 
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
