number-theory

9D The structure of a type 151below Ap accordingly. Since FT s Fp none of these modifications enlarges acontext, so no inconsistencies are introduced. ThereforeF H{T/P}pM:T.9C6 Second Replacement Lemma for Deductions Let A, M, P be as in the firstreplacement lemma, and let {T /P }PM be the result of replacing P at p by a term Tsuch that(i) rT 1-A T :apfor some FT such thatKK(ii) rT U IT U is consistent,where xl,... , xn (n 0) are the binding variables of the 2-abstracts in M whose bodiescontain P, and l;l,...,l;n are their types in A (see 9C4 for details). Then(iii)F U FT [-a {T/P}PM:T.Proof Just as in the previous lemma, by 2A11(i) we can assume Subjects(FT) =FV(T). Replace Ap by AT in A and modify the contexts and subjects in all theformulae below Op accordingly. No inconsistencies will be introduced, because by9C4(4) the only variables not in F that could cause problems are x l, ... , x, .9C6.1 Corollary The conclusion of Lemma 9C6 also holds if instead of 9C6(ii) weassume that FU FT is consistent and none of X1.... , xn occurs free in T.Proof If Subjects(FT) = FV(T) the above assumption implies that IT satisfies 9C6(ii).9D The structure of a typeThis section and the next give two alternative approaches to the structure of anarbitrary type. The approach in the next section is used in Chapter 8; that in thisone is more basic and is included mainly as a contrast and introduction to the otherapproach.From the viewpoint of logical order the present section fits between 2A2 and 2A3.9D1 Definition (Occurs, occurrence) Here positions are the same as in 9A1; however,positions containing * are not needed in the present section. The phrase "a occursin z at position p" is defined thus (cf. 9A2):(i) T occurs in T at position 0;(ii) if al --.v2 occurs in T at p, then a; occurs in T at pi (i = 1, 2).A triple (a, p, T) such that a occurs in T at p is called an occurrence of a in T, or acomponent of T. A type that occurs in T is called a subtype of T.9D1.1 Notation An occurrence (a, p,T) may be called simply "a" when no confusionis likely.Recall from 2A2 that the number of occurrences of variables in T is called DTI and