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This paper reports a Lean 4 formalization case study of an Aristotle API proof attempt for the Grasshopper problem, originally posed as IMO 2009 Problem 6. The generated artifact states a generalized Lean version of the theorem, contains four verified helper lemmas for local components of a maximality and adjacent-swap exchange strategy, and leaves the main theorem grasshopper closed directly by one unresolved sorry. The verified components establish that the final partial sum equals the total sum, that an adjacent transposition can affect only the relevant intermediate partial sum, that the changed partial sum has the expected form, and that maximality at a position admitting an adjacent successor swap forces a corresponding forbidde","title":"Using Aristotle API for AI-Assisted Theorem Proving in Lean 4: A Formalisation Case Study of the Grasshopper Problem","url":"https://arxiv.org/abs/2605.20120","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.20120v1 Announce Type: new \nAbstract: AI-assisted theorem proving can now generate substantial Lean developments for olympiad-level mathematics, but the evidential status of such developments depends on which declarations are actually verified. This paper reports a Lean 4 formalization case study of an Aristotle API proof attempt for the Grasshopper problem, originally posed as IMO 2009 Problem 6. The generated artifact states a generalized Lean version of the theorem, contains four verified helper lemmas for local components of a maximality and adjacent-swap exchange strategy, and leaves the main theorem grasshopper closed directly by one unresolved sorry. The verified components establish that the final partial sum equals the total sum, that an adjacent transposition can affect only the relevant intermediate partial sum, that the changed partial sum has the expected form, and that maximality at a position admitting an adjacent successor swap forces a corresponding forbidde","title":"Using Aristotle API for AI-Assisted Theorem Proving in Lean 4: A Formalisation Case Study of the Grasshopper Problem","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-20T04:43:44Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.20120"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:f33fcb1bfc19c94f48ccc7257b4e0ee59ad4d527a9556bf94bc32d8546b6ee863f35a42ec2a9e6a620efba4bfd04fd353ef98eb977d39dc147c621c8e84fbc02","signer":"crovia.substrate","subject":{"observed_at":"2026-05-20T04:43:44Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.20120"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"69c4881659d1dda922891ec1d54fc75d9fd365bbf8eef10bcb8207b0874b7090","leaf_index":144961,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"55080a852612ae4a2153599d3c892a6f15fcbb867f9d30e335b8b4e28b1da833","side":"left"},{"sibling":"86641f74d5549dfba28e4aa53f1e1e50d3f45e7b25969ba959d82e70a36a9ad3","side":"right"},{"sibling":"95bc5f4c149074df7c91a7c512bd88789a6f29a97c4fe41c4a8f84b005ac9fed","side":"right"},{"sibling":"c13a4ff711ed44cef142936cedb099faaeb4a5ff097572f4821a3c11fc3deeab","side":"right"},{"sibling":"792048c4180c5f0694e290f27c8772061668d06cc7c061876f8883321dfcda39","side":"right"},{"sibling":"f9a160dc5cf8feb3f1f50bad1e10d385a8a7fe4d264096626ff34be73ab3436c","side":"right"},{"sibling":"41f0af342418ef1081b5d968b56fe8290cf77b51073587d5abca55a1dcf5d23b","side":"left"},{"sibling":"5c26ec91378ced2564bd7218068bacda3ade3a0517e2dc081e9d169a340044ba","side":"right"},{"sibling":"9db04d2181fb869f16a5baf7dcfe999cc93a7e9beef8493d0cc71f57470fe5e1","side":"right"},{"sibling":"1526885f19d1fadf6955cf519dbc4e62d593a4bba99d741e8c301740a7068233","side":"left"},{"sibling":"e3a7d5c07f161682d61bd453ffc02ecdf87cfeda70f986d6650017f9d2d6b265","side":"left"},{"sibling":"edbc49f08e5b92291934c05c9e6efd270a6b0698d8d2fa474006366027dfe098","side":"right"},{"sibling":"3e4df6e7457cecbf36f350375e72dcab336a3984422e4c406ef809e4e2944e96","side":"left"},{"sibling":"8f4c0fbe56b6c010fbb8c782ebcd478079bb3f991d8704e2534209a075d9163c","side":"left"},{"sibling":"be08fedc4e72a6fb56606f66812fae7317e09690b9acb18385f4ab117a981238","side":"right"},{"sibling":"0534329a7475dc9df51998c83c16892126679dade0fa34182f21e869599386c7","side":"right"},{"sibling":"2d24720928ead0e7670650eb55f558c4f20e4c18df376f47ba72cfa8cf0ed344","side":"right"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":147301,"merkle_root":"08903d7159c3b38eeeeafc09eab15139ea417f1d94f02f1fbc87296b37db840a","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260521T183701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-21T18:37:33Z","sig_algorithm":"ed25519","signature":"905f2924632dfa2970c8690285f5b5d4a1d891d0e0ef1cbc404ebec2fd937215ac768e16f0a9f28b18977a55ae0bfd226db6833ae7ef588729054117d2da7303","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_4407f8d44e483d2a40955b19b62f4b26091fefb9fbec3d5c3f0df6be8909b750"}}