number-theory

72 5 A version using typed termsS. All the redexes in P, are either newly created or residuals of redexes not in 5;henced(P1) < d(P").Continue this procedure by making further minimal complete developments of setsof redexes with maximum degree; since each development strictly decreases d(P")we must eventually obtain a fl-normal form of P".5C1.1 Corollary The WN theorem for typable terms (2D5) holds.Proof By 5B7.4 and the WN theorem for typed terms.5C1.2 Historical notes The first known proof of WN for a type-theory equivalentto TAx was written around 1941 or '42 in unpublished notes by Alan Turing. (SeeGandy 1980a for a transcript of Turing's notes, and Gandy 1977 pp. 178-180 forsome comments on Turing's work in type-theory.)But the first proof to be actually published was carried out by Curry in the late1950's by a completely different technique (Curry and Feys 1958 §9F, especiallyCor. 9F9.2). Its key step was a cut-elimination theorem for a particular formulationof TACL.'Also there is a WN theorem in Prawitz 1965 for reducing proofs in logic which isessentially equivalent to WN for TA2.From the 1960's onwards the Turing method of proof was re-discovered, probablyindependently, several times (Morris 1968 §4F Thm. 2, Andrews 1971 Prop. 2.7.3,for example), and WN theorems were proved by various methods for many strongertype-theories than TA2.In particular, the mid-60's saw a spate of proofs of WN for typed .1-calculienhanced by primitive recursion operators (for example those in Tait 1965, Hanatani1966, Hinata 1967, Sanchis 1967, Tait 1967, Diller 1968, Dragalin 1968 and Howard1970); see Troelstra 1973 §§2.2.1-2.3.13 or HS86 Ch. 18 for descriptions of thebackground setting. At least two of these also included proofs of SN (see below).5C2 Strong Normalization (SN) Theorem (Sanchis 1967 Thm. 8, Diller 1968 §6,etc.) Let P" E TT(F) for some F; then, for /3- and for /hj-reductions,(i) all reductions of P" are finite,(ii) there is an algorithm which accepts P" as input and outputs a number k(P") suchthat all reductions of P" have length 5 k(P").Proof For (i) there is an accessible proof in HS86 Appendix 2: see Thm. A2.3 for/3, and Thm. A2.4 for fl q. Alternatively, see Barendregt 1992 Thm. 4.3.6.For (ii), simply construct a finitely branching tree of reductions starting at P" bydoing all possible contractions at each step. By (i) this tree's branches are all finiteCut-elimination originated in the study of predicate logic in Gentzen 1935, and the relation betweencut-elimination and normalization is explored in Zucker 1974. It depends on the correspondencebetween formulae and types to be described in the next chapter. Cut-elimination theorems for versionsof TAx can be found in Seldin 1977 and 1978.