number-theory

128 8 Counting a type's inhabitants.Tree forvm n Y0 1j n-11vYlmn-1O 10 1 2Tree forvYl ..YnX xl...xIn(vYI ...Yn)0Fig. 8131a.will lay the foundation by analysing the structure of an arbitrary long typed nfscheme.A key role will be played by a slightly strengthened form of the subformulaproperty (2B3.1). That property says in effect that the types of all the componentsof a closed fl-nf Mi are subtypes of r, and this implies that all the successesproduced by the search algorithm, growing deeper and deeper, have the types oftheir components drawn from the same fixed finite set. This limitation is the sourceof the bounds in the stretching and shrinking lemmas.8E1 Notation Recall that a nf-scheme is essentially a $-nf that may contain metavariablesunder certain restrictions, see Definitions 8C1 (untyped) and 8C3 (typed).The early parts of the present section will apply to both typed and untyped nfschemes,so types will be omitted when nf-schemes are written. But later parts willapply only to typed nf-schemes and types will then be displayed.We shall need the notation for positions, components and construction-treesintroduced in 9A1-4.In writing positions a sequence of n 0's may be written as on (with 00 andsimilarly for l's and 2's.Recall from 8A5 that every non-atomic nf-scheme X can be expressed uniquelyin the form(1) X - (m+n>_ 1),where v is one of xl,...,xn if X is closed. The construction-tree of such an X isshown in Fig. 8Ela. The head and arguments of X are v_ and Y1,...,Yn. (If X is anatom its head is X and it has no arguments.) Note that the position of L is(2) 0"`1n-`2 (1 < i < n).