number-theory

144 9 Technical details9A6.3 Note Of course replacing one component of a term can disturb or destroyother components. To see what happens in two important cases, let P - (P, p, M)and R - (R, r, M) and letM* _ {T/P}PM.(i) If P is disjoint from R it is not hard to show that M* contains an occurrenceof R at position r (i.e., roughly speaking, replacing P does not change its disjointneighbours.)(ii) If P is in R, then p - rq for some position q such that P occurs in R atposition q, and it is not hard to show that M* contains, at position r, the term{T/P}9R.9B ResidualsThis section summarises some properties of #-contractions needed in the proof ofthe weak normalisation theorem in 5C. The full theory of #-reduction is in factquite deep (see Barendregt 1984 Chapters 3 and 11-14), but none of it is used inthis book except the following few very basic ideas.Everything in this section is valid for both the untyped terms of Chapter 1 andthe typed terms of Chapter 5.9131 Notation Recall the definitions of ji-redex and #-contraction in 1B1. (In thissection "#" will usually be omitted.) A redex-occurrenceR _is a particular occurrence of a redex in a term. The notations function part andargument part of R will be used here for and N respectively.Recall from 1132 that a reduction is a finite or infinite sequence of contractions(P1, R1, Q1),(P2, R2, Q2)....where Pl _a P and Ql =a Pi+1 for i = 1,2,.... This definition allows a reductionto make a-conversions before or after each of its contractions, but the reader maysafely ignore these and concentrate on the contractions; the next lemma will saycontractions are unaffected by a-conversions in a certain precise sense.Recall from 1B3 that the length of a reduction is the number of its contractions.(And a-conversions are not counted.)In this section "c" will denote an arbitrary contraction and "r" an arbitraryreduction.If rl is a finite reduction from a term P to a term Q and r2 is a reduction of 0,the reduction consisting of rl followed by r2 will be calledrl + r2.9B1.1 Lemma (a-invariance) If P =a P' and P contains a /3-redex-occurrence R _(R, p, P) whose contraction changes P to Q, then P' contains a /3-redex-occurrence(R', p, P') whose contraction changes P' to a term Q' _a Q.