# Are minimal Groebner bases minimized bases?

With "minimal Groebner basis" I mean, fixed an ordering, a Groebner basis $$G$$ such that any proper subset of $$G$$ is no more a Groebner basis for the ideal $$I(G)$$ generated by $$G$$.

With "minimized basis" I mean a basis $$B$$ such that any proper subset of $$B$$ is no more a basis for $$I(B)$$.

So, can exist a minimal Groebner basis which is not minimized?

I can't find any contradiction but neither a counterexample.

A minimal lex Groebner basis which is not "minimized" is given by $$\{ x^2 + y, xy - y, y^2 + y \}$$ in $$\mathbb{Q}[x,y]$$.
This is clearly a minimal lex Groebner basis, but $$y^2 + y=y\cdot(x^2+y) - (x+1)\cdot(xy-y)$$ so the third element is superfluous as a generator of the ideal.