{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T14:21:04Z","timestamp":1754144464251,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T00:00:00Z","timestamp":1752537600000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T00:00:00Z","timestamp":1752537600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T00:00:00Z","timestamp":1752537600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,7,2]]},"abstract":"<jats:p>The adele ring of a number field is a central object in modern number theory. Its status as a locally compact topological ring is one of the key reasons why. We describe a formal proof that the adele ring of a number field is locally compact implemented in the Lean 4 theorem prover. Our work includes the formalisations of new types, including the completion of a number field at an infinite place, the infinite adele ring and the finite $S$-adele ring, as well as formal proofs that completions of a number field are locally compact and that their rings of integers at finite places are compact.<\/jats:p>","DOI":"10.46298\/afm.14840","type":"journal-article","created":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T14:55:14Z","timestamp":1752591314000},"source":"Crossref","is-referenced-by-count":0,"title":["Formalising the local compactness of the adele ring"],"prefix":"10.46298","volume":"Volume 1","author":[{"given":"Salvatore","family":"Mercuri","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2025,7,15]]},"container-title":["Annals of Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/2405.19270v4","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/2405.19270v4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T14:55:16Z","timestamp":1752591316000},"score":1,"resource":{"primary":{"URL":"https:\/\/afm.episciences.org\/14840"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,15]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/afm.14840","relation":{"has-preprint":[{"id-type":"arxiv","id":"2405.19270v3","asserted-by":"subject"},{"id-type":"arxiv","id":"2405.19270v2","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2405.19270","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2405.19270","asserted-by":"subject"}]},"subject":[],"published":{"date-parts":[[2025,7,15]]},"article-number":"14840"}}