A Proof of Strong Normalization for Generalized Labelled -reduction by Means of the Successor Relation Method Acknowledgements. I Want to Thank Femke Van Raamsdonk for Her Encouragement and For

Introduction The so-called successor relation method was introduced by Sanchis S] in order to prove results, in particular strong normalization, concerning a certain notion of reduction (in connection with GG odel's system T). The method has been further elaborated by Howard H] and in T] it has been exploited to prove semantical results concerning terms representing ordinal recursive functionals. The present paper contains another useful application of the method | now on the purely syntactical level again. The result to be proved applies to Hyland's and Wadsworth's system HW and certain generalizations of it. This system HW is a variant of the ordinary (untyped)-calculus with so-called labelled-reduction. Various proofs of strong normalization, \SN", are known for HW (see, e.g., B], K] and also V, Stelling I ]). Here the SN-property will be proved for any of the intended generalized versions. The nature of any of these generalized versions can globally be explained as follows. In the case of HW , the labels are the natural numbers and the reduction relation depends on the ordinary ordering < of these. In the generalization, the labels come from an arbitrarily given set L and the reduction relation depends on an arbitrarily given well-ordering of L | in a similar way as the original reduction relation depends on <. The set of terms of this generalized system (the so-called L-labelled-terms) is denoted by L. The corresponding version of generalized labelled-reduction is denoted by ! (L;) , or simply by !. Here follows a global description of the structure of the SN-proof for such a system ((L ;!). It is to be proved that every term A 2 L is strongly !-normalizing (or \!-SN", for short); this property means that there exists no innnite reduction sequence A 0 !A 1 !A 2 !A 3 !: : :. For the purpose of the proof we deene a so-called successor relation on the set L. This relation looks partly like ! and by methods that are fairly standard it is proved that the following holds for each A 2 L : (1) if A is-SN, then A is also !-SN. So it will be suucient to prove that each A 2 L is-SN. For this purpose we deene, for each 2 L, the notion semi-successor relation of rank. These semi-successor relations are to be considered as \approximations from above" of the successor relation : 1 (2) if …