Maximum Segments and Cuts

In the papers [1, 2, 4–6, 8] the similarities and differences between systems of sequents and systems of natural deduction for some fragments of intuitionistic logic were presented. The main goal of almost all of these papers was to study the connection between cut-free derivations and normal derivations. It is a well-known property that the natural deduction image of a cut-free derivation is a normal derivation (see, for example, Theorem 3 from Section 5 in [8]). In this paper we will show that even if a sequent derivation has some cuts, its natural deduction image is a normal derivation. Our systems will be the system of sequents (the system S) and the natural deduction system (the system N ) for intuitionistic predicate logic, which were defined in [8].(They are very similar to Gentzen’s systems LJ and NJ from [2].) In order to connect derivations of these systems we will use the map φ from the set of derivations of S onto the set of derivations of N (which was also defined in [8]). In derivations of the system S we will define a special kind of cuts, maximum cuts, which correspond to maximum segments in derivations of natural deduction. Maximum cuts will be the cuts whose left cut formula ”is connected” with the principal formula of a right rule (i.e. ”an introduction of a connective or a quantifier”) and its right cut formula ”is connected” with the principal formula of a left rule (i.e. ”an elimination of the connective or the quantifier”). The main result of the paper is the following: if a derivation D from the system S has some cuts which are not maximum cuts, then its image in the system N , the derivation φD, is a normal derivation.