number-theory

5B Reducing typed terms 695B5.1 Notes The restrictions in the above clauses may seem over-strong in that theyinvolve untyped variables and terms; but if they were weakened the last part of theuseful lemma 5B5.2 below would fail.Clause (iv) includes the case 1 # o. But in this case x° cannot occur free in P",because if it did thenwould not be a typed term (see Definition 5A1(iii)).Thus substituting for x° should change nothing. And this is exactly what (iv) says.No attempt has been made to define [NP/x'] when p # o or [N9/xa]xt when'r*0.5B5.2 Lemma If ((2xa.Mt)Na)t is a typed term with no bound-variable clashes, then(i) [Na/xa]Mt is defined and is a typed term with type r,(ii) [Na/x°]Mt - Mt ifxa 0 FV(Mt),(iii) Con([N'/xa]Mt) c (Con(Mt) - x) U Con(N'),(iv) ([Na/xa]Mt)f =a [N'1/x]Mf.Proof Parts (i)-(iv) are proved together by a straightforward but boring induction onIMt 1. (The assumption about ((2xa.Mt)Na)t implies in particular that Con(MI) - xis consistent with Con(Na) and that x does not occur in Nd.)5B6 Definition (Typed a-conversion) The relation -a is defined just as for untypedterms (see 1A8), using replacements with form(a)Axa'MtAY°'[Ya/xa]Mt(y FV(Mf))5B6.1 Lemma If Ax°.Mt is a typed term and y FV(Mf ), then [ya/xa]MT is definedand both sides of (a) are typed terms with the same type and minimum context. Hencethe class of all typed terms is closed under a-conversion, andPt =a Qt'i - i and Con(Pt) = Con (Q").5B6.2 Warning The condition "y FV(M1 )" in (a) cannot be weakened to "ya 0FV(Mt)". Because if it were, we could a-convert a typed term to an expression thatwas not one, thus:Axa-+b , xa-b ya=aAya-b, ya.b ya5B7 Definition (Typed redexes and reduction) Typed >p and v#,, are defined just likethe untyped relations in 1B and 1C. In particular typed f- and q-redexes have form((2x°'Mt)a~tNo)t,with x 0 FV(P'N'f), and their contracta are, respectively,[Nalxo]Mt,Pa-.tAnd, just as in Chapter 1, a contraction is a replacement of a redex by its contractum.A typed fl-nf (q-nf, /q-nf) is a typed term containing no fl-(rl-, $i -) redexes.