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\title{Locally Nameless Sigma Calculus}
\author{Ludovic Henrio and Florian Kamm\"uller and Bianca Lutz and Henry Sudhof}
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\begin{abstract}
We present a Theory of Objects based on the original functional
$\varsigma$-calculus by Abadi and Cardelli \cite{AC96a} but with an
additional parameter to methods. We prove confluence of the operational
semantics following the outline of Nipkow's proof of confluence for the
$\lambda$-calculus reusing his general \texttt{Commutation.thy}
\cite{nip:01} a generic diamond lemma reduction. 
We furthermore formalize a simple type system for our $\varsigma$-calculus
including a proof of type safety.
The entire development uses the concept of Locally Nameless representation
for binders \cite{ACPPW:POPL08}. We reuse an earlier proof of confluence
\cite{HK:FMOODS07} for a simpler $\varsigma$-calculus based on de Bruijn indices
and lists to represent objects. 
\end{abstract}

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\bibitem{AC96a}
M. Abadi and L. Cardelli.
\newblock {``A Theory of Objects''}.
\newblock Springer, New York, 1996.


\bibitem{ACPPW:POPL08}
B. Aydemir, A, Charguéraud, B.~C. Pierce, R. Pollack, and
  S. Weirich.
\newblock Engineering formal metatheory.
\newblock {\em Princ. of Programming Languages, POPL'08}, ACM, 2008. 

\bibitem{HK:FMOODS07}
L. Henrio and F. Kammüller.
\newblock A mechanized model of the theory of objects.
\newblock {\em Formal Methods for Open Object-Based Distributed Systems,}. 
LNCS \textbf{4468} Springer, 2007.


\bibitem{nip:01}
Tobias Nipkow.
\newblock{More Church Rosser Proofs.}
\newblock{\em Journal of Automated Reasoning}. 
\newblock{\textbf{26}:51--66, 2001.}
\end{thebibliography}

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