Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $\mathcal E:X\to(-\infty,+\infty]$ be an energy functional, and let $\tau>0$. Suppose that $(x_k^\tau)_{k\in\mathbb N_0}$ is a discrete minimizing movement for $\mathcal E$ with time step $\tau$, meaning that $x_k^\tau\in X$ satisfies $\mathcal E(x_k^\tau)<+\infty$ for every $k\in\mathbb N_0$ and, for every $k\in\mathbb N_0$, the point $x_{k+1}^\tau$ minimizes the map

\begin{align*} y\mapsto \mathcal E(y)+\frac{1}{2\tau}d^2(y,x_k^\tau) \end{align*}

over $X$. Then, for every pair of integers $m,\ell\in\mathbb N_0$ with $0\le m<\ell$,

\begin{align*} \frac{1}{2}\sum_{k=m}^{\ell-1}\frac{d^2(x_{k+1}^\tau,x_k^\tau)}{\tau}+\mathcal E(x_\ell^\tau)\le \mathcal E(x_m^\tau). \end{align*}