Equivalence of pushdown automata via first-order grammars

A decidability proof for bisimulation equivalence of first-order grammars is given. It is an alternative proof for a result by S'enizergues (1998, 2005) that subsumes his affirmative solution of the famous decidability question for deterministic pushdown automata. The presented proof is conceptually simpler, and a particular novelty is that it is not given as two semidecision procedures but it provides an explicit algorithm that might be amenable to a complexity analysis.