 
Summary: Untyped Algorithmic Equality for MartinL˜of's
Logical Framework with Surjective Pairs
Andreas Abel # and Thierry Coquand
Department of Computer Science, Chalmers University of Technology
abel,coquand@cs.chalmers.se
Abstract. An untyped algorithm to test ##equality for MartinL˜of's
Logical Framework with strong #types is presented and proven complete
using a model of partial equivalence relations between untyped terms.
1 Introduction
Type checking in dependent type theories requires comparison of expressions
for equality. In theories with #equality, an apparent method is to normalize
the objects and then compare their #normal forms syntactically. In the theory
we want to consider, an extension of MartinL˜of's logical framework with ##
equality by dependent surjective pairs (strong # types), which we call MLF# ,
a naive normalize and compare syntactically approach fails since ##reduction
with surjective pairing is known to be nonconfluent [Klo80].
We therefore advocate the incremental ##convertibility test which has been
given by the second author for dependently typed #terms [Coq91,Coq96], and
extend it to pairs. The algorithm computes the weak head normal forms of the
conversion candidates, and then analyzes the shape of the normal forms. In case
