number-theory

IC q- and iq-reductions93TFig. lC7a.(ii) If P =p, Q then there exists T such thatP DBn T, Q >p, T.Proof (i) From 1B5, 1C4, 1C6, 1C7. (ii) From (i) as in Fig. 1B5b.1C9 Definition (Jiq- and q-normal forms) A ft-normal form ($q-nf) is a termwithout frl-redexes. The class of all #q-nf's is called /Jq-nf. We say M has fq-nf N'ifMDRaN,NE f3j-nf.Similarly we define 1-normal form, q-nf, and M has q-nf N.1C9.1 Notation The f q-nf and q-nf of a term M are unique modulo =-,, by theChurch-Rosser theorems for fn and q ; they will be calledM*Pn, M*?1*1C9.2 Lemma (i) An q-reduction of a fl-nf cannot create new f3-redexes; more preciselyM E fl-nf and M >n NN E/3-nf.(ii) For every M,M*#,, is the q-nf of M*p; i.e. M*p, - (M*#)*,,.Proof (i) It is easy to check all possible cases. (ii) By 1C2, M*fl has an q-nf (M*#)*,,and this is a f3q-nf by (i).1C9.3 Corollary If N is a fl-nf then all the members of its ?I-family are fl-nf's andexactly one of them is a fq-nf, namely N*,,.1C9.4 Lemma A term has a $q-nf iff it has a f3-nf.Proof For "only if", see Curry et al. 1972 §11E Lemma 13.1 or Barendregt 1984Cor. 15.1.5. For "if", see 1C9.2. (By the way, do not confuse the present lemma witha claim that a term is in fl-nf if it is in Pq-nf, which is of course false!)