Extended Natural Deduction Images of Conversions from the System of Sequents

Maximum cuts in extended natural deduction

“…In [6] Gentzen introduced a system of sequents for intuitionistic predicate logic, the system LJ, and a natural deduction system for intuitionistic predicate logic, the system NJ. In the papers [2,4,6,8,9,10,11,15] 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 most of these papers was to connect the cut-elimination theorem from systems of sequents and the normalization theorem from natural deduction systems, the most important characteristics of these systems.…”

“…It is well known that there are problems to connect reduction steps of the cut-elimination procedure from a system of sequents and reduction steps of the normalization procedure from a natural deduction system when these systems cover full intuitionistic predicate logic (see, for example, the part 7 in [15]). To solve these problems the authors connected some modifications of Gentzen's systems LJ and NJ (see [2,8,9,10,15]), or they defined new reduction steps in cut-elimination and normalization procedures (see [2,4,10,15]). In some papers mentioned above (see, for example, [10,15]) it was concluded that "the cut-elimination theorem and the normalization theorem are equivalent".…”

“…In this paper a new pair systems will be considered. Our systems will be the systems from [2] which cover intuitionistic predicate logic: the system of sequents ℰ and the natural deduction system ℰ. It will be shown that the image of a sequent derivation without maximum cuts from the system ℰ is a normal derivation in the system ℰ.…”

“…The system ℰ is the system ℰ from [2] whose formulae have only upper indices. The system ℰ is a modification of the following systems: Gentzen's system NJ from [6] (i.e., Prawitz's system from [11]) and Zucker's system from [15].…”

Normal form theorem for systems of sequents

“…In [5] Gentzen introduced a natural deduction system for intuitionistic predicate logic, the system NJ, and a system of sequents for intuitionistic predicate logic, the system LJ. There are several papers [1,3,5,7,8,9,10,13] in which natural deduction systems and systems of sequents for some fragments of intuitionistic logic are compared. The most important connection between these systems is the connection between normal derivations, i.e.…”