  Decidability of definitional equality and conversion of terms into canonical
form play a central role in the meta-theory of a type-theoretic logical
framework. Most studies of definitional equality are based on a confluent,
strongly-normalizing notion of reduction. Coquand has considered a different
approach, directly proving the correctness of a practical equivalance algorithm
based on the shape of terms. Neither approach appears to scale well to richer
languages with unit types or subtyping, and neither directly addresses the
problem of conversion to canonical.
  In this paper we present a new, type-directed equivalence algorithm for the
LF type theory that overcomes the weaknesses of previous approaches. The
algorithm is practical, scales to richer languages, and yields a new notion of
canonical form sufficient for adequate encodings of logical systems. The
algorithm is proved complete by a Kripke-style logical relations argument
similar to that suggested by Coquand. Crucially, both the algorithm itself and
the logical relations rely only on the shapes of types, ignoring dependencies
on terms.
