De Vrijer has presented a proof of the finite developments theorem which, in addition to showing that all developments are finite, gives an effective reduction strategy computing longest developments as well as a simple formula computing their length. We show that by applying a rather simple and intuitive principle of duality to de Vrijer's approach one arrives at a proof that some developments are finite which in addition yields an effective reduction strategy computing shortest developments as well as a simple formula computing their length. The duality fails for general beta-reduction. Our results simplify previous work by Khasidashvili.

• 03B25 - Decidability of theories and sets of sentences
• 03C57 - Computable structure theory, computable model theory
• 03D05 - Automata and formal grammars in connection with logical questions