Singular

LIB "ncfrac.lib";
==> // ** redefining testNcfrac (LIB "ncfrac.lib";)
==> // ** redefining testNcloc (    LIB "ncloc.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;
list L = g1,g2;
ncloc loc0 = L;
poly g = g1^2*g2;
poly f = Dx;
ncfrac frac0 = [0,0,f,g];
frac0.loc = loc0;
ncfrac rm = ensureLeftNcfrac(frac0);
print(rm);
==> left repr.: (x^8*y^4+12*x^7*y^4+54*x^6*y^4+108*x^5*y^4+81*x^4*y^4,x^5*y^3\
   *Dx+6*x^4*y^3*Dx-3*x^4*y^3+9*x^3*y^3*Dx-12*x^3*y^3-9*x^2*y^3)
==> right repr.: (Dx,x^3*y+6*x^2*y+9*x*y)
rm.lnum*g-rm.lden*f;
==> 0
// geometric localization
ncloc loc1 = ideal(x-1,y-3);
f = Dx;
g = x^2+y;
ncfrac frac1 = [0,0,f,g];
frac1.loc = loc1;
ncfrac rg = ensureLeftNcfrac(frac1);
print(rg);
==> left repr.: (x^4+2*x^2*y+y^2,x^2*Dx+y*Dx-2*x)
==> right repr.: (Dx,x^2+y)
rg.lnum*g-rg.lden*f;
==> 0
// rational localization
intvec rat = 1;
ncloc loc2 = rat;
f = Dx+Dy;
g = x;
ncfrac frac2 = [0,0,f,g];
frac2.loc = loc2;
ncfrac rr = ensureLeftNcfrac(frac2);
print(rr);
==> left repr.: (x^2,x*Dx+x*Dy-1)
==> right repr.: (Dx+Dy,x)
rr.lnum*g-rr.lden*f;
==> 0