 Index-function representation of a list

Represent a finite list by a function from indices to optional values. There must be a length $m$ for which exactly the indices $0\le i<m$ are present; later indices are absent, so the representation has no holes. Prepending and removing a head shift indices, taking restricts the present domain, and dropping translates it. Functional composition can accumulate query costs, so array-like constant-time access does not follow from the representation type.