In this paper, we present an existing and formalized type theory
(UTT) as a logical framework.
We compare the resulting framework with LF and give
the representation of two significant type systems in
the framework: the typed lambda calculus which is closely related
to higher-order logic and a linear type system which is not
possible to encode in LF.
Mylonakis, N. "Adequate encodings of logical systems in UTT". 2003.