number-theory

156 9 Technical details9E6.3 Lemma If T is composite, then(i) #(Subtails(T))no. of composite s-components of r,(ii)1 + no. of composite sub premises ofT,(iii)ITI - no. of atomic sub premises of T,(iv)ITI -(v) #(Subpremises(T))ITI -(vi) no. of s-components of T < 21TI-1.Proof For (i): use 9E5.1. For (ii): each composite s-component is either a subpremiseor r itself. For (iii): use 9E6.2(ii). For (iv): use 9E6.2(iii). For (v): subtract (iii) from(ii). For (vi): use 9E6.2(ii), adding (iv) to (v) and adding 1 for T itself.9E7 Definition Order(T), the order of r, is 1 + the length of the longest position onthe condensed tree of T. In detail: Order(e) = 1 for atoms e, and for composite typesOrder9E7.1 Example Order((a->b-+c)-*(a-*b)--*a-*c) = 3.1+Max {Order(Ti),...,Order(T)}.9E8 Definition (Positive and negative s-components) An s-component a of r is calledpositive or negative according as the number of non-asterisk symbols in its positionis even or odd. If v is positive we say a occurs positively in T, otherwise a occursnegatively in T.9E8.1 Example If r - (a-+b-.c)->(a->b)-*a-*c, see Fig. 9E2.1a, its positive s-components are(r, 0, r), (c, *, r), (a, 32, r), (b, 31, r), (a, 21, r).9E8.2 Notes (i) It is straightforward to show that an s-component a is positive ornegative according as the corresponding component in the more usual sense (9D3)is positive or negative.(ii) A subpremise of T is positive if its position has even length. (Because theposition of a subpremise contains no *'s.)The following set plays a role in Chapter 8.9E9 Definition (NSS(T)) (cf. Ben-Yelles 1979 Def. 3.36.) If r is composite, NSS(T)is the set of all finite sequences(n >: 1) such that T contains a positivecomposite s-component with formfor some atom a. Each member of NSS (T) is called a negative subpremise-sequence(because it is a sequence of terms that have occurrences as negative subpremises inT).The set of all the members of the sequences in NSS(T) will be calledU NSS (r).