  Dependently typed programs contain an excessive amount of static terms which
are necessary to please the type checker but irrelevant for computation. To
separate static and dynamic code, several static analyses and type systems have
been put forward. We consider Pfenning's type theory with irrelevant
quantification which is compatible with a type-based notion of equality that
respects eta-laws. We extend Pfenning's theory to universes and large
eliminations and develop its meta-theory. Subject reduction, normalization and
consistency are obtained by a Kripke model over the typed equality judgement.
Finally, a type-directed equality algorithm is described whose completeness is
proven by a second Kripke model.
