Singular

LIB "olga.lib";
ring R = 0,(x,y,Dx,Dy),dp;
def S = Weyl();
setring S; S;
==> // coefficients: QQ
==> // number of vars : 4
==> //        block   1 : ordering dp
==> //                  : names    x y Dx Dy
==> //        block   2 : ordering C
==> // noncommutative relations:
==> //    Dxx=x*Dx+1
==> //    Dyy=y*Dy+1
// monoidal localization
poly g1 = x+3;
poly g2 = x*y+y;
list L = g1,g2;
poly s1 = g1;
poly s2 = g2;
poly r1 = Dx;
poly r2 = Dy;
vector frac1 = [s1,r1,0,0];
vector frac2 = [s2,r2,0,0];
vector rm = multiplyLeftFractions(frac1, frac2, 0, L);
print(rm);
==> [x^3*y^2+5*x^2*y^2+7*x*y^2+3*y^2,x*y*Dx*Dy+y*Dx*Dy-y*Dy]
// geometric localization
ideal p = x-1, y-3;
vector rg = multiplyLeftFractions(frac1, frac2, 1, p);
print(rg);
==> [x^3*y+5*x^2*y+7*x*y+3*y,x*Dx*Dy+Dx*Dy-Dy]
// rational localization
intvec v = 2;
s1 = y^2+y+1;
s2 = y-2;
r1 = Dx;
r2 = Dy;
frac1 = [s1,r1,0,0];
frac2 = [s2,r2,0,0];
vector rr1 = multiplyLeftFractions(frac1, frac2, 2, v);
print(rr1);
==> [y^3-y^2-y-2,Dx*Dy]
vector rr2 = multiplyLeftFractions(frac2, frac1, 2, v);
print(rr2);
==> [y^5-y^3-4*y^2-3*y-2,y^2*Dx*Dy+y*Dx*Dy-2*y*Dx+Dx*Dy-Dx]
areEqualLeftFractions(rr1, rr2, 2, v);
==> 0