Stratified Type Theory
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To exploit the expressivity of being able to refer to the type of types, such as for large elimination, dependent type systems will either employ a universe hierarchy or else contend with an inconsistent type-in-type rule. However, these are not be the only possible options. Taking inspiration from 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 function types to some fixed level strictly lower than that of the overall type. Even in the presence of type-in-type, this restriction suffices to enforce consistency of the system. We explore the expressivity of several extensions atop this design. First, the subsystem subStraTT employs McBride's crude-but-effective stratification (also known as displacement) as a simple form of level polymorphism where top-level definitions can be displaced uniformly to any higher level as needed, which is valid due to level cumulativity and plays well with stratified judgements. Second, to recover some expressivity lost due to the restriction on dependent function domains, the full StraTT system includes a separate nondependent function type with floating domains, whose level instead matches that of the overall type. Finally, we have implemented a prototype type checker for StraTT extended with datatypes along with a small type checked core library. While it's possible to show that the subsystem is consistent, showing consistency for the full system with floating nondependent functions remains open. Nevertheless, we believe that the full system is also consistent and have mechanized a syntactic proof of subject reduction. Furthermore, we use our implementation to investigate various well-known type-theoretic type-in-type paradoxes. These examples all fail to type check in expected ways as evidence towards consistency.
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