A Complete Proof Synthesis Method for the Cube of Type Systems

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Abstract
We present a complete proof synthesis method for the eight type systems of Barendregt's cube extended with $\eta$-conversion. Because these systems verify the proofs-as-objects paradigm, the proof synthesis method is a one level process merging unification and resolution. Then we present a variant of this method, which is incomplete but much more efficient. At last we show how to turn this algorithm into a unification algorithm.
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