Towards a Logical Framework with Intersection and Union Types

Claude Stolze 1 Luigi Liquori 1, 2 Furio Honsell 3 Ivan Scagnetto 3
1 KAIROS - Logical Time for Formal Embedded System Design
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : We present an ongoing implementation of a dependent-type theory (∆-framework) based on the Edinburgh Logical Framework LF, extended with Proof-functional logical connectives such as intersection , union, and strong (or minimal relevant) implication. Proof-functional connectives take into account the shape of logical proofs, thus allowing to reflect polymorphic features of proofs in formulae. This is in contrast to classical Truth-functional connec-tives where the meaning of a compound formula is only dependent on the truth value of its subformulas. Both Logical Frameworks and proof functional logics consider proofs as first class citizens. But they do it differently namely, explicitly in the former while implicitly in the latter. Their combination opens up new possibilites of formal reasoning on proof-theoretic semantics. We provide some examples in the extended type theory and we outline a type checker. The theory of the system is under investigation. Once validated in vitro, the proof-functional type theory can be successfully plugged in existing truth-functional proof assistants.
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Claude Stolze, Luigi Liquori, Furio Honsell, Ivan Scagnetto. Towards a Logical Framework with Intersection and Union Types. 11th International Workshop on Logical Frameworks and Meta-languages, LFMTP, Sep 2017, Oxford, United Kingdom. pp.1 - 9. ⟨hal-01534035v2⟩

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