  We provide a method, based on automata theory, to mechanically prove the
correctness of many numeration systems based on Fibonacci numbers. With it,
long case-based and induction-based proofs of correctness can be replaced by
simply constructing a regular expression (or finite automaton) specifying the
rules for valid representations, followed by a short computation. Examples of
the systems that can be handled using our technique include Brown's lazy
representation (1965), the far-difference representation developed by Alpert
(2009), and three representations proposed by Hajnal (2023). We also provide
three additional systems and prove their validity.
