number-theory

132 8 Counting a type's inhabitants8E7.1 Corollary If Xt is a closed long typed nf-scheme, the type of each meta-variablein XT either occurs as a positive subpremise of T or is T itself.Proof By 8E2.2(i) and 8E7(i).The main effect of 8E7 is to connect IAT(Z°), which in general depends on thestructure of ZI and hence implicitly on that of XT, with NSS(T) which depends on rand nothing else. The next corollary will use this to deduce reasonably neat boundsfor IA(Z°) and CA(Z,XT).8E7.2 Corollary If XT is a closed long typed nf-scheme and Z° is a subargument ofXT or Z° XT, then(i) Length(IA(Z°)) = Length(IAT(Z')) < ITI - 1,(ii) Length (CA(Z,XT)) < (1r - 1) x Depth(XT).Further, if 2v°',...,Av°' are all the abstractors in XT (not just its initial ones), then(iii) {pi...., pr} has < ITI - 1 distinct members.Proof For (i): Length (IAT(Z°)) < ITI - 1 by 8E7(iii) and 9E9.3(iv).For (ii): If Z - X the left side of (ii) is 0. If Z # X let (Zo,... , Z) (k >- 1) be theargument-branch from X to Z; then by 8E5.1(iv)Length(CA(Z,X)) = Length(IA(Zo)) +< k(JTJ - 1)+ Length(IA(Zk_l))by (i). But Depth(X) > k by 8E4.1(ii), so (ii) holds.For (iii): Each pi is in IAT(XT) or in IAT(YB) for some subargument YB of XT;and in both cases pi E U NSS (T) (in the former case trivially, and in the latter caseby 8E7(iii)). Then use 9E9.3(iii).8E7.3 Exercise* Show that if T is composite and d >_ 1 and .nI(,r,d) is defined, then(i) each XT E(ii) #(_q/(T, d)) < (d xd) contains < T1d meta-variables,ITI)(IT1"') x #(S/(T, d - 1)),(iii) #(-%/(T, d)) < 1 x 2ITI x 3(1T12) x ... x d(ITI`'-') x TJ(1+JTJ+IT12+ +1T1d-').8F Stretching, shrinking and completenessThis section fills in the three gaps that were left in the verification of the countingalgorithm in 8C-D: the stretching and shrinking lemmas and the "completeness"part of the search theorem.8F1 Search-Completeness Lemma Part (iii) of the search theorem 8C5 holds; i.e. ifT is composite and d > 0, thenLong (T,d) 9 d(T, < d + 1).