Peirce's Rule in a Full Natural Deduction System

Natural deduction (ND) for first order classical logic is obtainable from the intuitionist system by the addition of Peirce rule. In a previous paper [Luiz Carlos Pereira, Edward Hermann Haeusler, Vaston G. Costa, and Wagner Sanz. Normalization for the implicational fragment of classical propositional logic. Submited to a Journal, 2008], it was presented a normalization strategy for the implicational fragment with Pierce rule. The end normal form is divided in two parts: an intuitionist subdeduction followed by a series of Peirce rule applications, maybe empty. Here, we extend this normalization process to the system for first order classical logic NP system. NP normal derivations also present the same structure. This structure is the basis on which many properties for ND derivations can be presented. In particular, we present a form of Glivenko's theorem for the conjunction-implication fragment. Unfortunately, NP lacks strong normalization, although ND with classical absurdity rule doesn't lack it, as it's well-known.