Abstract

The problem of de ning iteration for higher-order nested datatypes of arbitrary ( nite) rank is solved within the framework of System F of higher-order parametric polymorphism. The proposed solution heavily relies on a general notion of monotonicity as opposed to a syntactic criterion on the shape of the type constructors such as positivity or even being polynomial. Its use is demonstrated for some rank-2 heterogeneous/nested datatypes such as powerlists and de Bruijn terms with explicit substitutions. An important feature is the availability of an iterative de nition of the mapping operation (the functoriality) for those rank-1 type transformers (i. e., functions from types to types) arising as least xed-points of monotone rank-2 type transformers. Strong normalization is shown by an embedding into F . The results dualize to greatest xed-points, hence to coinductive constructors with coiteration.

Citations