Nested datatypes are families of datatypes that are indexed over all types such that the constructors may relate different family members (unlike the homogeneous lists). Moreover, even the family name may be involved in the expression that gives the index the argument type of the constructor refers to. Especially in this case of true nesting, termination of functions that traverse these data structures is far from being obvious. A joint article with A. Abel and T. Uustalu (TCS 333(1–2), pp. 3–66, 2005) proposes iteration schemes that guarantee termination not by structural requirements but just by polymorphic typing. And they are generic in the sense that no specific syntactic form of the underlying datatype “functor” is required. However, there have not been induction principles for the verification of the programs thus obtained although they are well-known in the usual model of initial algebras on endofunctor categories.
The new contribution is a representation of nested datatypes in intensional type theory (more specifically, in the Calculus of Inductive Constructions) that is still generic, guarantees termination of all expressible programs and has induction principles that allow to prove functoriality of monotonicity witnesses (maps for nested datatypes) and naturality properties of iteratively defined polymorphic functions.