First Steps Towards Cumulative Inductive Types in CIC

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 [Figure not available: see fulltext.], ..., where [Figure not available: see fulltext.], .... In a cumulative type system, e.g., pCIC, for a term [Figure not available: see fulltext.] such that [Figure not available: see fulltext.] we also have that [Figure not available: see fulltext.]. 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.