number-theory

(M).4A The equality rule 554A3 Weak Normalization Theorem for TAA+p[n]a f[q]-nf M*, and furthermoreEvery TA,A+p[n]-typable term M hasF 1-p[n] M:'r t=om r F-no eqM* :'r.Proof Let F I-p[n] M:T. Then by 4A2 there exists M' =p[n] M such that(1) r 1-no eq M':T.By the weak normalization theorem for TA2 (2D5), M' has a of M* ; and by theChurch-Rosser theorem M' reduces to M* and M* is also the of of M. They by (1)and the subject-reduction theorem (2C1),(2) r F-no eq M*:T.Conversely, if (2) holds then r 1-p[n] M:'r by (Eq).4A3.1 Warning The weak normalization theorem cannot be strengthened to strongnormalization. For example letM =_Then M =p I so M is typable in both TA1+p and TA2+pn. But M has an infinitereduction, so strong normalization fails for both systems.4A3.2 Corollary The relation F I-p[nl M:-r is equivalent to each of the following:(i) F F-no eq M* :t,(ii) F r M* F-no eq M* :T,(iii) F F-p[n] M* :T,(iv) F r M* I-p[n] M* :'r,(v)r r M* I-p[nl M :T'(vi) F rM F-p[n1 M:T.Proof Each of (i)-(vi) implies F F-p[n1 M:T by (Eq) and the weakening property of" F-" (which holds for F-p[n] just as for I-2, see 2A9.1).For the converse, the relation F F-p[n] M:T implies (i) by 4A3. And (i) implies (ii)by 2A11. Next, (ii) implies (iv) trivially, and (iv) implies (iii), (v) and (vi) by (Eq)and weakening.4A3.3 Note When F F-#[n] M:T we cannot infer that Subjects(F) 2 FV(M), but bythe above corollary and 2A11 we can infer thatSubjects(F) 2 FV(M*).The next theorem will express the content of Corollary 4A3.2 very neatly for thecase that M is closed.4A4 Definition If M is closed, the set of all T such that I-g[,] M:T holds will becalledTypes p[n]