A rewriting logic approach to specification, proof-search, and meta-proofs in sequent systems

• Published in: Journal of logical and algebraic methods in programming Vol. 130; p. 100827
• Main Authors: Olarte, Carlos, Pimentel, Elaine, Rocha, Camilo
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2023; Elsevier
• Subjects: Computer Science; Proof theory; Rewriting logic; Sequent systems; Symbolic Computation; Sequent systems; Rewriting logic; Proof theory
• ISSN: 2352-2208
• DOI: 10.1016/j.jlamp.2022.100827

This paper develops an algorithmic-based approach for proving inductive properties of propositional sequent systems such as admissibility, invertibility, cut-admissibility, and identity expansion. Although undecidable in general, these structural properties are crucial in proof theory because they can reduce the proof-search effort and further be used as scaffolding for obtaining other meta-results such as consistency. The algorithms –which take advantage of the rewriting logic meta-logical framework– are explained in detail and illustrated with examples throughout the paper. They have been fully mechanized in the L-Framework, thus offering both a formal specification language and off-the-shelf mechanization of the proof-search algorithms coming together with semi-decision procedures for proving theorems and meta-theorems of the object system. As illustrated with case studies in the paper, the L-Framework achieves a great degree of automation when used on several propositional sequent systems, including single conclusion and multi-conclusion intuitionistic logic, classical logic, classical linear logic and its dyadic system, intuitionistic linear logic, and normal modal logics.