number-theory

8A Inhabitants113Fig. 8A12a.8A11.2 Lemma Let MT+ be the unique member of Long(T) to which MTri-expands(see 8A8). ThenMT E Nprinc(T) MT+ E Nprinc(T).Proof The it-expansion in the proof of 8A8 preserves principality because of theway the types given to zl, ... , zk are determined by the type v of the component thatis replaced.8A12 Comment (Sets of of s) Three sets of fl-nf's have been defined so far in thissection, namelyNhabs(T), Long(T), Nprinc(T),and the sets of all f t -nf's in these sets will be called respectivelyNhabs,(T), Long,(T), Nprinc,,(T).To clarify the relations between these six sets consider the typeT - (a-+a-->a)->a->a-*a.(See Fig. 8A12a.) For this T the six sets are all distinct and in fact there is a term inevery space in Fig. 8A12a except one. In detail:(i) Axa_a-+a.xa _ a-.a(ii)A,xa-'a-'aya.(xy)a-'aE Nhabs,, - (Nprinc U Long);E Nhabs - Nhabs,, - (Nprinc U Long);(iii)1xa-'a-+a yaza . (xyZ)aE Long - Nhabs, - Nprinc ;(iv) 1xa-.atiaya.(x(xyy))a-a E Nprincn - Long,,;(v) 2xa-.a-ayaZa.(x(xyy)z)a E Nprinc fl Long - Nhabs,;(vi)(vii).xa-.a-.ayaZa.(xZ(xyy))a EENprinc,, fl Long, ;Long, - Nprinc,,.