We introduce the ∆-framework, LF∆, a dependent type theory based on the Edinburgh Logical Framework LF, extended with the strong proof-functional connectives, i.e. strong intersection, minimal relevant implication and strong union. Strong proof-functional connectives take into account the shape of logical proofs, thus reflecting polymorphic features of proofs in formulæ. This is in contrast to classical or intuitionistic connectives where the meaning of a compound formula depends only on the truth value or the provability of its subformulæ. Our framework encompasses a wide range of type disciplines. Moreover, since relevant implication permits to express subtyping, LF∆ subsumes also Pfenning’s refinement types. We discuss the design decisions which have led us to the formulation of LF∆, study its metatheory, and provide various examples of applications. Our strong proof-functional type theory can be plugged in existing common proof assistants.
