We consider the EFO fragment of simple type theory, which restricts quantification and equality to base types but retains lambda abstractions and higher-order variables. We show that this fragment enjoys the characteristic properties of first-order logic: complete proof systems, compactness, and countable models. We obtain these results with an analytic tableau system and a concomitant model existence lemma. All results are with respect to standard models. The tableau system is well-suited for proof search and yields decision procedures for substantial fragments of EFO.
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