number-theory

8A Inhabitants 109A Pq-normal inhabitant of a type is an inhabitant in /3rl-nf. The sets of all typed anduntyped fl q-normal inhabitants of T will both be calledNhabs,(r ).A type with at least one inhabitant is said to be inhabited.As remarked earlier the aim of this chapter is to count /3-normal inhabitants.The following lemma will show that it does not matter whether we count typed oruntyped inhabitants.8A2 Lemma If MT E Nhabst(T) then Mf E Nhabs (T); further, the type-erasingmapping is a one-to-one correspondence between the typed and the untyped #-normalinhabitants Of T (modulo The same holds for /3,1-normal inhabitants.Proof Let M E $-nf; then M inhabits T if there exists a proof A of H M:T; andby 2B3 this A is uniquely determined by M. And such proofs correspond one-to-onewith typed closed terms by 5A7.8A2.1 Notation All terms in this chapter will be typed unless explicitly statedotherwise. But for ease of reading they will often be written with some or all oftheir types omitted.8A3 Definition The number (0, 1, 2.... or oo) of members of a set S, counted moduloma if § is a set of A-terms, is called the cardinality of S or#(S).For #(Nhabs(T)) and #(Nhabs,(i)) we shall usually say just#(T),#n(T).8A4 Definition (Counting, enumerating) A distinction will be made in this bookbetween counting and enumerating a set S of #-normal inhabitants of a type T:(i) to count S will mean to compute #(S) after a finite number of steps (evenwhen #(S) = oo);(ii) to enumerate or list S will mean to enumerate S in the usual recursiontheoreticsense, i.e. to output a sequence consisting of all the members of S (and nonon-members!), continuing for ever if S is infinite.8A4.1 Comment The aim of Ben-Yelles' algorithm is to count inhabitants as wellas enumerate them. Mere enumeration would be easy: we could simply list allclosed typed fl-nf's in some standard order and for each one decide whether it is aninhabitant of a given T by looking at its type. But counting is not so easy: we mustfind a way of enumerating Nhabs(T) which will tell us after only a finite number ofsteps whether the enumeration will continue for ever or not.The strategy will be to do the counting in order of increasing depth in a sense tobe defined below, and to first count certain inhabitants called long/3-nf's from which