Division Algorithm for non-global ordering in K[x,y]

Question

I'm trying to adapt the devision algorithm to non-global weight-ordering. Given $f=x^2$, $G=\{x-1\}$ and the weight vector $v=(-1,1)$. Now $K[x,y]$ is adopted with a weight-ordering ($deg_v(x^\alpha*y^\beta)=\beta - \alpha$). If I perform the standard version of the division algorithm used in groebner basis theory I'll fall in an infinite loop:
Since $lm_<(x-1)=-1$ I get $t_1:=spoly(x^2,x-1)=x^2-(-x^2)(x-1)=x^3$ and by induction $t_n=x^{n+2}$.
This is due to the fact that the ordering is a non-global ordering.

Like it is proposed in diffrent sources I wanted to adopt the homogenization theory which should garantee the termination for a homogenized imput. I get $Hom(x^2)=x^2$ and $Hom(x-1)=hx-1$, which helps so fare since I now have $lm_<(hx-1)=hx$ (for an underlying global order). But now I get $lm_<(x^2)=x^2 \nmid hx=lm_<(hx-1)$. Does that mean that due to my choosen order my algorithm terminates and returns $NF_<(f,G)=x^2$ although $x^2-(x+1)(x-1)=1$?