 
Summary: A ThirdOrder Representation of the
Calculus
Andreas Abel ?
Carnegie Mellon University, Department of Computer Science
5000 Forbes Avenue, Pittsburgh, PA 15213, USA
phone: +1(412)2682582, email: abel@cs.cmu.edu
Abstract. Higherorder logical frameworks provide a powerful technol
ogy to reason about object languages with binders. This will be demon
strated for the case of the calculus with two dierent binders which
can most elegantly be represented using a thirdorder constant. Since
cases of third and higherorder encodings are very rare in comparison
with those of second order, a secondorder representation is given as well
and equivalence to the thirdorder representation is proven formally.
1 Introduction
The calculus [Par92,OS97,Bie98], a proof theory for the implicational frag
ment of classical logic, has been established as a general tool to reason about
functional programming languages with control, e.g. continuations and excep
tions. It is basically an extension of the calculus by a second binder. Some of
its properties like strong normalization and con
uence are very fundamental for
its use in functional programming and proof systems; a formal verication of
