Proof-functional logical connectives allow reasoning about the structure of logical proofs, in this way giving to the latter the status of first-class objects. This is in contrast to classical truth-functional connectives where the meaning of a compound formula is dependent only on the truth value of its subformulas. In this paper we present a typed lambda calculus, enriched with products, coproducts, and a related proof-functional logic. This calculus, directly derived from a typed calculus previously defined by two of the current authors, has been proved isomorphic to the well-known Barbanera-Dezani-Ciancaglini-de'Liguoro type assignment system. We present a logic L featuring two proof-functional connectives, namely strong conjunction and strong disjunction. We prove the typed calculus to be isomorphic to the logic L and we give a realizability semantics using Mints' realizers and a completeness theorem. A prototype implementation is also described.

Proof-functional logical connectives allow reasoning about the structure of logical proofs, in this way giving to the latter the status of first-class objects. This is in contrast to classical truth-functional connectives where the meaning of a compound formula is dependent only on the truth value of its subformulas. We present a proof-functional logic and we give a semantics using Mints' realizers accounting for intersection types, union types, and subtyping. The semantics interprets the type ω as the set universe, the -> type as a function space, the /\ and V types as set intersection and set union, respectively, and the subtype relation as a subset operator. Using the proof-as-types and terms-as-propositions paradigms, we extend the typed calculus previously defined by the authors with a decidable subtyping relation and we show this calculus to be isomorphic to the Barbanera-Dezani-Ciancaglini-de'Liguoro type assignment system. A subtyping algorithm is presented and proved to be sound and complete. Hindley gave a subtyping algorithm for intersection types but, as far as we know, there is no system in the literature also including union types.

Using Curry-Howard isomorphism, we extend the typed lambda-calculus with intersection and union types, and its corresponding proof-functional logic, previously defined by the authors, with subtyping and explicit coercions.We show the extension of the lambda-calculus to be isomorphic to the Barbanera-Dezani-de'Liguoro type assignment system and we provide a sound interpretation of the proof-functional logic with the NJ(β) logic, using Mints' realizers.We finally present a sound and complete algorithm for subtyping in presence of intersection and union types. The algorithm is conceived to work for the (sub)type theory Ξ.

We present an ongoing implementation of a dependent-type theory (Δ-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.

We introduce a dependent-type theory Δ-framework, LF-Δ , 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-Δ 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.

We present the Delta-calculus, an explicitly typed lambda-calculus with strong pairs, projections and explicit type coercions. The calculus can be parametrized with different intersection type theories T, e.g. the Coppo-Dezani, the Coppo-Dezani-Sallé, the Coppo-Dezani-Venneri and the Barendregt-Coppo-Dezani ones, producing a family of Delta-calculi with related intersection type systems. We prove the main properties like Church-Rosser, unicity of type, subject reduction, strong normalization, decidability of type checking and type reconstruction. We state the relationship between the intersection type assignment systems à la Curry and the corresponding intersection type systems à la Church by means of an essence function translating an explicitly typed Delta-term into a pure lambda-term one. We finally translate a Delta-term with type coercions into an equivalent one without them; the translation is proved to be coherent because its essence is the identity. The generic Delta-calculus can be parametrized to take into account other intersection type theories as the ones in the Barendregt et al. book.