Simple normal crossings divisors and simultaneous normal crossings position

Let $X$ be a regular locally Noetherian scheme of pure dimension $n$ (embedding dimension and regular local ring). A reduced effective Cartier divisor $D\subseteq X$ (Effective cartier divisor, Cartier divisor) is a strict normal crossings divisor (an SNC divisor) if for every point $p\in X$, writing $d_p=\dim {\mathcal{O}}_{X,p}$, there is a regular system of parameters $u_1,\dots ,u_{d_p}$ of ${\mathcal{O}}_{X,p}$ and a subset $J\subseteq \{1,\dots ,d_p\}$ such that the ideal of $D$ in ${\mathcal{O}}_{X,p}$ is generated by ${\prod }_{j\in J}{u}_j$; equivalently, every irreducible component of $D$ is regular and the local equations of the components through $p$ extend to a regular system of parameters. A finite family $E=\{D_1,\dots ,D_m\}$ of reduced effective Cartier divisors on $X$ is in simultaneous SNC position if for every subset $J\subseteq \{1,\dots ,m\}$ the union of components of the divisors in $J$ is an SNC divisor; equivalently, at every point $p$ the union of all components of members of $E$ passing through $p$ is cut out by a subset of a regular system of parameters of ${\mathcal{O}}_{X,p}$. For the marked-ideal algorithm on this page, the irreducible components within each individual member of $E$ must be pairwise disjoint, and no irreducible component is repeated in different members. Thus at a point each member contributes at most one distinct parameter equation. This is the additional boundary convention of Włodarczyk, Definition 2.1.1. Such a family $E$ always carries a fixed total order, and the members of $E$ play the role of the exceptional divisors of a multiple test blow-up.