By [edited by] Haskell B. Curry, J. Roger Hindley, Jonathan P. Seldin.

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In Q 4 we shall reconsider the classification of obs according to order and degree. A specialization of weak reduction, called canonical reduction, will next be treated in Q 5. Finally in Q 6 we shall introduce some techniques for abbreviating complex combinators when equality is not postulated ; these are occasionally useful in the more technical portions of Q C and Q E. The discussion of the bracket prefix, which belongs in principle under weak reduction, will be reserved for Q C . 1. Formulation We shall begin by recapitulating the main features of the system and conventions about it, at the same time making some minor changes.

If i < n, let 2;+lbe formed by contracting the head redex in Z:. + by contracting R'; in either case Z;+ has properties analogous to those postulated for Z;. Thus we continue until we reach Zi, when DL stops. Let D; be the internal reduction from 2: to 2, (= Y ) . d. , THEOREM 5. ) If X Z Y (1 1) then there is a stundurd reduction from X to Y. Proof. By Theorem 4, we may suppose that the given reduction (1 1) is semistandard. Let Z be the last stage which is reached by head contractions alone. up, 3 UOV,.

When and only when there is a sequence X,, X , , (8) x =X, >1 X, ... > 1 n 2 0, such that Such a sequence will be called a reduction; X , will be called the kth stage, and the contraction forming X , (k > 0) its kth step. The number of steps, n, 15. Due to Lercher [SRR]. 11B] THE WEAK THEORY OF COMBINATORS 29 will be called the length of the reduction. A Y for which a reduction (8) exists will sometimes be called a reductum of X . e. 16 Let R be a redex in X with contracturn T, and let Y be the result of replacing R by T i n X.