number-theory

46 3 The principal-type algorithmSubcase IVa2: the pair (3) has a unifier. Then the unification algorithm gives a mostgeneral unifier uu; apply a renaming if necessary (3D2.5) to ensure that(4) Dom(u) = Vars(ipl, Xr, T),(5) Range(u) fl V = 0,where(6) V = (Vars(AP) U Vars(AQ)) - Dom(u).Then apply u to Ap and AQ; this changes their conclusions to, respectively,u1:6* ,...,uP:BP, Wl :tpi ,...,Wr:tpr r-* P:p*-.v*,vl:(al*,...,vq:(a9*,WI:XI*,...,Wr:Xr*F-iQ:2*,where Bi =- u(Bi), etc. And by the definition of uu we have* * * * *Wi = Xl , . tr - Xr , P = 7Hence (-->E) can be applied to the conclusions of u(Ap) and u(AQ).resulting combined deduction APQ; its conclusion isCall the(7) u1:0*.... I UP OPJustification of Subcase IVa. For IVal we must prove that if PQ is typable then(wl, . , Wr, p) and (Xi, ... , Xr, i) have a unifier, and for IVa2 we must prove that theabove APQ is principal for PQ.Justification of IVal. If PQ is typable, there is a deduction A whose conclusionhas form(8) U1:2r1,...,UP:7rP, vl:µl,...,vy:uq, wl:vl,...,Wr:Vr F-s PQ:/3for some types 7t1, ... , 7tp, etc. By the subject-construction theorem (2B2), A must havebeen built by applying rule (--* E) to two deductions Al and A2 whose conclusionsare(9) UI:7tl,...,up:7tp, W1:V1,...,Wr:Vr f-a P:(10) v1:1,11,...,vy:zq, W1:Vl,...,Wr:Vr I--.. Q:afor some type a. But Ap and AQ are principal deductions for P and Q, soAl = ri(Ap),for some substitutions r1 and r2 such thatDom(rl) = Vars(Ap),A2 = r2(AQ)Dom(r2) = Vars(AQ).FRoughly speaking, (5) and (6) say that when u is applied to Ap and AQ any new variables it introduceswill differ from all those already in Ap and AQ, and hence no unnecessary identifications of variableswill be made. (Cf. the motivation in 3C4.) By the way, if the aim of the algorithm had been toconstruct merely a principal type and not a principal deduction, (5) could have been weakened byre-defining V to consist of just the variables (if any) in a that do not occur in the types in (3). (Cf. (1)in 3C4.)