Blanchette, J.C., Popescu, A. orcid.org/0000-0001-8747-0619 and Traytel, D. (2017) Soundness and completeness proofs by coinductive methods. Journal of Automated Reasoning, 58 (1). pp. 149-179. ISSN 0168-7433
Abstract
We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL’s recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL’s new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion.
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Copyright, Publisher and Additional Information: | © 2016 Springer Science+Business Media Dordrecht. This is an author-produced version of a paper subsequently published in Journal of Automated Reasoning. Uploaded in accordance with the publisher's self-archiving policy. | ||||
Keywords: | Codatatypes; Lazy evaluation; First-order logic; Soundness; Completeness; Gentian systems; Proof assistants; Isabelle/HOL | ||||
Dates: |
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Institution: | The University of Sheffield | ||||
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) | ||||
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Depositing User: | Symplectic Sheffield | ||||
Date Deposited: | 30 Sep 2022 14:47 | ||||
Last Modified: | 30 Sep 2022 15:19 | ||||
Status: | Published | ||||
Publisher: | Springer Science and Business Media LLC | ||||
Refereed: | Yes | ||||
Identification Number: | https://doi.org/10.1007/s10817-016-9391-3 |