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Decidability of language equivalence of deterministic pushdown automata (DPDA) was established by G. Senizergues (1997), who thus solved a famous long-standing open problem. A simplified proof, also providing a primitive recursive complexity upper bound, was given by C. Stirling (2002). In this paper, the decidability is re-proved in the framework of first-order terms and grammars (given by finite sets of root-rewriting rules). The proof is based on the abstract ideas used in the previous proofs, but the chosen framework seems to be more natural for the problem and allows a short presentation which should be transparent for a general computer science audience.