Sub-extensional type theory

Martin-Löf’s type theory is the basis of a variety of formal systems underlying many proof assistants. There are a lot of variants of type theory, among which two classes can be identified: intensional and extensional type theory. The difference between the two lies in the manner in which the equality is dealt with, but this results in huge differences. Intensional type theory requires the user to write explicit coercions, which can lead to a big overhead when programming with dependent types. It also lacks some desirable properties such as function extensionality. On the other hand, extensional type theory deals more smoothly with these issues, but as a result it suffers from undecidable typechecking. Moreover, it allows to type non-normalizing terms. There have been many different attempts to reconcile these two views, either by reducing the overhead in intensional type theory, by trying to put up with the undecidability of extensional type theory, or by proposing new kinds of type theories which combine the good aspects of the two.