Let $\mathcal{C}_{u}$ be an unsafe set and $\mathcal{C}_{0}$ be the set we start in. Let $B:\mathbb{R}^{n}\to \mathbb{R}$ where $B(x)\le 0$, $\forall x\in \mathcal{C}_{0}$ and $B(x)>0$, $\forall x\in \mathcal{C}_{u}$. Then $B$ is a barrier certificate if $$\dot{B}(x)\le 0\implies \mathcal{C}\text{ is invariant}.$$i.e. $B$ is a barrier certificate if it being strictly decreasing implies $\mathcal{C}$ is an Invariant Set.