Monotone fixed-point types and strong normalization

Abstract

Several systems of fixed-point types (also called retract types or recursive types with explicit isomorphisms) extending system F are considered. The seemingly strongest systems have monotonicity witnesses and use them in the definition of beta reduction for those types. A more naïve approach leads to non-normalizing terms. All the other systems are strongly normalizing because they embed in a reduction-preserving way into the system of non-interleaved positive fixed-point types which can be shown to be strongly normalizing by an easy extension of some proof of strong normalization for system F.