We propose a new relation called strong conformance in the context of
Dill's trace theory, and define B strongly conforms to A to be true
exactly when B conforms to A and the success set of B contains the
success set of A. When B strongly conforms A, module B operated in
module A's maximal environment exhibits all the traces that A ||
\overline{A} exhibits. In addition, if A has a success trace x, B can
have additional success traces of the form xi\alpha^{*} where i is an
input and \alpha is the alphabet of the trace structure.  This means
that B can have additional capabilities that A does not. We show that
strong conformance is capable of detecting certain errors in
asynchronous circuits that cannot be detected through conformance
(defined by Dill).  Strong conformance also helps justify circuit
optimization rules that replace a component A by another component B
that may have extra capabilities ({\em e.g.} can accept more inputs).
The structural operators {\em compose}, {\em rename}, and {\em hide}
of Dill's trace theory are shown to be monotonic with respect to {\em
strong conformance}.  Experiments using a {\em modified} version of
Dill's trace theory verifier that implements the check for strong
conformance are presented.