We present a type-based specification for useless-variable elimination for a higher-order, call-by-value functional language. Utilizing a weak form of dependent types, we introduce a mechanism for eliminating at runtime useless code that is not detectable statically. We prove the specifications sound and safe with respect to an operational semantics, ensuring that eliminated expressions contributed no observable behavior to the original program. We define an algorithm that implements useless-variable elimination without dependent types, and we prove this algorithm correct with respect to the specification.