Extending Type Theory with Forcing

This paper presents an intuitionistic forcing translation for the Calculus of Constructions (CoC), a translation that corresponds to an internalization of the presheaf construction in CoC. Depending on the chosen set of forcing conditions, the resulting type theory can be extended with extra logical principles. The translation is proven correct---in the sense that it preserves type checking---and has been implemented in Coq. As a case study, we show how the forcing translation on integers (which corresponds to the internalization of the topos of trees) allows us to define general inductive types in Coq, without the strict positivity condition. Using such general inductive types, we can construct a shallow embedding of the pure lambda-calculus in Coq, without defining an axiom on the existence of an universal domain. We also build another forcing layer where we prove the negation of the continuum hypothesis.