  This paper provides an induction rule that can be used to prove properties of
data structures whose types are inductive, i.e., are carriers of initial
algebras of functors. Our results are semantic in nature and are inspired by
Hermida and Jacobs' elegant algebraic formulation of induction for polynomial
data types. Our contribution is to derive, under slightly different
assumptions, a sound induction rule that is generic over all inductive types,
polynomial or not. Our induction rule is generic over the kinds of properties
to be proved as well: like Hermida and Jacobs, we work in a general fibrational
setting and so can accommodate very general notions of properties on inductive
types rather than just those of a particular syntactic form. We establish the
soundness of our generic induction rule by reducing induction to iteration. We
then show how our generic induction rule can be instantiated to give induction
rules for the data types of rose trees, finite hereditary sets, and
hyperfunctions. The first of these lies outside the scope of Hermida and
Jacobs' work because it is not polynomial, and as far as we are aware, no
induction rules have been known to exist for the second and third in a general
fibrational framework. Our instantiation for hyperfunctions underscores the
value of working in the general fibrational setting since this data type cannot
be interpreted as a set.
