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## Section: New Results

### Logical frameworks with Union and Intersection constraints and Oracles

Participants : Luigi Liquori, Claude Stolze.

In [13], we introduced the $\Delta$-framework, DLF, 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, DLF subsumes also Pfenning's refinement types. We discuss the design decisions which have led us to the formulation of DLF, study its metatheory, and provide various examples of applications. Our strong proof-functional type theory can be plugged in existing common interactive proof assistants.

Moreover, in [7], we introduced two further extensions of LF, featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. Such calls are the basic tool for plugging-in, i.e. gluing together, different type theories and proof development environments.