number-theory

8B Examples of the search strategy 1158B1 Lemma Every type r can be expressed uniquely in the following form, wherem >_ 0 and e is an atom:2 =Proof Easy induction on ITI8131.1 Notation The occurrences of and e displayed above will be calledthe premises and conclusion (or tail) of T respectively, and m will be called the arityof T. (See Definition 9E5 for more details.)Two type-occurrences will be called isomorphic if they are occurrences of thesame type. (cf. Definition (iv).) Iff the tail-components of o and T are isomorphicwe may sayTail(a) = Tail(T).8B2 Comments (Long typed P-nf s) Let T be any type; say T has formT -and let MI be any fl-nf with type T. By 8A5, MT has form(2xll(m >_ 0, e an atom)11 fnn1T*)(TI-+ ...Tk->T*)where 0 < k:5 m and T* __ If MT is long (see 8A7), then(i) k = m,(ii) T* __ e,(iii) the types of xl,... , xm coincide with the premises of T,(iv) the tail of the type of v is isomorphic to that of T,(v) if MT is closed then m >_ 1 and v is an xi (1 < i< m) andTi - PV->...-+Pn-+e.The following examples show how the above comments are applied.8B3 Example (A type T with #(T) = 1) (Ben-Yelles 1979 p. 42.) The following typehas exactly one normal inhabitant:T -And its normal inhabitant (which is also both long and principal) isSTAxa-.b-.cya-»bza.xZ(yz).Proof We shall start by proving that Long(T) = {ST}.Step 1. First look at the structure of T: in the notation of 8B2 we have m = 3 ande - c, Tl =_ a--+b--+c, T2 - a--+b, T3 - a.Hence any MT E Long(T) must have just three initial abstracted variables, sayMT(AX 1X zx33.(v(v1-...P"-c)M°' ... Mn")c)(TI-T2-T3~C).