Generalized - reduction and explicit substitutions

Extending the-calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitutions have never been studied. This paper presents such a calculus sg and shows that it is a desirable extension of the-calculus. In particular, we show that sg preserves strong normalisation, is sound and it simulates classical-reduction. Furthermore, we study the simply typed-calculus extended with both generalised reduction and explicit substitution and show that well-typed terms are strongly normalising and that other properties such as subtyping and subject reduction hold. 1 Introduction 1.1 The-calculus with generalised reduction In ((x :: y :N)P)Q, the function starting with x and the argument P result in the redex (x :: y :N)P which when contracted will turn the function starting with y and Q into a redex. This fact has been exploited by many researchers and reduction has been extended so that the future redex based on the matching y and Q is given the same priority as the other redex. Attempts at generalising reduction can be summarized by three axioms: These (related) rules attempt to make more redexes visible. C for example, makes sure that y and Q form a redex even before the redex based on x and P is contracted. Due to compatibility, implies C. Moreover, ((x :: y :N)P)Q ! (x :(y :N)Q)P and hence both and C put adjacently next to its matching argument. moves the argument next to its matching whereas C moves the next to its matching argument. can be equally applied to explicitly and implicitly typed systems. The transfer of or C to explicitly typed systems is not straightforward however, since in these systems, the type of y may be aaected by the reducible pair x ; P. For example, it is ne to write ((x: :: y:x :y)z)u ! (x: :(y:x :y)u)z but not to write ((x: :: y:x :y)z)u ! C (y:x :(x: :y)z)u. For this reason, we study-like rules in this paper. Now, we discuss where generalised reduction has been used (cf. 25]). 31] introduces the notion of a premier redex which is similar to the redex based on y and Q above (which we call generalised redex). 32] uses and (and calls the combination