Sims-reduced generating set

Let $G$ be a group acting on a set $S = \{ 1, 2, 3, \ldots, n \}$ of size $n$ (equivalent $G$ is equipped with an embedding in the symmetric group on $n$ elements). A Sims-reduced generating set is a generating set for $G$ where for any elements $i < j$, there is at most one elements of $G$ which fixes all elements less than $i$ and sends $i$ to $j$.
Any generating set for $G$ can be trimmed down to a Sims-reduced generating set using the Sims filter.