Mohri's pushing strategy-we will see others-is always to extract some sort of "maximum left factor" from each suffix function F q and push it backwards.
he expresses F q = k ⊗ G for as "large" a k ∈ K as possible-a maximal common prefix-then pushes factor k back out of the suffix function so that it is counted earlier on paths through q (i.e., before reaching q).