Stratified Type Theory

A hierarchy of type universes is a rudimentary ingredient in the type theories of many proof assistants to prevent the logical inconsistency resulting from combining dependent functions and the type-in-type rule. In this work, we argue that a universe hierarchy is not the only option for a type theory with a type universe. Taking inspiration from Leivant's Stratified System F, we introduce Stratified Type Theory (StraTT), where rather than stratifying universes by levels, we stratify typing judgements and restrict the domain of dependent functions to strictly lower levels. Even with type-in-type, this restriction suffices to enforce consistency. In StraTT, we consider a number of extensions beyond just stratified dependent functions. First, the subsystem subStraTT employs McBride's crude-but-effective stratification (also known as displacement) as a simple form of level polymorphism where global definitions with concrete levels can be displaced uniformly to any higher level. Second, to recover some expressivity lost due to the restriction on dependent function domains, the full StraTT includes a separate nondependent function type with a "floating" domain whose level matches that of the overall function type. Finally, we have implemented a prototype type checker for StraTT extended with datatypes and inference for level and displacement annotations, along with a small core library. We have proven subStraTT to be consistent and StraTT to be type safe, but consistency of the full StraTT remains an open problem, largely due to the interaction between floating functions and cumulativity of judgements. Nevertheless, we believe StraTT to be consistent, and as evidence have verified the failure of some well-known type-theoretic paradoxes using our implementation.