Proof-relevant Horn clauses for dependent type inference and term synthesis

First-order resolution has been used for type inference for many years, including in Hindley-Milner type inference, type-classes, and constrained data types. Dependent types are a new trend in functional languages. In this paper, we show that proof-relevant first-order resolution can play an important role in automating type inference and term synthesis for dependently typed languages. We propose a calculus that translates type inference and term synthesis problems in a dependently typed language to a logic program and a goal in the proof-relevant first-order Horn clause logic. The computed answer substitution and proof term then provide a solution to the given type inference and term synthesis problem. We prove the decidability and soundness of our method.