A short proof of Curry’s normal form theorem

Abstract

In Chapter 6 of their book [l] Curry and Feys define a notion of reduction (strong reduction) for the extensional theory of equality in combinatory logic, show [l, Theorem 3, p. 221 ] that strong reduction has the Church-Rosser property, and define a notion of normal form in analogy with the corresponding concept in lambda-conversion. Curry's normal form theorem [l, Theorem 7, p. 230] asserts that if a term ("ob") of combinatory logic is in normal form, it is irreducible, so that if X has normal form X*, then X reduces to X* by a process (namely, strong reduction) that cannot be continued further.Curry's proof of his theorem in [l] is quite long and difficult (see [3, p. 228] for comment). There is another lengthy proof in the current draft of [2] and the first author has discovered a proof using his axiomatization of strong reduction [3]. The present proof follows the same general line as the latter proof, but it is considerably shorter and simpler. Definitions and notation are as in [3], except for the symbol E> which is used here for weak reduction.1. Substitution and abstraction. The first result is essentially from [4, Lemmas 1 and 4]. The proofs, by induction, are easy.Lemma 1. Let P be a redex scheme. Then: (a) P contains at most one occurrence of each meta-variable; (b) If M is a meta-variable occurring in P, there is an N such that NM occurs in P; (c) If P is not basic (i.e. is the result of at least one application of scheme (viii) of [3, p. 233]), then P is weakly irreducible;(d) If P is not basic, then either P = SPiP2 or P=SPi where Pi contains at least one occurrence of an atomic combinator.The hypotheses of the next lemma are, in essence, the properties of redex schemes asserted in Lemma 1(a), (b), and (c).