number-theory

1D Restricted A-terms 11(ii) The BCKA-terms are so called because the closed terms in this class correspondto combinations of three combinators called "B", "C" and "K" in combinatory logic(see 9F for details). They have also sometimes been called linear A-terms but thisname is nowadays usually applied to the following class.1D3 Definition (BCIA-terms) A BCIA-term or linear A-term is a A-term P such that(i) for each subterm of P, x occurs free in M exactly once,(ii) each free variable of P has just one occurrence free in P.Clearly every BCIA-term is a BCKA-term, but the BCKA-term K is not a BCI1.-term; in fact a term is a BCIA-term if it is both a Al-term and a BCKA-term. Theclosed BCIA-terms correspond to combinations of the combinatory called B, C andI in combinatory logic; details are in 9F.1D4 Lemma Each of the three classes (Al-terms, BCKA-terms and BCIA-terms) isclosed under f q-reduction, i.e. every term obtained by flu-reducing a member of theclass is also in the class.Proof Straightforward.1D5 Definition A f3-contraction[N/x]M is said to cancel N if x doesnot occur free in M; it is said to duplicate N if x has at least two free occurrencesin M.A #-reduction is non-duplicating if none of its contractions duplicates; it isnon-cancelling if none cancels.1D6 Lemma Every f3-reduction of a AI-term is non-cancelling; every one of a BCKAtermis non-duplicating, and every one of a BCIA-term is both.