Second Order Isomorphic Types: A Proof Theoretic Study on Second Order λ-Calculus with Surjective Pairing and Terminal Object

• Published in: Information and computation Vol. 119; no. 2; pp. 176 - 201
• Main Author: Dicosmo, R.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.06.1995; Elsevier
• Subjects: Applied sciences; Computer science; control theory; systems; Exact sciences and technology; General logic; Language theory and syntactical analysis; Logic and foundations; Mathematical logic, foundations, set theory; Mathematics; Sciences and techniques of general use; Theoretical computing; Second order; Type; Intuitionistic logic; Soundness; Decidability; Completeness; Lambda calculus; Isomorphism; Proof theory
• ISSN: 0890-5401; 1090-2651
• DOI: 10.1006/inco.1995.1085

We investigate invertible terms and isomorphic types in the second order lambda calculus extended with surjective pairs and terminal (or Unit) type. These two topics are closely related: on one side, the study of invertibility is a necessary tool for the characterization of isomorphic types; on the other hand, we need the notion of isomorphic types to study the typed invertible terms. The result of our investigation is twofold: we give a constructive characterization of the invertible terms, extending previous work by Dezani and Bruce-Longo, and a decidable equational theory of the isomorphisms of types which hold in all models of the calculus, which is a conservative extension to the second order case of the results previously achieved for the case of first order typed calculi. Via the Curry-Howard correspondence, this work also provides a decision procedure for strong equivalence of formulae in second order intuitionistic positive propositional logic, that is suitable to search equivalent proofs in automated deduction systems.