On projection in CAD-based quantifier elimination with equational constraint

Collins [4] observed t.hat quant.ifirr climimtion problen~s oft,en have cquat,ional constraints. and he asserted that. such const,raints ~a.11 be used to reduce the projection wt,s required for cylindrical algebraic decomposition (cad) lmwd qumtifier climinat.ion. This paper provides a. dct,ailed partial validity proof for t,lic: restricted equatiom.1 projection method out,lirwd by Collins [Ll]. The proof is sufficicmt t.0 valitMo the use of t.he nwt~hod for the first. projection of the projection phase. A further consequence is t,hat the 111cth0d cm 1.~ used for both t,he first, and second projections in the case of a trivarintc qua.nt.ificr eliniination prolAcu~ llil.viIlg sufficient e~UatiOlli~l constraints.