摘要：We define a shallow embedding of logical proof-irrelevant Pure Type Systems (piPTSs) into minimal first-order logic. In logical piPTSs a distinguished sort *^p of propositions is assumed. Given a context Gamma and a Gamma-proposition tau, i.e., a term tau such that Gamma |- tau : *^p, the embedding translates tau and Gamma into a first-order formula F_Gamma(tau) and a set of first-order axioms Delta_Gamma. The embedding is not complete in general, but it is strong enough to correctly translate most of piPTS propositions (by completeness we mean that if Gamma |- M : tau is derivable in the piPTS then F_Gamma(tau) is provable in minimal first-order logic from the axioms Delta_Gamma). We show the embedding to be sound, i.e., if F_Gamma(tau) is provable in minimal first-order logic from the axioms Delta_Gamma, then Gamma |- M : tau is derivable in the original system for some term M. The interest in the proposed embedding stems from the fact that it forms a basis of the translations used in the recently developed CoqHammer automation tool for dependent type theory.
关键词：pure type systems; first-order logic; hammers; proof automation; dependent type theory
