  Logical frameworks are successful in modeling proof systems. Recently, CoLF
extended the logical framework LF to support higher-order rational terms that
enable adequate encoding of circular objects and derivations. In this paper, we
propose CoLF$^\omega$ as an alternative interpretation of CoLF-style signatures
where terms are taken to be all possibly infinitary terms that are consistent
with a given signature. In particular, we propose the notion of productive
B\"ohm trees, a particular kind of typed $\bot$-free B\"ohm trees that are
closed under hereditary substitution. We show that the productive B\"ohm trees
are capable of meta-encoding their own structure. Overall, we hope to establish
CoLF$^\omega$ as a new formal framework for the encoding of infinitary regular
and non-regular structures.
