\section{Computational Appendices}

We believe in implementing as much as possible. Actually, we hardly believe ourselves unless we implement.

All code in these appendices is written in {\sl Mathematica}~\cite{Wolfram:Mathematica}.

\def\face{\input{figs/face.pdf_t}}
\def\human{\input{figs/human.pdf_t}}
\def\machine{\input{figs/machine.pdf_t}}

\def\cellscale{1}
\def\nbpdfInput#1{{\noindent{\imagetop{\face}\ \imagetop{\includegraphics[scale=\cellscale]{#1}}\vskip 2mm}}}
\def\nbpdfOutput#1{{\noindent{\imagetop{\machine}\ \imagetop{\includegraphics[scale=\cellscale]{#1}}\vskip 2mm}}}
\def\nbpdfliInput#1{{\noindent{\imagetop{\human}\ \imagetop{\includegraphics[scale=\cellscale]{#1}}\vskip 2mm}}}
\def\nbpdfliOutput#1{{\noindent{\imagetop{\machine}\ \imagetop{\includegraphics[scale=\cellscale]{#1}}\vskip 2mm}}}

\subsection{Computational Verification of Theorem~\ref{thm:GDO},~(i)} \label{app:GDOCompositions}

\input GDOCompositions.tex