Generic Bidirectional Typing for Dependent Type Theories

Bidirectional typing is a discipline in which the typing judgment is decomposed explicitly into inference and checking modes, allowing one to control the flow of type information in typing rules and to specify algorithmically how they should be used. Bidirectional typing has been fruitfully studied and bidirectional systems have been developed for many type theories. However, the formal development of bidirectional typing has until now been kept confined to specific theories, with general guidelines remaining informal. In this work, we give a generic account of bidirectional typing for a general class of dependent type theories. This is done by first giving a general definition of bidirectional type theories, each giving rise to both a declarative and a bidirectional type system. We then show, in a theory-independent fashion, that the two systems are equivalent. Finally, we establish the decidability of bidirectional typing for normalizing theories, yielding a generic typing algorithm that has been implemented in a prototype and used in practice with many theories.