number-theory

9E The condensed structure of a type 1559E4 Definition (S-components) Iff a node on the condensed tree of T is labelled witha type a and a position p we call the triple (a, p, T) an s-component of -r. (Thus ans-component is a particular occurrence of an s-subtype.)9E4.1 Notation S-components are distinguished from s-subtypes by underliningtheir names. The phrases "p contains g", "p properly contains a" etc. are definedhere as in 9A5.9E5 Definition (Premises, tail) If p is a composite s-component of a type T andp =_(m >_ 1), the s-components pl,..., p,, are called the premises ofp and a is called the conclusion or tail-component of p, or justTail(p).9E5.1 Lemma Two distinct s-components of a type T cannot have the same tailcomponent.Proof Induction on itt, using the fact that if T = Tl--*... the only s-components containing a are r and a and a is not the tail of itself because atomsdo not have tails (by 9E5).9E5.2 Warning The above lemma does not say that the tails of two distinct s-components p and g cannot be occurrences of the same atom. That is, usingthe =-notation of (iv), the lemma forbids Tail(p) = Tail(g) but does not forbidTail(p) = Tail(g).9E6 Definition (Subpremises, subtails) An s-component of T is called a subpremiseor subtail of T according as it is a premise or tail of another s-component of T. Thesets of all subpremises and all subtails of T will be called, respectively,Subpremises(T),Subtails(r).9E6.1 Example The type T - (a--+b--+c)--+(a-+b)-+a-+c in Fig. 9E2.1a has six subpremises,namely all three a's andIt has three subtails, namely3, T), (a-*b, 2, T), (b, 31, T).(b, 2*, T), (c, 3*, T), (c, *, T)9E6.2 Notes (i) A proper s-component g = (a, p, T) (p * 0) is a subpremise iff p is apremise position and a subtail iff p is a tail position.(ii) Each s-component of T is either a subtail, an atomic subpremise, a compositesubpremise, or T itself, and cannot be more than one of these.(iii) If T is composite, its leftmost atom-occurrence is a subpremise and itsrightmost is a subtail.(iv) An atom has no subpremises or subtails.