Having the type of all types in a type system results in paradoxes like Russel's paradox. Therefore type theories like predicative calculus of inductive constructions (pCIC) — the logic of the Coq proof assistant — have a hierarchy of types Type₀, Type₁, Type₂, ···, where Type₀ : Type₁, Type₁ : Type₂, ···. In a cumulative type system, e.g., pCIC, for a term t such that t: Type_i we also have that t: Type_i+1. The system pCIC has recently been extended to support universe polymorphism, i.e., definitions can be parametrized by universe levels. This extension does not support cumulativity for inductive types. For example, we do not have that a pair of types at levels i and j is also considered a pair of types at levels i + 1 and j + 1.

In this paper, we discuss our on-going research on making inductive types cumulative in the pCIC. Having inductive types be cumulative alleviates some problems that occur while working with large inductive types, e.g., the category of small categories, in pCIC.

We present the pCuIC system which adds cumulativity for inductive types to pCIC and briefly discuss some of its properties and possible extensions. We, in addition, give a justification for the introduced cumulativity relation for inductive types.