{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:36:37Z","timestamp":1753889797370,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2017,11,27]],"date-time":"2017-11-27T00:00:00Z","timestamp":1511740800000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2017,11,27]],"date-time":"2017-11-27T00:00:00Z","timestamp":1511740800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2017,11,27]],"date-time":"2017-11-27T00:00:00Z","timestamp":1511740800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>In the Simply Typed $\\lambda$-calculus Statman investigates the reducibility relation $\\leq_{\\beta\\eta}$ between types: for $A,B \\in \\mathbb{T}^0$, types freely generated using $\\rightarrow$ and a single ground type $0$, define $A \\leq_{\\beta\\eta} B$ if there exists a $\\lambda$-definable injection from the closed terms of type $A$ into those of type $B$. Unexpectedly, the induced partial order is the (linear) well-ordering (of order type) $\\omega + 4$.   In the proof a finer relation $\\leq_{h}$ is used, where the above injection is required to be a B\\&amp;quot;ohm transformation, and an (a posteriori) coarser relation $\\leq_{h^+}$, requiring a finite family of B\\&amp;quot;ohm transformations that is jointly injective.   We present this result in a self-contained, syntactic, constructive and simplified manner. En route similar results for $\\leq_h$ (order type $\\omega + 5$) and $\\leq_{h^+}$ (order type $8$) are obtained. Five of the equivalence classes of $\\leq_{h^+}$ correspond to canonical term models of Statman, one to the trivial term model collapsing all elements of the same type, and one does not even form a model by the lack of closed terms of many types.<\/jats:p>","DOI":"10.23638\/lmcs-13(4:19)2017","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:35:00Z","timestamp":1743701700000},"source":"Crossref","is-referenced-by-count":0,"title":["Statman's Hierarchy Theorem"],"prefix":"10.23638","volume":"Volume 13, Issue 4","author":[{"given":"Bram","family":"Westerbaan","sequence":"first","affiliation":[]},{"given":"Bas","family":"Westerbaan","sequence":"additional","affiliation":[]},{"given":"Rutger","family":"Kuyper","sequence":"additional","affiliation":[]},{"given":"Carst","family":"Tankink","sequence":"additional","affiliation":[]},{"given":"Remy","family":"Viehoff","sequence":"additional","affiliation":[]},{"given":"Henk","family":"Barendregt","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2017,11,27]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1711.05497v2","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1711.05497v2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:35:00Z","timestamp":1743701700000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/4074"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,27]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-13(4:19)2017","relation":{"has-preprint":[{"id-type":"arxiv","id":"1711.05497v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1711.05497","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1711.05497","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2017,11,27]]},"article-number":"4074"}}