NFP + Axiom of sumset = NF.
Note that one-step reduction does not need to decrease the total number of redexes that a term contains, hence, it does not follow that every term can be turned into one in nf via finitely many one-step reductions.
The term S KS K is in nf, however, to be in nf is not a requirement for definability.
Not all terms have an nf, however, many do.
Examples, first of all, include S and K. (The variables, if included, of which there are ℵ0 many, are all in nf).
A term is in normal form (nf) when it contains no redexes.
It is more convenient to work with defining terms that are in nf, because an application of a combinator that is not in nf could be started with reducing the combinator to its normal form.
However, note that the preference for choosing combinators in nf is not meant to imply that a combinator cannot be defined by two or more terms in nf.
The contraction of the only redex in WWW returns the same term, which shows that uniqueness does not imply that the term is in nf.
Y KI) reduces to KI(Y KI)), KI KI Y KI, KI KI KI Y KI(KI)))), … as well as to I. A term weakly normalizes when it has an nf, whereas a term strongly normalizes when all its reduction sequences lead to an nf (hence, to the nf) of the term.
Moreover, in NF the universal set exists.
