number-theory

5B Reducing typed terms(ii) if MT E TT(F) then Con(MT) c F and L(MT) is a deduction ofCon(MT) F-+ M{:'r.67Proof (i) By 5A5.1 and 5A6. (ii) Easy induction on IMfI.5B Reducing typed termsMost of the term-structure concepts introduced in Section 1A, and those in 9A, canbe extended from untyped to typed terms in an obvious way. For example position,subterm, occurs and component are all defined for typed terms just as for untypedterms in 9A.The only difference is that in a typed term every subterm is decorated with atype-superscript; in particular an atom has form xT not just x. For example ifMT = (Xa-'by°)b we shall say that x°-b and ya occur in M1 but not that x and yoccur in MT.Some of the concepts in IA and 9A do not translate to typed terms quite sounambiguously however, so their typed definitions will be given below and theirmain properties will be stated.5B1 Definition The length, IMTI, of a typed term MT is the same as JM11, see 1A2.5B2 Definition (Replacement) Let P° _ (Pa, p, MT) be a component of a typed termMT (see 9A2-3), and let T° be any typed term with the same type as P°. Define{T°/Pa}PMTto be the result of replacing P° by T° at position p in MT, just as for untyped terms(see 9A6), but only when (i) or (ii) below holds:'(i) Con(Tc) s Con(P°);(ii) the setis consistent, where L1,',...,.lx"are the abstractors in MT whose scopes contain P°.5B2.1 Lemma (i) If 5B2(i) holds then {Ta/P°}PMT is a well-defined typed term andCon({Ta/P°}PMT) 9 Con (MT).(ii) If 5B2(ii) holds then {T°/P°}PMT is well-defined andProof Like the proofs of 9C5 and 9C6.Con({Ta/P°}PMT) c Con(MT) U Con(T°).5B2.2 Note If neither (i) nor (ii) holds in 5B2 we do not define {Ta/P°}PMT. Inthis case the replacement of P° by T° in MT will still produce an expression ofsome kind but it might not be a typed term. For example letMT = (xa-bya)b, P° = yQ, Ta =xa.Compare the conditions in the replacement lemmas for deductions, 9C5-6.