Proper Second-Order Equation Operator

Let $n\ge1$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the set of real numbers with the order $\le$ of its ordered field structure, let $\mathcal{S}(n)$ be the set of symmetric real $n\times n$ matrices, and let $F$ be a second-order equation operator on $U$.

We say that $F$ is proper if the following two conditions hold.

1. $F$ is degenerate elliptic.

2. For every $x\in U$, every $p\in\mathbb{R}^n$, every $X\in\mathcal{S}(n)$, and all $r,s\in\mathbb{R}$ with $r\le s$,