Propositional equality for gradual dependently typed programming

Gradual dependent types can help with the incremental adoption of dependently typed code by providing a principled semantics for imprecise and proofs, where some parts have been omitted. Current theories gradual types, though, lack central feature type theory: propositional equality. Lennon-Bertrand et al. show that, when reflexive proof refl is only closed value an equality type, extension Calculus Inductive Constructions (CIC) violates static observational equivalences. Extensionally-equal functions should be indistinguishable at run time, but they distinguished using combination imprecision. This work presents language that supports We avoid above issues devising which not inhabitant. Instead, each accompanied term least as precise equated terms, acting witness their plausible These witnesses track partial information program runs, raising errors shows two terms are undeniably inconsistent. Composition internalized construct language, deferred function bodies whose evaluation blocked variables. thus ensure extensionally-equal compose without error, thereby preventing contexts from distinguishing them. describe challenges designing consistency precision relations this system, along solutions to these challenges. Finally, we prove important metatheory: safety, conservative embedding CIC, weak canonicity, guarantees Siek al., reducing program’s introduces no new or dynamic errors.