Weak resolution of singularities

Let $L_0$ be a field and let $L_1$ be a nontrivial finitely generated extension of $L_0$. Suppose there exists a valuation $v$ on $L_1$ with residue field $L_0$. Weak resolution of singularities. There also exists a nontrivial discrete valuation $w:L_1\to\mathbb{Z}$ with residue field $L_0$. This condition is presented as a hypothesis relevant to reductions of Hilbert's tenth problem for formal power series rings; its resolution status is not specified in the source.