +1936.4 2069.9 1935.8 2061.8 1935.6 2056.4 C
  7.4673 +1935.8 2055.9 1936.3 2055.7 1936.1 2055.2 C
  7.4674 +1935.7 2054.7 1935 2055 1934.4 2054.9 C
  7.4675 +1934.4 2061.5 1934.4 2068.7 1934.4 2074.6 C
  7.4676 +1935.7 2075.1 1936 2073.7 1935.2 2073.2 C
  7.4677 +[0 0.01 1 0]  vc
  7.4678 +f 
  7.4679 +S 
  7.4680 +n
  7.4681 +vmrs
  7.4682 +1939 2030.7 m
  7.4683 +1940.3 2027.4 1939.7 2019.3 1939.5 2013.9 C
  7.4684 +1939.7 2013.5 1940.1 2013.2 1940 2012.7 C
  7.4685 +1939.5 2012.3 1938.8 2012.5 1938.3 2012.4 C
  7.4686 +1938.3 2019 1938.3 2026.2 1938.3 2032.1 C
  7.4687 +1939.5 2032.7 1939.8 2031.2 1939 2030.7 C
  7.4688 +[0 0.01 1 0]  vc
  7.4689 +f 
  7.4690 +0.4 w
  7.4691 +2 J
  7.4692 +2 M
  7.4693 +S 
  7.4694 +n
  7.4695 +1975.2 2077.2 m
  7.4696 +1975.3 2077.3 1975.4 2077.4 1975.5 2077.5 C
  7.4697 +1974.7 2073.2 1974.9 2067.5 1975.2 2063.6 C
  7.4698 +1975.4 2064 1974.6 2063.9 1974.8 2064.3 C
  7.4699 +1974.9 2069.9 1974.3 2076.5 1975.2 2081.1 C
  7.4700 +1974.9 2079.9 1974.9 2078.4 1975.2 2077.2 C
  7.4701 +[0.92 0.92 0 0.67]  vc
  7.4702 +f 
  7.4703 +S 
  7.4704 +n
  7.4705 +1930.8 2067.4 m
  7.4706 +1931.5 2070.1 1929.6 2072.1 1930.6 2074.6 C
  7.4707 +1931 2072.6 1930.8 2069.8 1930.8 2067.4 C
  7.4708 +f 
  7.4709 +S 
  7.4710 +n
  7.4711 +2010 2050.1 m
  7.4712 +2009.8 2050.5 2009.5 2050.9 2009.3 2051.1 C
  7.4713 +2009.5 2056.7 2008.9 2063.3 2009.8 2067.9 C
  7.4714 +2009.5 2062.1 2009.3 2054.7 2010 2050.1 C
  7.4715 +f 
  7.4716 +S 
  7.4717 +n
  7.4718 +1930.1 2060.9 m
  7.4719 +1929.3 2057.1 1930.7 2054.8 1929.9 2051.3 C
  7.4720 +1930.2 2050.2 1931.1 2049.6 1931.8 2049.2 C
  7.4721 +1931.4 2049.6 1930.4 2049.5 1930.1 2050.1 C
  7.4722 +1928.4 2054.8 1933.4 2063.5 1925.3 2064.3 C
  7.4723 +1927.2 2063.9 1928.5 2062.1 1930.1 2060.9 C
  7.4724 +[0.07 0.06 0 0.58]  vc
  7.4725 +f 
  7.4726 +S 
  7.4727 +n
  7.4728 +1929.6 2061.2 m
  7.4729 +1929.6 2057.6 1929.6 2054.1 1929.6 2050.6 C
  7.4730 +1930 2049.9 1930.5 2049.4 1931.1 2049.2 C
  7.4731 +1930 2048.6 1930.5 2050.2 1929.4 2049.6 C
  7.4732 +1928 2054.4 1932.8 2063 1925.3 2064 C
  7.4733 +1926.9 2063.3 1928.3 2062.4 1929.6 2061.2 C
  7.4734 +[0.4 0.4 0 0]  vc
  7.4735 +f 
  7.4736 +S 
  7.4737 +n
  7.4738 +1930.8 2061.6 m
  7.4739 +1930.5 2058 1931.6 2054 1930.8 2051.3 C
  7.4740 +1930.3 2054.5 1930.9 2058.5 1930.4 2061.9 C
  7.4741 +1930.5 2061.2 1931 2062.2 1930.8 2061.6 C
  7.4742 +[0.92 0.92 0 0.67]  vc
  7.4743 +f 
  7.4744 +S 
  7.4745 +n
  7.4746 +1941.2 2045.1 m
  7.4747 +1939.7 2042.6 1937.3 2041.2 1935.4 2039.3 C
  7.4748 +1934.2 2040 1933.7 2036.4 1934 2039.3 C
  7.4749 +1934.9 2040.1 1936.1 2039.9 1936.8 2040.8 C
  7.4750 +1935.3 2044.2 1942.3 2041.7 1939.5 2046 C
  7.4751 +1937.1 2048.5 1940.5 2045.6 1941.2 2045.1 C
  7.4752 +f 
  7.4753 +S 
  7.4754 +n
  7.4755 +1910 2045.8 m
  7.4756 +1910 2039.4 1910 2033 1910 2026.6 C
  7.4757 +1910 2033 1910 2039.4 1910 2045.8 C
  7.4758 +f 
  7.4759 +S 
  7.4760 +n
  7.4761 +1978.8 2022.3 m
  7.4762 +1979.1 2021.7 1979.4 2020.4 1978.6 2021.6 C
  7.4763 +1978.6 2026.9 1978.6 2033 1978.6 2037.6 C
  7.4764 +1979.2 2037 1979.1 2038.2 1979.1 2038.6 C
  7.4765 +1978.7 2033.6 1978.9 2026.8 1978.8 2022.3 C
  7.4766 +f 
  7.4767 +S 
  7.4768 +n
  7.4769 +vmrs
  7.4770 +2026.1 2041.2 m
  7.4771 +2026.1 2034.8 2026.1 2028.3 2026.1 2021.8 C
  7.4772 +2026.1 2028.5 2026.3 2035.4 2025.9 2042 C
  7.4773 +2024.4 2042.9 2022.9 2044.1 2021.3 2044.8 C
  7.4774 +2023.1 2044 2025.1 2042.8 2026.1 2041.2 C
  7.4775 +[0.07 0.06 0 0.58]  vc
  7.4776 +f 
  7.4777 +0.4 w
  7.4778 +2 J
  7.4779 +2 M
  7.4780 +S 
  7.4781 +n
  7.4782 +2026.4 2021.8 m
  7.4783 +2026.3 2028.5 2026.5 2035.4 2026.1 2042 C
  7.4784 +2025.6 2042.8 2024.7 2042.7 2024.2 2043.4 C
  7.4785 +2024.7 2042.7 2025.5 2042.7 2026.1 2042.2 C
  7.4786 +2026.5 2035.5 2026.3 2027.9 2026.4 2021.8 C
  7.4787 +[0.4 0.4 0 0]  vc
  7.4788 +f 
  7.4789 +S 
  7.4790 +n
  7.4791 +2025.6 2038.4 m
  7.4792 +2025.6 2033 2025.6 2027.6 2025.6 2022.3 C
  7.4793 +2025.6 2027.6 2025.6 2033 2025.6 2038.4 C
  7.4794 +[0.92 0.92 0 0.67]  vc
  7.4795 +f 
  7.4796 +S 
  7.4797 +n
  7.4798 +1934 2023.5 m
  7.4799 +1934 2024.7 1933.8 2026 1934.2 2027.1 C
  7.4800 +1934 2025.5 1934.7 2024.6 1934 2023.5 C
  7.4801 +f 
  7.4802 +S 
  7.4803 +n
  7.4804 +1928.2 2023.5 m
  7.4805 +1928 2024.6 1927.4 2023.1 1926.8 2023.2 C
  7.4806 +1926.2 2021 1921.4 2019.3 1923.2 2018 C
  7.4807 +1922.7 2016.5 1923.2 2019.3 1922.2 2018.2 C
  7.4808 +1924.4 2020.4 1926.2 2023.3 1928.9 2024.9 C
  7.4809 +1927.9 2024.2 1929.8 2023.5 1928.2 2023.5 C
  7.4810 +[0.18 0.18 0 0.78]  vc
  7.4811 +f 
  7.4812 +S 
  7.4813 +n
  7.4814 +1934 2019.2 m
  7.4815 +1932 2019.6 1930.8 2022.6 1928.7 2021.8 C
  7.4816 +1924.5 2016.5 1918.2 2011.8 1914 2006.7 C
  7.4817 +1914 2005.7 1914 2004.6 1914 2003.6 C
  7.4818 +1913.6 2004.3 1913.9 2005.8 1913.8 2006.9 C
  7.4819 +1919 2012.4 1924.1 2016.5 1929.2 2022.3 C
  7.4820 +1931 2021.7 1932.2 2019.8 1934 2019.2 C
  7.4821 +f 
  7.4822 +S 
  7.4823 +n
  7.4824 +1928.7 2024.9 m
  7.4825 +1926.3 2022.7 1924.1 2020.4 1921.7 2018.2 C
  7.4826 +1924.1 2020.4 1926.3 2022.7 1928.7 2024.9 C
  7.4827 +[0.65 0.65 0 0.42]  vc
  7.4828 +f 
  7.4829 +S 
  7.4830 +n
  7.4831 +1914.3 2006.7 m
  7.4832 +1918.7 2011.8 1924.5 2016.4 1928.9 2021.6 C
  7.4833 +1924.2 2016.1 1919 2012.1 1914.3 2006.7 C
  7.4834 +[0.07 0.06 0 0.58]  vc
  7.4835 +f 
  7.4836 +S 
  7.4837 +n
  7.4838 +1924.8 2020.8 m
  7.4839 +1921.2 2016.9 1925.6 2022.5 1926 2021.1 C
  7.4840 +1924.2 2021 1926.7 2019.6 1924.8 2020.8 C
  7.4841 +[0.92 0.92 0 0.67]  vc
  7.4842 +f 
  7.4843 +S 
  7.4844 +n
  7.4845 +1934 2018.4 m
  7.4846 +1933.2 2014.7 1934.5 2012.3 1933.7 2008.8 C
  7.4847 +1934 2007.8 1935 2007.2 1935.6 2006.7 C
  7.4848 +1935.3 2007.1 1934.3 2007 1934 2007.6 C
  7.4849 +1932.2 2012.3 1937.2 2021 1929.2 2021.8 C
  7.4850 +1931.1 2021.4 1932.3 2019.6 1934 2018.4 C
  7.4851 +[0.07 0.06 0 0.58]  vc
  7.4852 +f 
  7.4853 +S 
  7.4854 +n
  7.4855 +vmrs
  7.4856 +1933.5 2018.7 m
  7.4857 +1933.5 2015.1 1933.5 2011.7 1933.5 2008.1 C
  7.4858 +1933.8 2007.4 1934.3 2006.9 1934.9 2006.7 C
  7.4859 +1933.8 2006.1 1934.3 2007.7 1933.2 2007.2 C
  7.4860 +1931.9 2012 1936.7 2020.5 1929.2 2021.6 C
  7.4861 +1930.7 2020.8 1932.2 2019.9 1933.5 2018.7 C
  7.4862 +[0.4 0.4 0 0]  vc
  7.4863 +f 
  7.4864 +0.4 w
  7.4865 +2 J
  7.4866 +2 M
  7.4867 +S 
  7.4868 +n
  7.4869 +1934.7 2019.2 m
  7.4870 +1934.3 2015.6 1935.4 2011.5 1934.7 2008.8 C
  7.4871 +1934.1 2012 1934.7 2016 1934.2 2019.4 C
  7.4872 +1934.4 2018.7 1934.8 2019.8 1934.7 2019.2 C
  7.4873 +[0.92 0.92 0 0.67]  vc
  7.4874 +f 
  7.4875 +S 
  7.4876 +n
  7.4877 +1917.6 2013.6 m
  7.4878 +1917.8 2011.1 1916.8 2014.2 1917.2 2012.2 C
  7.4879 +1916.3 2012.9 1914.8 2011.8 1914.3 2010.8 C
  7.4880 +1914.2 2010.5 1914.4 2010.4 1914.5 2010.3 C
  7.4881 +1913.9 2008.8 1913.9 2011.9 1914.3 2012 C
  7.4882 +1916.3 2012 1917.6 2013.6 1916.7 2015.6 C
  7.4883 +1913.7 2017.4 1919.6 2014.8 1917.6 2013.6 C
  7.4884 +f 
  7.4885 +S 
  7.4886 +n
  7.4887 +1887.2 2015.3 m
  7.4888 +1887.2 2008.9 1887.2 2002.5 1887.2 1996.1 C
  7.4889 +1887.2 2002.5 1887.2 2008.9 1887.2 2015.3 C
  7.4890 +f 
  7.4891 +S 
  7.4892 +n
  7.4893 +1916.7 2014.4 m
  7.4894 +1917 2012.1 1913 2013 1913.8 2010.8 C
  7.4895 +1912.1 2009.8 1910.9 2009.4 1910.7 2007.9 C
  7.4896 +1910.4 2010.6 1913.4 2010.4 1914 2012.4 C
  7.4897 +1914.9 2012.8 1916.6 2012.9 1916.4 2014.4 C
  7.4898 +1916.9 2015.1 1914.5 2016.6 1916.2 2015.8 C
  7.4899 +1916.4 2015.3 1916.7 2015 1916.7 2014.4 C
  7.4900 +[0.65 0.65 0 0.42]  vc
  7.4901 +f 
  7.4902 +S 
  7.4903 +n
  7.4904 +1914 2009.3 m
  7.4905 +1912.8 2010.9 1909.6 2005.3 1911.9 2009.8 C
  7.4906 +1912.3 2009.6 1913.6 2010.2 1914 2009.3 C
  7.4907 +[0.92 0.92 0 0.67]  vc
  7.4908 +f 
  7.4909 +S 
  7.4910 +n
  7.4911 +1951.2 1998.8 m
  7.4912 +1949 1996.4 1951.5 1994 1950.3 1991.8 C
  7.4913 +1949.1 1989.1 1954 1982.7 1948.8 1981.2 C
  7.4914 +1949.2 1981.5 1951 1982.4 1950.8 1983.6 C
  7.4915 +1951.9 1988.6 1947.1 1986.5 1948.1 1990.4 C
  7.4916 +1948.5 1990.3 1948.7 1990.7 1948.6 1991.1 C
  7.4917 +1949 1992.5 1947.3 1991.9 1948.1 1992.5 C
  7.4918 +1947.1 1992.7 1945.7 1993.5 1945.2 1994.7 C
  7.4919 +1944.5 1996.8 1947.7 2000.5 1943.8 2001.4 C
  7.4920 +1943.4 2002 1943.7 2004 1942.4 2004.5 C
  7.4921 +1945.2 2002.2 1948.9 2000.9 1951.2 1998.8 C
  7.4922 +f 
  7.4923 +S 
  7.4924 +n
  7.4925 +1994.9 1993 m
  7.4926 +1995.1 1996.5 1994.5 2000.3 1995.4 2003.6 C
  7.4927 +1994.5 2000.3 1995.1 1996.5 1994.9 1993 C
  7.4928 +f 
  7.4929 +S 
  7.4930 +n
  7.4931 +1913.8 2003.3 m
  7.4932 +1913.8 1996.9 1913.8 1990.5 1913.8 1984.1 C
  7.4933 +1913.8 1990.5 1913.8 1996.9 1913.8 2003.3 C
  7.4934 +f 
  7.4935 +S 
  7.4936 +n
  7.4937 +1941.9 1998 m
  7.4938 +1940.5 1997.3 1940.7 1999.4 1940.7 2000 C
  7.4939 +1942.8 2001.3 1942.6 1998.8 1941.9 1998 C
  7.4940 +[0 0 0 0]  vc
  7.4941 +f 
  7.4942 +S 
  7.4943 +n
  7.4944 +vmrs
  7.4945 +1942.1 1999.2 m
  7.4946 +1942.2 1998.9 1941.8 1998.8 1941.6 1998.5 C
  7.4947 +1940.4 1998 1940.7 1999.7 1940.7 2000 C
  7.4948 +1941.6 2000.3 1942.6 2000.4 1942.1 1999.2 C
  7.4949 +[0.92 0.92 0 0.67]  vc
  7.4950 +f 
  7.4951 +0.4 w
  7.4952 +2 J
  7.4953 +2 M
  7.4954 +S 
  7.4955 +n
  7.4956 +1940 1997.1 m
  7.4957 +1939.8 1996 1939.7 1995.9 1939.2 1995.2 C
  7.4958 +1939.1 1995.3 1938.5 1997.9 1937.8 1996.4 C
  7.4959 +1938 1997.3 1939.4 1998.6 1940 1997.1 C
  7.4960 +f 
  7.4961 +S 
  7.4962 +n
  7.4963 +1911.2 1995.9 m
  7.4964 +1911.2 1991.6 1911.3 1987.2 1911.4 1982.9 C
  7.4965 +1911.3 1987.2 1911.2 1991.6 1911.2 1995.9 C
  7.4966 +f 
  7.4967 +S 
  7.4968 +n
  7.4969 +1947.2 1979.1 m
  7.4970 +1945.1 1978.8 1944.6 1975.7 1942.4 1975 C
  7.4971 +1940.5 1972.6 1942.2 1973.7 1942.4 1975.7 C
  7.4972 +1945.8 1975.5 1944.2 1979.8 1947.6 1979.6 C
  7.4973 +1948.3 1982.3 1948.5 1980 1947.2 1979.1 C
  7.4974 +f 
  7.4975 +S 
  7.4976 +n
  7.4977 +1939.5 1973.3 m
  7.4978 +1940.1 1972.6 1939.8 1974.2 1940.2 1973.1 C
  7.4979 +1939.1 1972.8 1938.8 1968.5 1935.9 1969.7 C
  7.4980 +1937.4 1969.2 1938.5 1970.6 1939 1971.4 C
  7.4981 +1939.2 1972.7 1938.6 1973.9 1939.5 1973.3 C
  7.4982 +f 
  7.4983 +S 
  7.4984 +n
  7.4985 +1975.2 2073.2 m
  7.4986 +1975.2 2070.2 1975.2 2067.2 1975.2 2064.3 C
  7.4987 +1975.2 2067.2 1975.2 2070.2 1975.2 2073.2 C
  7.4988 +[0.18 0.18 0 0.78]  vc
  7.4989 +f 
  7.4990 +S 
  7.4991 +n
  7.4992 +1929.9 2065.7 m
  7.4993 +1928.1 2065.6 1926 2068.8 1924.1 2066.9 C
  7.4994 +1918.1 2060.9 1912.9 2055.7 1907.1 2049.9 C
  7.4995 +1906.7 2047.1 1906.9 2043.9 1906.8 2041 C
  7.4996 +1906.8 2043.9 1906.8 2046.8 1906.8 2049.6 C
  7.4997 +1913.2 2055.5 1918.7 2061.9 1925.1 2067.6 C
  7.4998 +1927.1 2067.9 1928.6 2064.4 1930.1 2066.2 C
  7.4999 +1929.7 2070.3 1929.9 2074.7 1929.9 2078.9 C
  7.5000 +1929.6 2074.4 1930.5 2070.1 1929.9 2065.7 C
  7.5001 +[0.07 0.06 0 0.58]  vc
  7.5002 +f 
  7.5003 +S 
  7.5004 +n
  7.5005 +1930.1 2061.6 m
  7.5006 +1928.1 2062.1 1927 2065.1 1924.8 2064.3 C
  7.5007 +1920.7 2058.9 1914.4 2054.3 1910.2 2049.2 C
  7.5008 +1910.2 2048.1 1910.2 2047.1 1910.2 2046 C
  7.5009 +1909.8 2046.8 1910 2048.3 1910 2049.4 C
  7.5010 +1915.1 2054.9 1920.3 2059 1925.3 2064.8 C
  7.5011 +1927.1 2064.2 1928.4 2062.3 1930.1 2061.6 C
  7.5012 +[0.18 0.18 0 0.78]  vc
  7.5013 +f 
  7.5014 +S 
  7.5015 +n
  7.5016 +1932 2049.9 m
  7.5017 +1932.3 2050.3 1932 2050.4 1932.8 2050.4 C
  7.5018 +1932 2050.4 1932.2 2049.2 1931.3 2049.6 C
  7.5019 +1931.4 2050.5 1930.3 2050.4 1930.4 2051.3 C
  7.5020 +1931.1 2051.1 1930.7 2049.4 1932 2049.9 C
  7.5021 +f 
  7.5022 +S 
  7.5023 +n
  7.5024 +1938.3 2046 m
  7.5025 +1936.3 2046.8 1935.2 2047.2 1934.2 2048.9 C
  7.5026 +1935.3 2047.7 1936.8 2046.2 1938.3 2046 C
  7.5027 +[0.4 0.4 0 0]  vc
  7.5028 +f 
  7.5029 +S 
  7.5030 +n
  7.5031 +vmrs
  7.5032 +1938.3 2047 m
  7.5033 +1937.9 2046.9 1936.6 2047.1 1936.1 2048 C
  7.5034 +1936.5 2047.5 1937.3 2046.7 1938.3 2047 C
  7.5035 +[0.18 0.18 0 0.78]  vc
  7.5036 +f 
  7.5037 +0.4 w
  7.5038 +2 J
  7.5039 +2 M
  7.5040 +S 
  7.5041 +n
  7.5042 +1910.2 2043.2 m
  7.5043 +1910.1 2037.5 1910 2031.8 1910 2026.1 C
  7.5044 +1910 2031.8 1910.1 2037.5 1910.2 2043.2 C
  7.5045 +f 
  7.5046 +S 
  7.5047 +n
  7.5048 +1933.5 2032.1 m
  7.5049 +1933.7 2035.2 1932.8 2035.8 1933.7 2038.6 C
  7.5050 +1933.3 2036.6 1934.6 2018 1933.5 2032.1 C
  7.5051 +f 
  7.5052 +S 
  7.5053 +n
  7.5054 +1907.3 2021.8 m
  7.5055 +1906.6 2025.9 1909.4 2032.6 1903.2 2034 C
  7.5056 +1902.8 2034.1 1902.4 2033.9 1902 2033.8 C
  7.5057 +1897.9 2028.5 1891.6 2023.8 1887.4 2018.7 C
  7.5058 +1887.4 2017.7 1887.4 2016.6 1887.4 2015.6 C
  7.5059 +1887 2016.3 1887.2 2017.8 1887.2 2018.9 C
  7.5060 +1892.3 2024.4 1897.5 2028.5 1902.5 2034.3 C
  7.5061 +1904.3 2033.6 1905.7 2032 1907.3 2030.9 C
  7.5062 +1907.3 2027.9 1907.3 2024.9 1907.3 2021.8 C
  7.5063 +f 
  7.5064 +S 
  7.5065 +n
  7.5066 +1933.7 2023.2 m
  7.5067 +1932 2021.7 1931.1 2024.9 1929.4 2024.9 C
  7.5068 +1931.2 2024.7 1932.4 2021.5 1933.7 2023.2 C
  7.5069 +f 
  7.5070 +S 
  7.5071 +n
  7.5072 +1989.2 2024.4 m
  7.5073 +1987.4 2023.7 1985.8 2022.2 1985.1 2020.4 C
  7.5074 +1984.6 2020.1 1986 2018.9 1985.1 2019.2 C
  7.5075 +1985.6 2020.8 1984.1 2019.4 1984.6 2021.1 C
  7.5076 +1986.3 2022.3 1988.1 2025.3 1989.2 2024.4 C
  7.5077 +f 
  7.5078 +S 
  7.5079 +n
  7.5080 +1904.4 2031.9 m
  7.5081 +1903 2029.7 1905.3 2027.7 1904.2 2025.9 C
  7.5082 +1904.5 2025 1903.7 2023 1904 2021.3 C
  7.5083 +1904 2022.3 1903.2 2022 1902.5 2022 C
  7.5084 +1901.3 2022.3 1902.2 2020.1 1901.6 2019.6 C
  7.5085 +1902.5 2019.8 1902.6 2018.3 1903.5 2018.9 C
  7.5086 +1903.7 2021.8 1905.6 2016.8 1905.6 2020.6 C
  7.5087 +1905.9 2020 1906.3 2020.8 1906.1 2021.1 C
  7.5088 +1905.8 2022.7 1906.7 2020.4 1906.4 2019.9 C
  7.5089 +1906.4 2018.5 1908.2 2017.8 1906.8 2016.5 C
  7.5090 +1906.9 2015.7 1907.7 2017.1 1907.1 2016.3 C
  7.5091 +1908.5 2015.8 1910.3 2015.1 1911.6 2016 C
  7.5092 +1912.2 2016.2 1911.9 2018 1911.6 2018 C
  7.5093 +1914.5 2017.1 1910.4 2013.6 1913.3 2013.4 C
  7.5094 +1912.4 2011.3 1910.5 2011.8 1909.5 2010 C
  7.5095 +1910 2010.5 1909 2010.8 1908.8 2011.2 C
  7.5096 +1907.5 2009.9 1906.1 2011.7 1904.9 2011.5 C
  7.5097 +1904.7 2010.9 1904.3 2010.5 1904.4 2009.8 C
  7.5098 +1905 2010.2 1904.6 2008.6 1905.4 2008.1 C
  7.5099 +1906.6 2007.5 1907.7 2008.4 1908.5 2007.4 C
  7.5100 +1908.9 2008.5 1909.7 2008.1 1909 2007.2 C
  7.5101 +1908.1 2006.5 1905.9 2007.3 1905.4 2007.4 C
  7.5102 +1903.9 2007.3 1905.2 2008.5 1904.2 2008.4 C
  7.5103 +1904.6 2009.9 1902.8 2010.3 1902.3 2010.5 C
  7.5104 +1901.5 2009.9 1900.4 2010 1899.4 2010 C
  7.5105 +1898.6 2011.2 1898.2 2013.4 1896.5 2013.4 C
  7.5106 +1896 2012.9 1894.4 2012.9 1893.6 2012.9 C
  7.5107 +1893.1 2013.9 1892.9 2015.5 1891.5 2016 C
  7.5108 +1890.3 2016.1 1889.2 2014 1888.6 2015.8 C
  7.5109 +1890 2016 1891 2016.9 1892.9 2016.5 C
  7.5110 +1894.1 2017.2 1892.8 2018.3 1893.2 2018.9 C
  7.5111 +1892.6 2018.9 1891.1 2019.8 1890.5 2020.6 C
  7.5112 +1891.1 2023.6 1893.2 2019.8 1893.9 2022.5 C
  7.5113 +1894.1 2023.3 1892.7 2023.6 1893.9 2024 C
  7.5114 +1894.2 2024.3 1897.4 2023.8 1896.5 2026.1 C
  7.5115 +1896 2025.6 1897.4 2028.1 1897.5 2027.1 C
  7.5116 +1898.4 2027.4 1899.3 2027 1899.6 2028.5 C
  7.5117 +1899.5 2028.6 1899.4 2028.8 1899.2 2028.8 C
  7.5118 +1899.3 2029.2 1899.6 2029.8 1900.1 2030.2 C
  7.5119 +1900.4 2029.6 1901 2030 1901.8 2030.2 C
  7.5120 +1903.1 2032.1 1900.4 2031.5 1902.8 2033.1 C
  7.5121 +1903.3 2032.7 1904.5 2032 1904.4 2031.9 C
  7.5122 +[0.21 0.21 0 0]  vc
  7.5123 +f 
  7.5124 +S 
  7.5125 +n
  7.5126 +1909.2 2019.4 m
  7.5127 +1908.8 2020.3 1910.2 2019.8 1909.2 2019.2 C
  7.5128 +1908.3 2019.3 1907.6 2020.2 1907.6 2021.3 C
  7.5129 +1908.5 2021 1907.6 2019 1909.2 2019.4 C
  7.5130 +[0.18 0.18 0 0.78]  vc
  7.5131 +f 
  7.5132 +S 
  7.5133 +n
  7.5134 +1915.5 2015.6 m
  7.5135 +1913.5 2016.3 1912.4 2016.8 1911.4 2018.4 C
  7.5136 +1912.5 2017.2 1914 2015.7 1915.5 2015.6 C
  7.5137 +[0.4 0.4 0 0]  vc
  7.5138 +f 
  7.5139 +S 
  7.5140 +n
  7.5141 +1915.5 2016.5 m
  7.5142 +1915.1 2016.4 1913.8 2016.6 1913.3 2017.5 C
  7.5143 +1913.7 2017 1914.5 2016.2 1915.5 2016.5 C
  7.5144 +[0.18 0.18 0 0.78]  vc
  7.5145 +f 
  7.5146 +S 
  7.5147 +n
  7.5148 +vmrs
  7.5149 +1887.4 2012.7 m
  7.5150 +1887.3 2007 1887.2 2001.3 1887.2 1995.6 C
  7.5151 +1887.2 2001.3 1887.3 2007 1887.4 2012.7 C
  7.5152 +[0.18 0.18 0 0.78]  vc
  7.5153 +f 
  7.5154 +0.4 w
  7.5155 +2 J
  7.5156 +2 M
  7.5157 +S 
  7.5158 +n
  7.5159 +1935.9 2007.4 m
  7.5160 +1936.2 2007.8 1935.8 2007.9 1936.6 2007.9 C
  7.5161 +1935.9 2007.9 1936.1 2006.7 1935.2 2007.2 C
  7.5162 +1935.2 2008.1 1934.1 2007.9 1934.2 2008.8 C
  7.5163 +1935 2008.7 1934.6 2006.9 1935.9 2007.4 C
  7.5164 +f 
  7.5165 +S 
  7.5166 +n
  7.5167 +1942.1 2003.6 m
  7.5168 +1940.1 2004.3 1939.1 2004.8 1938 2006.4 C
  7.5169 +1939.1 2005.2 1940.6 2003.7 1942.1 2003.6 C
  7.5170 +[0.4 0.4 0 0]  vc
  7.5171 +f 
  7.5172 +S 
  7.5173 +n
  7.5174 +1942.1 2004.5 m
  7.5175 +1941.8 2004.4 1940.4 2004.6 1940 2005.5 C
  7.5176 +1940.4 2005 1941.2 2004.2 1942.1 2004.5 C
  7.5177 +[0.18 0.18 0 0.78]  vc
  7.5178 +f 
  7.5179 +S 
  7.5180 +n
  7.5181 +1914 2000.7 m
  7.5182 +1914 1995 1913.9 1989.3 1913.8 1983.6 C
  7.5183 +1913.9 1989.3 1914 1995 1914 2000.7 C
  7.5184 +f 
  7.5185 +S 
  7.5186 +n
  7.5187 +1941.6 1998.3 m
  7.5188 +1943.4 2001.9 1942.4 1996 1940.9 1998.3 C
  7.5189 +1941.2 1998.3 1941.4 1998.3 1941.6 1998.3 C
  7.5190 +f 
  7.5191 +S 
  7.5192 +n
  7.5193 +1954.8 1989.9 m
  7.5194 +1953.9 1989.6 1954.7 1991.6 1953.9 1991.1 C
  7.5195 +1954.5 1993.1 1953.6 1998 1954.6 1993.2 C
  7.5196 +1954 1992.2 1954.7 1990.7 1954.8 1989.9 C
  7.5197 +f 
  7.5198 +S 
  7.5199 +n
  7.5200 +1947.6 1992.5 m
  7.5201 +1946.2 1993.5 1944.9 1993 1944.8 1994.7 C
  7.5202 +1945.5 1994 1947 1992.2 1947.6 1992.5 C