The $∆-$framework

Furio Honsell 1 Luigi Liquori 2, 3 Ivan Scagnetto 1 Claude Stolze 2
2 KAIROS - Logical Time for Formal Embedded System Design
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : We introduce a dependent-type theory ∆-framework, LF-∆ , based on the Edinburgh Logical Framework LF, extended with the strong proof-functional connectives intersection, union, and relevant implication. Proof-functional connectives take into account the shape of logical proofs, thus allowing the user to reflect polymorphic features of proofs in formulae. This is in contrast to classical and intuitionistic connectives where the meaning of a compound formula is dependent only on the truth value or the provability of its subformulae. Both Logical Frameworks and Proof Functional Logics consider proofs as first class citizens albeit differently. The former mention proofs explicitly, while the latter mention proofs implicitly. Their combination therefore opens up new possibilites of formal reasoning on proof-theoretic semantics. We study the metatheory of LF-∆ and provide various examples of applications. Moreover, we discuss a prototype implementation of a type checker and a refiner allowing the user to accelerate and possibly automate, the proof development process. This proof-functional type theory can be plugged in existing common proof assistants.
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Furio Honsell, Luigi Liquori, Ivan Scagnetto, Claude Stolze. The $∆-$framework. 38th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, 2018, Dec 2018, Ahmedabad, India. pp.37:1--37:21, ⟨10.4230/LIPIcs.FSTTCS.2018.37⟩. ⟨hal-01701934⟩

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