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Untyped Algorithmic Equality for Martin-Lof's Logical Framework with Surjective Pairs

Summary: Untyped Algorithmic Equality for Martin-L¨of's
Logical Framework with Surjective Pairs
Andreas Abel and Thierry Coquand
Department of Computer Science, Chalmers University of Technology
Abstract. An untyped algorithm to test -equality for Martin-L¨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 Martin-L¨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 non-confluent [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


Source: Abel, Andreas - Theoretische Informatik, Ludwig-Maximilians-Universität München


Collections: Computer Technologies and Information Sciences