First-order logic with dependent types

We present DFOL, an extension of classical first-order logic with dependent types, i.e., as in Martin-Lof type theory, signatures may contain type-valued function symbols. A model theory for the logic is given that stays close to the established first-order model theory. The logic is presented as an institution, and the logical framework LF is used to define signatures, terms and formulas. We show that free models over Horn theories exist, which facilitates its use as an algebraic specification language, and show that the classical first-order axiomatization is complete for DFOL, too, which implies that existing first-order theorem provers can be extended. In particular, the axiomatization can be encoded in LF.