upper_trivial

Specification

From [Aa19] p.577: the trivial upper bound is $n^2$ (non asymptotic)

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Lean 4 Statement

theorem upper_trivial {n : ℕ} : max013AffineTranslates n ≤ n ^ 2

ID: GreensOpenProblems__24__upper_trivial

Evaluation Results

Recent Submissions

Model	Status	Goals Left	Submitted	Lean Snippet	Output
GPT-5.3 Codex Agent v8	Partial	3	Feb 25, 2026	unfold max013AffineTranslates rw [Nat.sSup_def] by_cases hbounded : BddAbove {k : ℕ \| ∃ A : Finset ℤ, A.card = n ∧ k = ((A ×ˢ A).filter (fun (x, y) ↦ x ≠ y ∧ x + 3 * (y - x) ∈ A)).card} · simp [hbounded] · simp [hbounded]	View { "agent_telemetry": { "compilations": 3, "iterations": 10, "duration_s": 181.31, "best_status": "partial", "best_goals": 3, "max_iterations": 10, "model": "openai/gpt-5.3-codex", "error": null, "provider": "openrouter_primary", "finish_reason": "tool_calls", "usage": { "prompt_tokens": 51881, "completion_tokens": 2979, "total_tokens": 54860, "reasoning_tokens": 2316, "cost_usd": 0.08330735 }, "reasoning_effort": "high", "response_trace_count": 13 }, "response_trace": [ { "iteration": 1, "event": "search_mathlib", "query": "Nat.sSup_le", "results_count": 8 }, { "iteration": 2, "event": "search_mathlib", "query": "sSup_le", "results_count": 1 }, { "iteration": 3, "event": "search_mathlib", "query": "Finset.card_filter_le", "results_count": 1 }, { "iteration": 4, "event": "search_mathlib", "query": "Finset.card_product", "results_count": 1 }, { "iteration": 5, "event": "search_mathlib", "query": "pow_two", "results_count": 1 }, { "iteration": 6, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1772001594-yp21rW0OQGVJbZCROxVV", "api_elapsed_s": 9.03, "usage": { "prompt_tokens": 2713, "completion_tokens": 514, "total_tokens": 3227, "reasoning_tokens": 325, "cost_usd": 0.00791175 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_text": "Drafting cardinality proof skeleton", "reasoning_details": [ { "format": "openai-responses-v1", "index": 0, "type": "reasoning.summary", "summary": "Drafting cardinality proof skeleton" }, { "id": "rs_0cc9539925cfafc001699e993b2d948190b4411616f37a9984", "format": "openai-responses-v1", "index": 0, "type": "reasoning.encrypted", "data": "gAAAAABpnplCkU_lRfisSsct1-5gPk36I00Plx9P_IXI-H_wkR9zs-mgjaWoHbBcYY2YIyx2BAd08yIQP_Ma2bnQOEi2wvrKZz--XYRGBfojKw-KoLNoZz5-7DycN9DPgR48zh-EnoktOKAYZ3hzH-uV5lmAs6zPm5nx4pYAA8-p2rnbgUO3_5IpJuS2ClTlRCsfzrizkJIXWkOCDMAsYlHU4E_Maks2ekq7wNDOGBmRIJXgjLvH0qhl3sxGQ0I3ebUauqAWEFU-iuVe2IhqjQKH6523nSHJhDGLyabfFFf4hT1Y9DiSxXi3ctFs_oOurhaqEvYJjfoArDNiORi6jqEqKDl7SpkPS0jX0EjpHLCu6G5LUGabTSSTuB1eIA8tIVEbEcPbxRTEcHys1SVlUUUVBiPmputmCcNa3FhgSBQo3uM8xYTOfKQniU1oeV784wYbRu00Q87IwFqkRANTzxgY50lA2WpMvMc9s-7KuK1KhgQsxqHjI2hT-9SDEkNu9STh3n3X484UreW16YI2y8Q1-IEH3hezFGYfOJuZ4awJgWr_zfR72NADAx5k0c01ILhAYFv5yc79wvxm-A7gzOJuCKaX1nmu0LRgJSBuN_ePAsq37rmWPVp1PAotB56UHG6dw79yQtYpvc-HF-xUQ2Nh4q2NdkYbTVYDBCrS8UCc-ptW2iQ39bORvyqp14ntPJbCYYu_TZI1CnloXIKT_liy4TgoYLk5wqpkasOCFm4mcrdwZ5MVgeLSNgyTWFH3Q67f9KMzsil0DSgBQdRdI4dBHunhnxCbys5_9jV_MKEknk5YqxuQ60YzT58hkORpl6dzLXdmP7BMCDvnInBbcSWraNR34DYDoNV9ZN2TAt_zWNpbSAWpGSrgMYj4mWQiayIcUTXVnoj_yuMr7Vp-jehPqYPJYW0FOLHlA0uGtlVEo983nNZMD8AFMvyyi56-yEc7o9H374YofJ7-pc4e47Uezj8Lhzf3kRxjSoa5W-oYCCn2aNWU9BaBDcc5e2RiYjhkLlR8Un0YBgwxtJNyyWJBb9tT98nyWCObBoE4xhLThxrhmd4qGoJwVJI2BmC356LRweTASKXOHvWkj1d1bjRhl3ALehMqa5RieS-FhsYTu5F9ER_U7DT2h3oLofdO--7FIBS0WPXCzoQ8yaymrDUUq3nIP2xPmURz-2mBL8uzB-DgP3SsnxUiZdEA3bGZZyuahZnTjCznHEFz_mNFL5KYYGBySIjW4jlDlFiaHpYOFKxZaOea_8b_bYwLQKbbB-xr1yp4RVMU58fMtMuIEr4kt9oNegpoDWuR8oP9qeLIe4JOQm53pkjVq-TxTbbJxtHjZO9hMDVWTw9Ek9m4SIle1lyjLBZRo7CPSvSDV_wFA8KqAOxPAFjfSrhi2-7aANC25M54EojJ_a1tr9gcDgP9fWYebbIaKhY6LSlu3eZgogghmH1N8CIYUu4Gk9dOn0lhzksQVeQVGD5vhlGKt_-G2LbQXSCAPUow5gx40LSap8tBUyS8kJ0jy_49Ws8-8kqtztYNgXr_zSbrFrGUDTFtsQkXkdRWpGaT7XyREn-lCzEd25x6aPhJ4KN4wpmyRdnN6ZZWt2d2uJXi102PTMZL-Bc_cO2hjml_eT4-FwHcfb3wP7WtnRUl0OfZz85QR2TYeM7T2IEY7H0m0HZPoe1p6KBMQMSRHC8_zSOzrRlvZYCkc5LGaBm7bpAbTzD9FYtgpMMaj3NXs-mfST4DUznpsQIp8mYkrvdl7HFsw2GHS-jc4HowlcsqwWCR56ZUeFjrBE13p_poF9xubE_uNdNKgOPMc9Ii7mu4ujv94K6xyVe6Cv76CwPA6rfmyTAaLXVdw0I_LuN_DVKcYqgJjd01k4J1-tVWAO4XxOUqwTlLlqPY18qF7Zpp-1pFblA-2RA_19yEEaJ3sB4JCy61L4gN_v856GIWsvb0qcYoi-n8CwqNF4e9VIaRrFznM6VanGmZ9oA0bKNNTr-gRY63BPWaGZUxrZVQKHksuV6Gr_TPfWkr6kDBgtAX8FY9cW5pR5lXcsqkD9fIwq5ANF9CDGUqF0rTrcBBJEbhbZjENdIj1DZ7n-7HQxB8toNlwBHgNsfvSWX93jWChih08NAfvxQbPecLJtpSAdKgL_CoCVOMPTS6_AMiCkeWJ9ixH3xv8keMMS8ph9JmpyoaW1vhQxzlOFrJzedQR_wd8WX-8GAwGDdpOFXxO7s6npkG7bQ1hHJIosWHrzcTYoqT02g-kNFkvpNxS_CmXSFvjPjg6EDTbhxJWUvCUsttX9LOvPDAf1sVul1mW8s6" } ] }, { "iteration": 6, "event": "compile_proof", "provider": "openrouter_primary", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "refine sSup_le ?_\nintro k hk\nrcases hk with \u27e8A, hA, rfl\u27e9\nhave h1 : ((A \u00d7\u02e2 A).filter (fun (x, y) \u21a6 x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)).card \u2264 (A \u00d7\u02e2 A).card := ?_\nhave h2 : (A \u00d7\u02e2 A).card = A.card * A.card := ?_\nhave h3 : A.card * A.card = n * n := ?_\nhave h4 : n * n = n ^ 2 := ?_\nexact le_trans h1 (le_trans (by simpa [h2]) (by simpa [h3, h4]))", "compiler_output_preview": "/app/asi_portal/verifier/quick_1772001603_5689.lean:36:9: error: typeclass instance problem is stuck\n CompleteSemilatticeSup ?m.15\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `CompleteSemilatticeSup` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For exa\n...<truncated>", "compile_elapsed_s": 49.071, "tactic_replay": { "tactic_count": 8, "successful_tactics": 0, "first_failure_step": 1, "goal_trajectory": [ 1, 1, 1, 1, 1, 1, 1, 1 ], "max_goals": 1, "final_goal_count": 1, "goals_opened": 0, "goals_closed": 0 }, "lsp_feedback": { "diagnostics": [ { "fullRange": { "end": { "character": 19, "line": 35 }, "start": { "character": 9, "line": 35 } }, "message": "typeclass instance problem is stuck\n CompleteSemilatticeSup ?m.15\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `CompleteSemilatticeSup` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck.", "range": { "end": { "character": 19, "line": 35 }, "start": { "character": 9, "line": 35 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 6, "line": 44 }, "start": { "character": 0, "line": 44 } }, "message": "'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]", "range": { "end": { "character": 6, "line": 44 }, "start": { "character": 0, "line": 44 } }, "severity": 3, "source": "Lean 4" } ], "error_line": 35, "error_char": 9, "error_msg": "typeclass instance problem is stuck\n CompleteSemilatticeSup ?m.15\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `CompleteSemilatticeSup` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck.", "goal_before_error": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "full_timeline": [ { "line": 35, "code": "refine sSup_le ?_", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 36, "code": "intro k hk", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 37, "code": "rcases hk with \u27e8A, hA, rfl\u27e9", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 38, "code": "have h1 : ((A \u00d7\u02e2 A).filter (fun (x, y) \u21a6 x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)).card \u2264 (A \u00d7\u02e2 A).card := ?_", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 39, "code": "have h2 : (A \u00d7\u02e2 A).card = A.card * A.card := ?_", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 40, "code": "have h3 : A.card * A.card = n * n := ?_", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 41, "code": "have h4 : n * n = n ^ 2 := ?_", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 42, "code": "exact le_trans h1 (le_trans (by simpa [h2]) (by simpa [h3, h4]))", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 } ], "tactic_count": 8, "successful_tactics": 0, "first_failure_step": 1, "cps": 0.0, "cps_peak_step": 0, "error_code": "refine sSup_le ?_" } }, { "iteration": 7, "event": "search_mathlib", "query": "Nat.sSup", "results_count": 8 }, { "iteration": 8, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1772001657-Iu29DlniAYE302swJv2F", "api_elapsed_s": 16.265, "usage": { "prompt_tokens": 5902, "completion_tokens": 869, "total_tokens": 6771, "reasoning_tokens": 680, "cost_usd": 0.0138257 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_text": "Planning Nat.sSup bounding proof", "reasoning_details": [ { "format": "openai-responses-v1", "index": 0, "type": "reasoning.summary", "summary": "Planning Nat.sSup bounding proof" }, { "id": "rs_0cc9539925cfafc001699e9979aa588190a3b1cc767308d678", "format": "openai-responses-v1", "index": 0, "type": "reasoning.encrypted", "data": "gAAAAABpnpmH3ObggJUXvWL8MCD3lE_9MD6VuqZhM5_0bTSzXUfy1lHnbMQz8am-P9MGfVTCR6xPlr9LSFtg_6Yq0oiJOFeJOPB6fUYU9Pw6nJ2altxy3SEkyqPtOHVbm6yxAUOgy4PwQ42FS38j1hqogJk4tpDCwgiCzh_0Q05RFGSffCWeqzAOMIphiKrttw0y_pa3Q_50jW0EWOc0JGUPTtewZo-gG6lWjIHVEQjGXaP1ZpgkzSGk1yv6IE4ePIepAUKd4V3nLEJiCQb6phIedKb70lhc71t2mGl2r6SOtqVsPWojUouHUmr_A1rNxea1WUodQGdCicyyoAJqlgx8iWbO84xMWTwNwHD_ElV3-Gdj_3ExbYwLUe8uiwsijCJeqLEihapYu9oPmJWd9WoR0EtGchhyrAu3ZelbymXm7J-bXpqXJVwTXheeoSXYaHBYRWTZGFER5HWGl6VE2fc6qbQrII9wiMD0WswX1lWW2wyRfEsAYGPoWHHNVQIGWG1iysO-fr-ukF5oqEMsn9o8ZJm1saYuYxZ5-8Ynwa_-Udyc7cJCCnM09deUV0YUOTVzcDBShsJBg17a97i1KlFfffiiUmz9TFGd9t3nR9iHBe7AqqSK13oSiUvckpjLU3XAYlk3zEybF_k0bBuon7kB67PVNdeKDe305f6WfkGma7OnCrvALeU0CxI3EJBaqGASRLl-rXSLV2r3Ijzz76E8zY6vFFKtcHQdffsOyZ60Yva-35sbadRDw8yEQ7g3b9AYEbPuOuUHOAP4moSDHnr3DjDQaVHqQNe6ZAL4So6qSHLh0EuECM8scUwyEVDHHJNhEta6n4XV7Q6yprSVgatIfpGN3hQZeaf77sSqhsaD7mqLh6OIyO0H2WeYzskAAe6pVWf-TfqheCxBjlHumaVo2pMh4j6_0BScRVvuWjT7meBKMWEFfM-Y6luSLMjsR8t8eTCS4NnSknKjCVWOOZMpp4D0KJZLjpgzw259h11w43WrlUHDXIUZZADCLanGNpmmUoJwwiwvE0rvxVA21pGS6uWMsLorQfW9l8VUmGHIf-Km9Y3wSKuANsjJ3BQyQk4d6UKWpYrkw5HSczk6Soh-n_UqJ39kEFb9jLh67TnuOICcvAmItpMdUKRibMlGmc74_T1Xg3qtK5K1cxS-Uj1gkz2Iuoa_drZck9D7DF_QC7NTpMXJBDe0ZsIR56v5ebsY81EkXKnT56Rbm0qZCuo1T1cRf2Sy9wevvPviZEcdg0UNNk8iV7Tf-VKBHcgwXwOWLvFMYFr-2E12ZYg6qNg6tIlFZ63dcAda9aphdI44fe2C0zOipl48hER94m0PI8Lt0D5aHuT2f17HyLG8hzH6Y225DJK6SsZ961m6i_NiJd99AdianKRl1eEw5AsLuRh3DHhB1Zvz4r4EqbIAFjqGSn0ZJE-oKL2kHZFz_NL5XvUSPznD4Dg0JsHDayC0lrqBMFgLTJ2Mz7Mbz5Elbymc2anyVAcLtHqORJ4xq6ZzuAtty8OTvsK5pWB30Xu6c8p2NONleni55EIh4dn1e17R_2DVPFth1ktTaH7Cn6EH-5t2hgtqWYQ4S-jx5vNokBBEB-A-qjvoH88o1rHKQR477hXE1TgGui5uWvksKhM810iRQXOKnCq5CaA8shZw0JNKmbiNvnVMeiTGWuqu52Hma1_-9_iX4iqBOicIfTu5TvpGcTp2Zl1TtcsvpVbKZmAxAnIIsK8KsYOKdfi3--7YQSUZZVC1Ek2wN_b55OgB3poIOW9oodJfHKClThhQBOXzVyuQ8_M5xofEOLQxJrfn6MAqlMzZXQ5Q5Td3VH6ibcHPoMZCWC8PGb8T4S3KbI5b0BoYT_nZ2P9o3Jn53Rl8E2GGNWTJQsMv4ZzHsW5r5FK4tLfQZVD912lDdZ8DNZpefnDSMa2xBq7ykPbyY28FhWEfISDs3CCWIYNSctu9JXBC8clNKldOHjDfDCu5aV-yu-1Vpm9xopQH2-q4lBK2YJMxz2Sei2WmY_hjfKOUUl1Ujlw7p5evxildY1Hrd-h5x4SbEOnBI-SX4bcsgbx_6WnLBDBvBXB8Eh6UuVjhQim_ciCEdFDPmg1N89Dq2Gh6eBKpeL97s-Cw8SlDumK3gtO4W4zQw6MEcz2AYg48b67UfsX1_qI-HwIVGVDr5ah28mlOJ6B9CJzNRbjD78Y6xr9ENczIYtRXLtXnE0nrqh5Zn0TaYGAj2l2dOBbaRaTHxUy-iXUbud69UYjAZLJ39q-Leu6GcswTPczgkRAZ9O9OKhKOiPBVTHcQC0UIYUHyYHSbn7uQZ9UmB59n7PvWzJU-2aG8X5_8Tklu8UFoguZlkqDHE_K9_9pV6zaQJ4hlNUmy2zjjL7579J3WU5_FaegIe-LifqioyH7oPqUMHeP8S0W_iVZ8ptLtNWFqTTBUCvoEFcw0isYcnj5kCvQvw_IUCYSt0Tp9jBo7AIcEU3Yk5TnZBiRUud1lVWrPhURxFQFXP31I0osuf8kfUNrVAzOX4_htd18KP35xtFqqcFXD9lPGqo1L_uHGWJ25MLrqykv5VEMxKfwMQNnRjOLB5KgkGaTbKHNLc4wkaRzJOzgKmZ1YIcQH9k-R59n791eNn4Yo6w_agvfvjDwNyRsp0Aqsbw1cjdVLOR2RT6OsCZx9XgH_Y04CSRM-x4IKskZLwCXBCqAqg2PSK2PGlquTRjTAhCOYyfUnFXNIGTqgjx2Wql5mEnmx5v1W18JVj0AqoqBCFGO8Rrc7Pw1wYCZJM5agw7UsGvqxyWhplqB29DAFcmMQJFXgujjTO2DDdlAY5kyrMyBGxD_RUX3nY8ybAaftFLPM5VN_gLgh0erRpgK1hd3laTb1f5_3-NGjyG-NWNe4CkK3eaQaCkxAvxcwS_pAg9ax3eo5QUTVRu27tP4Vl1JNvh-Z5GHkn1br0f7r3AnvJeCzgu9jhrpmuzHBau_ypdwXbep7f_liwbPf5QZ2z7l3-AK0m-Pj2Kdq8jkDiSsDGMfOShG_kuc7gVRz7lngy8W0vHBGzYeymu4RDUsEzMgsG8r5qO-iyE9hqzEH6r7_iqauQcJ9wwrRopheRWgj-wk4FfyL1bJWyYbKX_3v5hYG_VAhWJqZoA-LDa_XrvL33PjCgr2iaSKPKoapZjIDTE5lFMAiuB_WuyzOV6DyqOxoqYVdowaJrVmYyTkmoQCleGP8wiMMc7hu4mbty-fOUrWBcylHhcPfwBGvQi_NFiIxSYVOtNhBidDc-V9mMldl9GXR05n2j8B132eJNPcTqc6ZV_XcWcKRFL5XX3bcYVZO9A1PuMh1ITFimvRhy66hnyfgvc0_nd2_g8O9G8NZyf9KI6XXHkiP9ASyqqXD2LsjirLrkiEeBh10blXxL2tWPTiYaRtvn_ilYOIuFd_UmRrpWwoNWAIqguEgvrwCHgcMI1VYFhbzRhy2I20XgJRqzR1iu0QlCAfDHHUzH15u7Y9Z5vKVhTgcoVGnlHi0Id77KqL1xxRWKor9rx0OEe0s1xI4tDnrE3Jv-B_4kPIs1bZBWcYTAB9UX9f2nFAKB6kyi5JctDryPrANOr-LMVW-gHiUlPUFBjHz8rN1Fqta1IN2xbs2jZpH-Ix1xlYHJ06xZ2FojJf9Pykv2igogAZ-n_QJ7qMJVJ65TDWTpqZfrm3ncUJAI7ZWn27OnbklsgaloeKABdvbzj1bCtL-aL0knvceuBtpF9NatoCHn-ZLBBXmCcDhMD55TLcG3YobUz-VtE0GvBZDpjsGkFSyyVgXfTtUeZR_Axa3kzZ2tVktt6nwejpI6oW4IhvReHYRyRkBwiDL-YVSB42Et7TbFBN-5N0hZXWBP6yCWpUJkag2ZG2iEI-5zFYqhV4PKS5kHu7OpEfSzbTmlfgu5ZkgaIlyphaa_a-YoWMvouvXE7iPYZglJUD1t26p8Uhu5DxcVx8uExgF4gylMoObx9ihNlWBr5577hXtJQ-3DK2hrWzsxYny6gowZUW-s01lrd_zEKPqz2kpVmX80YD0JMupbA3EHXcDftQfJkyl0ZmqZnzqwUO6U62xYuQ_avhnklkyNveTfRYG4D7RwhrR4HMraQqeRXLjrGd6K1TI3LfIP-fvat3QHNmMbMt-sFeDpBS3bHh5eNSLeY_b3HOyfTu-udw_CvNIV5Q5lqjiIO9Hiq-KHtgPtRfKrwQqugSSuwQdjaqx1IcaLxPmEdaiumBMCqDssvPMIlyEftKcfQyMylggPsG5FbZ4B_GQ8bk_H3o1OwZydmUV77jtFX4DtEXXZ0AHTKiR1LoZXGkRqcgwFNS3VAhkdLfPcqDKi4ohn7rUQNboNNKKGoC8EHg5nUPN1rBPIvPj1tpFzuZznaisERUzZuS5BMeNZXiSNj3ozGv27ToKXpJdNk8yaXvCkqaUvo_G52aWcFnCsvrX4p-cz0FtdYSSlAlfKtSMT9bk8gTKb1wyoH9r2IKIyM5Se6Bj9X626b1zZVdJmPgkcYkfUnz-X0eMEQ==" } ] }, { "iteration": 8, "event": "compile_proof", "provider": "openrouter_primary", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "unfold max013AffineTranslates\nrefine sSup_le ?_\nintro k hk\nrcases hk with \u27e8A, hA, rfl\u27e9\nhave h1 : ((A \u00d7\u02e2 A).filter (fun (x, y) \u21a6 x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)).card \u2264 (A \u00d7\u02e2 A).card := ?_\nhave h2 : (A \u00d7\u02e2 A).card = A.card * A.card := ?_\nhave h3 : A.card * A.card = n * n := ?_\nhave h4 : n * n = n ^ 2 := ?_\nexact le_trans h1 (by simpa [h2, h3, h4])", "compiler_output_preview": "/app/asi_portal/verifier/quick_1772001673_2313.lean:37:9: error: typeclass instance problem is stuck\n CompleteSemilatticeSup ?m.16\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `CompleteSemilatticeSup` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For exa\n...<truncated>", "compile_elapsed_s": 31.159, "tactic_replay": { "tactic_count": 9, "successful_tactics": 1, "first_failure_step": 2, "goal_trajectory": [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ], "max_goals": 1, "final_goal_count": 1, "goals_opened": 0, "goals_closed": 0 }, "lsp_feedback": { "diagnostics": [ { "fullRange": { "end": { "character": 19, "line": 36 }, "start": { "character": 9, "line": 36 } }, "message": "typeclass instance problem is stuck\n CompleteSemilatticeSup ?m.16\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `CompleteSemilatticeSup` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck.", "range": { "end": { "character": 19, "line": 36 }, "start": { "character": 9, "line": 36 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 6, "line": 45 }, "start": { "character": 0, "line": 45 } }, "message": "'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]", "range": { "end": { "character": 6, "line": 45 }, "start": { "character": 0, "line": 45 } }, "severity": 3, "source": "Lean 4" } ], "error_line": 36, "error_char": 9, "error_msg": "typeclass instance problem is stuck\n CompleteSemilatticeSup ?m.16\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `CompleteSemilatticeSup` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck.", "goal_before_error": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "full_timeline": [ { "line": 35, "code": "unfold max013AffineTranslates", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 36, "code": "refine sSup_le ?_", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 37, "code": "intro k hk", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 38, "code": "rcases hk with \u27e8A, hA, rfl\u27e9", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 39, "code": "have h1 : ((A \u00d7\u02e2 A).filter (fun (x, y) \u21a6 x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)).card \u2264 (A \u00d7\u02e2 A).card := ?_", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 40, "code": "have h2 : (A \u00d7\u02e2 A).card = A.card * A.card := ?_", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 41, "code": "have h3 : A.card * A.card = n * n := ?_", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 42, "code": "have h4 : n * n = n ^ 2 := ?_", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 43, "code": "exact le_trans h1 (by simpa [h2, h3, h4])", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 } ], "tactic_count": 9, "successful_tactics": 1, "first_failure_step": 2, "cps": 11.11111111111111, "cps_peak_step": 1, "error_code": "refine sSup_le ?_" } }, { "iteration": 9, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1772001704-sW4poJgqaHXhS8NuYWrJ", "api_elapsed_s": 4.621, "usage": { "prompt_tokens": 10622, "completion_tokens": 219, "total_tokens": 10841, "reasoning_tokens": 98, "cost_usd": 0.0216545 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_text": "Exploring alternative rewrite strategy", "reasoning_details": [ { "format": "openai-responses-v1", "index": 0, "type": "reasoning.summary", "summary": "Exploring alternative rewrite strategy" }, { "id": "rs_0cc9539925cfafc001699e99aa24288190be5db2e7e7d424fb", "format": "openai-responses-v1", "index": 0, "type": "reasoning.encrypted", "data": "gAAAAABpnpmsAr6UwP0rp0xWXbPWD94WiJxfE4rORsel2GsURbTZP9AA5npoW6qU0rBk5DXNgBWP1X1-8mSVzMl8HdXRKis7Ey4g4iNA7d5nO_FUgUOU188m-s3SRenRHmQ-L66ADHbv1GPqBggrpEsj7Xlo95vnahP5Zo2Ry8f-x4GQfeCsosqMuidsR5xhKoHk-SQalu4zFelpXvTSFmJIzoAxG3U6m5eV1fPLJGAFuq4h_ypRu4rWDjrF4ail7Pj0h-7UIlMGv76OtgBBwd5pA3W9wYscL7cQ-Rj0KrCBUuJK39TCytV2ye2gbNonaXaGs_2kBxrVvKymnehWElo63p_wHFNLAjjOmTKp_S9bbkou2jrO-ZXRX2Kknw64xg8zgy2bioQMscH3OPVA2mdQ7uzefFCtUy8EwydhI1cyOIlBvDIrb2yjB_7l_w4PXQDNMInYd9EDnEXi2A799CRDgytc1Kx-z_YRzSW1EWe9ngpPG9IibxBd1HvQErL3Vh8w_JqaQJRKrrciC3lCl83CZyUR2AqUZ_DG0kzg9n3rv04H_gbowAtE1tP1lQ6G9yZY0m4g35dVU0xv-o160OLlnnbi5ysRMC-aPSc_HouPhUahmh8yM4q8RJ1v8ahb2j_a0FD0q0K7X_nNlBIvCvcGC7FRJcAv8P5SEkBOeQH3zcwYxhaqKs76TB0opSuoKyEkUio9a7sACDPkU7w5QRe4KX6pQXMZ4R3eqcP1-eMS2qtR_g6VXnzQcwnFcOLU8X4251ihMLmsZ5iP4ta0gqx2WwZZcatWlyo1_URqhpFoCb6_jc0Oy1hCRxH3sJF20JnCpxCGi4HZE80m9kL9STnAv3diTV3aHhk1jlthISoF9kCHzD-X4n92sl4li1oN_oixOhD6O5P1xlTxH8DgeLUBQPl0UO9p15evMMho2exhGdSzga_81h3xqN1XBP3EWoTWl-j-9I2t8cpx0cx8uHcIq-Px05QAwXQzHclAn-3zmXi_1nx2gkZUTSmvrbYhehYqBjd0jwPVW-Wlyo1xp6YW-YtbNzfn3FIhLeorBaHdsY3F2ScSLZXS36cTXVwDcZ_No563lyLJHbTUOnYnzPnu_ygFr9gmi6dRaVcTXsCEtZKYOnrJdcDvenWxMoIs3cyOQ0rUyC6mE0SaeNKOrADHWDuFITXFuzW1bqEzooPNK4XjAU4LZ3TI49D0ge9tgZlSAXaPC9C2H_yvjsZ2VCzORmbUyG6YrdibRjMob5naHdESltuRKWbxnGOrI7hT2-O1eZAApAres51mBotUA3CyXPfAiqyFI-QeCULTVBcX_77aZxfcUmaATm3qAUDk_JzdCr1mLgWQRnrWN-cjdcJSuseYVaGsgw==" } ] }, { "iteration": 9, "event": "compile_proof", "provider": "openrouter_primary", "status": "partial", "goals_remaining": 3, "timed_out": false, "banned_tokens": [], "proof_code": "unfold max013AffineTranslates\nrw [Nat.sSup_def]\nby_cases hbounded : BddAbove {k : \u2115 \| \u2203 A : Finset \u2124, A.card = n \u2227 k = ((A \u00d7\u02e2 A).filter (fun (x, y) \u21a6 x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)).card}\n\u00b7 simp [hbounded]\n\u00b7 simp [hbounded]", "compiler_output_preview": "/app/asi_portal/verifier/quick_1772001709_7386.lean:39:2: error: unsolved goals\ncase pos\nn : \u2115\nhbounded :\n BddAbove\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 Nat.find ?m.17 \u2264 n ^ 2\n/app/asi_portal/verifier/quick_1772001709_7386.lean:40:2: error: unsolved goals\ncase neg\nn : \u2115\nhbounded :\n \u00acBddAbove\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n \n...<truncated>", "compile_elapsed_s": 38.951, "tactic_replay": { "tactic_count": 5, "successful_tactics": 0, "first_failure_step": 1, "goal_trajectory": [ 1, 1, 2, 3, 2 ], "max_goals": 3, "final_goal_count": 2, "goals_opened": 2, "goals_closed": 1 }, "lsp_feedback": { "diagnostics": [ { "fullRange": { "end": { "character": 19, "line": 38 }, "start": { "character": 2, "line": 38 } }, "leanTags": [ 1 ], "message": "unsolved goals\ncase pos\nn : \u2115\nhbounded :\n BddAbove\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 Nat.find ?m.17 \u2264 n ^ 2", "range": { "end": { "character": 19, "line": 38 }, "start": { "character": 2, "line": 38 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 19, "line": 39 }, "start": { "character": 2, "line": 39 } }, "leanTags": [ 1 ], "message": "unsolved goals\ncase neg\nn : \u2115\nhbounded :\n \u00acBddAbove\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 Nat.find ?m.17 \u2264 n ^ 2", "range": { "end": { "character": 19, "line": 39 }, "start": { "character": 2, "line": 39 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 19, "line": 39 }, "start": { "character": 68, "line": 34 } }, "leanTags": [ 1 ], "message": "unsolved goals\nn : \u2115\n\u22a2 \u2203 n_1,\n \u2200\n a \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n a \u2264 n_1", "range": { "end": { "character": 0, "line": 35 }, "start": { "character": 68, "line": 34 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 18, "line": 38 }, "start": { "character": 10, "line": 38 } }, "message": "This simp argument is unused:\n hbounded\n\nHint: Omit it from the simp argument list.\n simp \u0335[\u0335h\u0335b\u0335o\u0335u\u0335n\u0335d\u0335e\u0335d\u0335]\u0335\n\nNote: This linter can be disabled with `set_option linter.unusedSimpArgs false`", "range": { "end": { "character": 18, "line": 38 }, "start": { "character": 10, "line": 38 } }, "severity": 2, "source": "Lean 4" }, { "fullRange": { "end": { "character": 18, "line": 39 }, "start": { "character": 10, "line": 39 } }, "message": "This simp argument is unused:\n hbounded\n\nHint: Omit it from the simp argument list.\n simp \u0335[\u0335h\u0335b\u0335o\u0335u\u0335n\u0335d\u0335e\u0335d\u0335]\u0335\n\nNote: This linter can be disabled with `set_option linter.unusedSimpArgs false`", "range": { "end": { "character": 18, "line": 39 }, "start": { "character": 10, "line": 39 } }, "severity": 2, "source": "Lean 4" }, { "fullRange": { "end": { "character": 6, "line": 41 }, "start": { "character": 0, "line": 41 } }, "message": "'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]", "range": { "end": { "character": 6, "line": 41 }, "start": { "character": 0, "line": 41 } }, "severity": 3, "source": "Lean 4" } ], "error_line": 34, "error_char": 68, "error_msg": "unsolved goals\nn : \u2115\n\u22a2 \u2203 n_1,\n \u2200\n a \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n a \u2264 n_1", "goal_before_error": null, "full_timeline": [ { "line": 35, "code": "unfold max013AffineTranslates", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 36, "code": "rw [Nat.sSup_def]", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 37, "code": "by_cases hbounded : BddAbove {k : \u2115 \| \u2203 A : Finset \u2124, A.card = n \u2227 k = ((A \u00d7\u02e2 A).filter (fun (x, y) \u21a6 x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)).card}", "goal": "```lean\nn : \u2115\n\u22a2 Nat.find ?m.17 \u2264 n ^ 2\n```\n---\n```lean\nn : \u2115\n\u22a2 \u2203 n_1,\n \u2200\n a \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n a \u2264 n_1\n```", "goal_count": 2, "ast_complexity": 305 }, { "line": 38, "code": "\u00b7 simp [hbounded]", "goal": "```lean\ncase pos\nn : \u2115\nhbounded :\n BddAbove\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 Nat.find ?m.17 \u2264 n ^ 2\n```\n---\n```lean\ncase neg\nn : \u2115\nhbounded :\n \u00acBddAbove\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 Nat.find ?m.17 \u2264 n ^ 2\n```\n---\n```lean\nn : \u2115\n\u22a2 \u2203 n_1,\n \u2200\n a \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n a \u2264 n_1\n```", "goal_count": 3, "ast_complexity": 775 }, { "line": 39, "code": "\u00b7 simp [hbounded]", "goal": "```lean\ncase neg\nn : \u2115\nhbounded :\n \u00acBddAbove\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 Nat.find ?m.17 \u2264 n ^ 2\n```\n---\n```lean\nn : \u2115\n\u22a2 \u2203 n_1,\n \u2200\n a \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n a \u2264 n_1\n```", "goal_count": 2, "ast_complexity": 524 } ], "tactic_count": 5, "successful_tactics": 0, "first_failure_step": 1, "cps": 0.0, "cps_peak_step": 0, "error_code": "theorem upper_trivial {n : \u2115} : max013AffineTranslates n \u2264 n ^ 2 := by" } }, { "iteration": 10, "event": "search_mathlib", "query": "Nat.find_spec", "results_count": 1 } ], "verification": { "banned_tokens": [], "command": [ "[path] "lean", "/app/asi_portal/verifier/quick_1772001760_6659.lean" ], "comparator_passed": null, "duration_s": 15.153506994247437, "goals_remaining": 3, "is_verified": false, "lean_file_path": "", "problem": { "axioms_check_name": "upper_trivial", "axioms_check_skipped": false, "benchmark_id": "GreensOpenProblems__24__upper_trivial", "category": "research_open", "endings": [ "end variants", "end Green24" ], "imports": [ "FormalConjectures.Util.ProblemImports" ], "preamble_chars": 950, "source_file": "FormalConjectures/GreensOpenProblems/24.lean", "title": "upper_trivial" }, "returncode": 1, "status": "partial", "stderr": "", "stdout": "/app/asi_portal/verifier/quick_1772001760_6659.lean:39:2: error: unsolved goals\ncase pos\nn : \u2115\nhbounded :\n BddAbove\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 Nat.find ?m.17 \u2264 n ^ 2\n/app/asi_portal/verifier/quick_1772001760_6659.lean:40:2: error: unsolved goals\ncase neg\nn : \u2115\nhbounded :\n \u00acBddAbove\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 Nat.find ?m.17 \u2264 n ^ 2\n/app/asi_portal/verifier/quick_1772001760_6659.lean:35:68: error: unsolved goals\nn : \u2115\n\u22a2 \u2203 n_1,\n \u2200\n a \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n a \u2264 n_1\n/app/asi_portal/verifier/quick_1772001760_6659.lean:39:10: warning: This simp argument is unused:\n hbounded\n\nHint: Omit it from the simp argument list.\n simp \u0335[\u0335h\u0335b\u0335o\u0335u\u0335n\u0335d\u0335e\u0335d\u0335]\u0335\n\nNote: This linter can be disabled with `set_option linter.unusedSimpArgs false`\n/app/asi_portal/verifier/quick_1772001760_6659.lean:40:10: warning: This simp argument is unused:\n hbounded\n\nHint: Omit it from the simp argument list.\n simp \u0335[\u0335h\u0335b\u0335o\u0335u\u0335n\u0335d\u0335e\u0335d\u0335]\u0335\n\nNote: This linter can be disabled with `set_option linter.unusedSimpArgs false`\n'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n", "timed_out": false }, "phase2_error": "ProgressAnalysisResult.__init__() missing 1 required positional argument: 'progress_score'" }
Gemini 3.1 Pro Agent v8	Failed	--	Feb 24, 2026	rw [max013AffineTranslates] apply csSup_le · ?_ · ?_	View { "agent_telemetry": { "compilations": 7, "iterations": 10, "duration_s": 756.15, "best_status": "failed", "best_goals": null, "max_iterations": 10, "model": "google/gemini-3.1-pro-preview", "error": null, "provider": "openrouter_primary", "finish_reason": "tool_calls", "usage": { "prompt_tokens": 163264, "completion_tokens": 32561, "total_tokens": 195825, "reasoning_tokens": 30665, "cost_usd": 0.555872 }, "reasoning_effort": "high", "response_trace_count": 17 }, "response_trace": [ { "iteration": 1, "event": "search_mathlib", "query": "sSup_le", "results_count": 1 }, { "iteration": 2, "event": "search_mathlib", "query": "csSup_le", "results_count": 1 }, { "iteration": 3, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1771902982-kSYavuv6kvBB1oISko37", "api_elapsed_s": 5.531, "usage": { "prompt_tokens": 2492, "completion_tokens": 49, "total_tokens": 2541, "reasoning_tokens": 7, "cost_usd": 0.005572 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_details": [ { "id": "tool_compile_proof_Jzv7fyruYmV4anItnhZR", "format": "google-gemini-v1", "index": 0, "type": "reasoning.encrypted", "data": "ElwKWgG+Pvb7FNDmzuDD1mxgapGtdXD+8CjDWkLiCp36kJs6FxnvS9z2tSEdlhWqIocUvx/CH2MXTFiZkhykH1DP+yApVVWkPriJo+rA9uoFu7x0uAlC4t1IRYX+Rg==" } ] }, { "iteration": 3, "event": "compile_proof", "provider": "openrouter_primary", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [ "sorry" ], "proof_code": "rw [max013AffineTranslates]\n apply csSup_le\n \u00b7 sorry\n \u00b7 sorry", "compiler_output_preview": "\nBanned tokens found: sorry", "compile_elapsed_s": 0.001 }, { "iteration": 4, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1771902987-VCWluZkuGRh7H0W3jPLH", "api_elapsed_s": 4.26, "usage": { "prompt_tokens": 2902, "completion_tokens": 55, "total_tokens": 2957, "reasoning_tokens": 14, "cost_usd": 0.006464 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_details": [ { "id": "tool_compile_proof_H14haW0beApG8dS30f0D", "format": "google-gemini-v1", "index": 0, "type": "reasoning.encrypted", "data": "Em4KbAG+Pvb7rFW5vJiqepG3tSTT7ppdGq/rCbKNE1U1tPcgBqzKqnJm0+v8974Rt6xvoWEk8xH2bgVB0hJYxgBd3pb5GxdxhzKPEEnTZ7lOR/WCYtyA3ZRqFdMxnMACKz6xwT4MAz5NN/LWkG9PhQ==" } ] }, { "iteration": 4, "event": "compile_proof", "provider": "openrouter_primary", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "rw [max013AffineTranslates]\napply csSup_le\n\u00b7 ?_\n\u00b7 ?_", "compiler_output_preview": "/app/asi_portal/verifier/quick_1771902991_6932.lean:38:3: error: unexpected token '?'; expected '{' or tactic\n/app/asi_portal/verifier/quick_1771902991_6932.lean:38:2: error: unsolved goals\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n/app/asi_portal/verifier/quick_1771902991_6932.lean:35:68: error: unsolved goals\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \n...<truncated>", "compile_elapsed_s": 30.745, "tactic_replay": { "tactic_count": 3, "successful_tactics": 0, "first_failure_step": 1, "cps": 0.0, "cps_peak_step": 0, "goal_trajectory": [ 1, 1, 2 ], "max_goals": 2, "final_goal_count": 2, "goals_opened": 1, "goals_closed": 0 }, "lsp_feedback": { "diagnostics": [ { "fullRange": { "end": { "character": 5, "line": 37 }, "start": { "character": 3, "line": 37 } }, "message": "unexpected token '?'; expected '{' or tactic", "range": { "end": { "character": 5, "line": 37 }, "start": { "character": 3, "line": 37 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 3, "line": 37 }, "start": { "character": 2, "line": 37 } }, "leanTags": [ 1 ], "message": "unsolved goals\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty", "range": { "end": { "character": 3, "line": 37 }, "start": { "character": 2, "line": 37 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 3, "line": 37 }, "start": { "character": 68, "line": 34 } }, "leanTags": [ 1 ], "message": "unsolved goals\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2", "range": { "end": { "character": 0, "line": 35 }, "start": { "character": 68, "line": 34 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 6, "line": 40 }, "start": { "character": 0, "line": 40 } }, "message": "'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]", "range": { "end": { "character": 6, "line": 40 }, "start": { "character": 0, "line": 40 } }, "severity": 3, "source": "Lean 4" } ], "error_line": 34, "error_char": 68, "error_msg": "unsolved goals\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2", "goal_before_error": null, "full_timeline": [ { "line": 35, "code": "rw [max013AffineTranslates]", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 36, "code": "apply csSup_le", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 37, "code": "\u00b7 ?_", "goal": "```lean\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```\n---\n```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 2, "ast_complexity": 449 } ], "tactic_count": 3, "successful_tactics": 0, "first_failure_step": 1, "cps": 0.0, "cps_peak_step": 0, "error_code": "theorem upper_trivial {n : \u2115} : max013AffineTranslates n \u2264 n ^ 2 := by" } }, { "iteration": 5, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1771903022-2xJUHq4pX8B6HXdVW2ND", "api_elapsed_s": 23.206, "usage": { "prompt_tokens": 5277, "completion_tokens": 1650, "total_tokens": 6927, "reasoning_tokens": 1382, "cost_usd": 0.030354 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_text": "Analyzing Goal Properties\n\nI'm currently focused on the goals, specifically their properties. I've noted the error messages, which point to the goals: Nonempty and a bound condition on `b`. I'm also considering the implications of `csSup_le` and whether it applies to `Nat`. I am testing with a witness for `Nonempty`.\n\n\nExploring Set Representations\n\nI'm now exploring different set representations for demonstrating non-emptiness. I'm considering `Finset.range n` and its variations, aiming for a set of size `n`. It looks like `Finset.map` could be useful here. Additionally, I am clarifying the conditions for `csSup_le` and considering the role of `BddAbove`. I have also considered that `csSup_le` works differently for `Nat`.\n\n\nExamining Non-emptiness Further\n\nI'm now focusing specifically on the non-emptiness requirement. I've re-examined the goals generated after applying `csSup_le`, and I'm questioning the necessity of `S.Nonempty` when considering `Nat` and `ConditionallyCompleteLinearOrderBot`. I'm exploring the implications of an empty set `S` and whether it satisfies the condition when `0 \u2264 n^2`. I'm starting to build a proof using `csSup_le`, considering cases for `n` to prove non-emptiness. I'm noting that an empty `A` where n=0 may be the key.\n\n\nInvestigating Set Construction\n\nI'm now investigating how to construct a `Finset \u2124` of size `n`. I've realized `n` in `\u2124` is trivial, and have considered `Finset.Icc 1 n` and `Finset.image` of `Finset.range n`. I'm now leaning toward using `Finset.range n` and mapping it to `\u2124` with `Int.ofNat` and its injectivity, specifically using `(Finset.range n).map \u27e8(\u2191), Int.ofNat.injective\u27e9`. I'm verifying the use of `Finset.Ico 0 (n:\u2124)` and its cardinality.\n\n\n", "reasoning_details": [ { "format": "google-gemini-v1", "index": 0, "type": "reasoning.text", "text": "Analyzing Goal Properties\n\nI'm currently focused on the goals, specifically their properties. I've noted the error messages, which point to the goals: Nonempty and a bound condition on `b`. I'm also considering the implications of `csSup_le` and whether it applies to `Nat`. I am testing with a witness for `Nonempty`.\n\n\nExploring Set Representations\n\nI'm now exploring different set representations for demonstrating non-emptiness. I'm considering `Finset.range n` and its variations, aiming for a set of size `n`. It looks like `Finset.map` could be useful here. Additionally, I am clarifying the conditions for `csSup_le` and considering the role of `BddAbove`. I have also considered that `csSup_le` works differently for `Nat`.\n\n\nExamining Non-emptiness Further\n\nI'm now focusing specifically on the non-emptiness requirement. I've re-examined the goals generated after applying `csSup_le`, and I'm questioning the necessity of `S.Nonempty` when considering `Nat` and `ConditionallyCompleteLinearOrderBot`. I'm exploring the implications of an empty set `S` and whether it satisfies the condition when `0 \u2264 n^2`. I'm starting to build a proof using `csSup_le`, considering cases for `n` to prove non-emptiness. I'm noting that an empty `A` where n=0 may be the key.\n\n\nInvestigating Set Construction\n\nI'm now investigating how to construct a `Finset \u2124` of size `n`. I've realized `n` in `\u2124` is trivial, and have considered `Finset.Icc 1 n` and `Finset.image` of `Finset.range n`. I'm now leaning toward using `Finset.range n` and mapping it to `\u2124` with `Int.ofNat` and its injectivity, specifically using `(Finset.range n).map \u27e8(\u2191), Int.ofNat.injective\u27e9`. I'm verifying the use of `Finset.Ico 0 (n:\u2124)` and its cardinality.\n\n\n" }, { "id": "tool_compile_proof_zVOUwwpRzgkBhaT8IZ8W", "format": "google-gemini-v1", "index": 0, "type": "reasoning.encrypted", "data": "EpYlCpMlAb4+9vs5SUZHGPzODr+rnr/GAPNsuJxzVDrtFmoHKhwc727urdviIYvhqaA2fcvi9QO//o14Zl4AtSFGd5yg4C8X3hlBwDrayOZYwJDrJK42vMxX493iHyXuzqGvF99uPOUEJuCMoZXXkAG0jx5z2p+x9JPeNCHzSIrCI78HZ2LwHrOOgoJBedmT9Kr9RTM5ws7u5QOTB9Z0mUFQYhgJ3iNOmcC+HU6M226mEr6hghQJHONyn8E7JGZJ6z9IcPrMMwIWd7GPTVC9gUvW/Dc2RiZInswKtGoknP5Yz1IpPyKb05k79rNGCl3GwC/al748pfiUHFCqwA9c4UaXM2zPnRGaQ7MqNDW+RM2qCfRd4cQMmFV7OewXVwHLetfIyUCsHzdVMGPY1er2K17oK2Ru+3fNZgIHtWxKv2w45yrdh+dEXLFpwkSw57FJ+j+JMATjbUOy9h+SpAcwP/zNU+p+OTI0s0Xp19WVKYzQSZWZExDgMVZTCNUYEdaB8YCh1a6CqDPxp8tafu33rL/qwLC01d0YG/nC1LEHYQbiJgU7fegMLuWaYhrETy9wYvvMzXCZfR8aQzZ4FRo8XjlDzIC1xfDXj2jVvn4kvJVeKeHf0QbHsIoIgy/xA/hLTjubmWEDHHPyILqdkJ6SGdH+cyZ1xbjfTIjaH10xbKYwcNOyBq4Gngg3+1qc5RA42EqXHZY0ZZWiGBPHjdIo4t2wow7OJBKOYO6v9pi1KYxsOpzzcPUqGoUXBq0aWGX9msUFIu6Blqj5XAJdTZPWmqxxDz1pOqwFHHKIcqOwS3n8csAN/YJ0YHEx1qkb+jiD3bZg5GFbciFyUNr2NwkD/yI59LPpF2mZnr4Q59vs3y0YWn2aorktHD/6iDT+xtl0Ni5PQiXREzdiOTZfvG4nUzi1kFpoLfegj7GIp7X0DutJ7Xx6xgdII5H1QyGhkvQ1iCppb1LXRE1RwID8TeWNRbvXrPV/qDaQAuHNKXmn2A6ivxSUvytDStMY3IXm2y3frt43AyE4l0rCB/VPHcCG0HOApknZ7VijXLPvOBmQp2jE77xChsj9njzhC3RNs/aLO7RJibrSu/UugfFr+ZBsLINX0fS/DyUi+JpVVAQhuU7AlyFpiDTyjGwrZnFIOj63NJ55hKA1vihpyLIUWWsfUntQbvwKElPSaI6LT0J1yeUOCzf+ylpWKUJN/q1tELn9lFbWAEvbhjSyQGJwUlizEsmFmUKSuTHvd7Q9TQQ4gSPaDuIJJGaCbqqGaAdptM8r5NVfT92DlBD3TjEkXrFjgsIhlirWUaFD+/12/7ueNwB4ymX/xYtYXGKFDtc6EWRmYz9AhZrYtG9F+0hwedUGKAlQ/g+bT+BonnvIYpVBPAlBYpuWs+LVI9rFMIYg05I3gzKh9a0SpWFC9g0tzegnNqCMTnWnBTkHUjFL+b7Q/Ygf/xnCd47Nrb38Lzk0lrnrAkaCS3ORZfaMjYjHh+qThohn5YmdPaefNE1i3AJTSX3y/NEm4uGXNkhhvTRYr6fcOqWmkhcMAJLedYllv4A3tmV6vI4DtbvSUb67shDgKbJl/Rx2GIPMu2cXqAA5kpwritnOXC8+ZmTbVJEQP9w+RxLnMYrYJy2w3HMhaMsLugWw+tjmu+dpSJOka3NzinysdcxSe+aDEzV4/aQ1hxsR3W7kH28Ub7CX6Gu9WPTHbs7VgvLL+pR+O8gYlkUXRQYPcj/qp3P1Dk/TFwgzpuq8Jj0Y6PExbDd//UzIDFqw6KBswHdH7VsisLplEKGIi15FzYx//kCh/bSDXR34wB8ngSJdp4dbr4hZtZd9PEMWitAdKPPcX9WPcWM/bJBFbASmjHYR7IN7f+xMJzH0d5UT+CENUWYzJZIUbgVIlcGFjnjL+EmEo+irM4AqzmwwJTYCxczdEEnV3+Zq4+Qi4FKu4gbddx3Q6hP2Wb5LonXdpJud76rteqvMiclY5vatSS/8BUVnp2ex2oh3ZQh+8VjFUSKFnOxjF7/f0VtIez0tpvJaPKq8ULkx2Tgkfdv/geRiBavTj5//wtzmA0CjBHZvGvNb4jgHjGhkPxPsjoFysjbDkpLfsz7OQVhpqKbcefWkxbQTcEZ0A9dakBKFqYAepT9ZwwonCnyqJ60UIAQrZqyD/oGjv2fsZEyUIIOTGrokNFc4mgR72qtnDFue41YGGTB6o1bitAACoknDKgNlxyRSWlFvxe3OAXLAUw/rkTxPzOX/2mKlMAFvFU6lFLnB1UNb0kTx5/ksMVA6DhT2lePXCSZWMjCgU4pOh4vZaKxGSaE6xtplL5MV8DgNhpkFK9TsG2Qg76i+Pfxa+XfAZPdHmIKANJkpGx0t4W/1db5a51kMkD32NGcyIhg3V6+/+J6qXCX+w3DR5idevL8iZNkeLicFcm1IWuQVdaffD6G963U/qiS1YukIAsKQuAN4D50lPxKqX8QRw8hpR4101vWDtsIX6uSaXSQgiNyVM4t6SO1hxkr5fwMgA3Xzg4U5iDJxmKvRR81TYLXF+6TtTuzPhYSOrBXwQvBuDGuNFm5yLEfJT3EkVg/G+x3+L6ylO/sXf2Mw7Bmcw4EXv9TDbk1mcYDL4tuo8cKXwT7grQqbQtp7Wqf9Yea9IeSST8jsp2Ah6NxqvSeZH70QhZrUdNGtGP5r6+v55pWklEnOM1g1cNnFRpAhj4A5BLLrJLwtVi3Be1S9Lqs57mfXF2kgQBw6mt4mKjjdeilOm09NMuiPuvTIAkspF6i0bopg8rVtpVwNCJ4HdHAN2Kd//z03h3WxHc9k7G20PonZQ6L25RRGXcriTaFaxEhdHMZdcwOAEQnTm6cInBsyQA/d1nD/XAU5LlW9zxz9+NZtHcmSZbJ5y301W1omWnKLT8YP1h1vJhuy/3cQbGEa2aXs0RiJ2TbrrCmIX4TBDEFfB3DHPoLxOfPE61bUpc+xCY05XcI7arO70GhDG3PaSL3YfSleday02EyLXakQD0QE+M95kyZKlfJJKhQBc8dHxT42tPsuGAOg4rwIhFrGESovXI+esGlIquL2cH1f0NjEQG26zJacHuCPWRJiWI4ULoxRA5D81Gzs6JFB5yGY6f+kxzhicgOgSKfW85Vjop40HwiIvZPAdlxigU0rzpdy/i3tCIiZInnlYAcs+6nza/3NrtKRZbBMPTQ+ZpptcKnTjTIf9qBe5m78wjZmybXrChbFM6AlO7hDqMnRc8Z1dWk0t53dEhgmak4BXO8EGPe8hCmvZRsUjwDz8H2TjjXXCw7OjjtelVvw4n8BiU63RMf+7nOnwpJWiJEh7DJ6N4ptXyNqZt6Nw7uGB1Yd5mynLLg+bw/mbwv42pJutk4ggoeg+wnEYnhp8O2FMOI29D7twb8lsN+WkUmB/uEWN5rrxalFo9TGYCS7Pd/gyzDzuO0aIYFu3L+R7zcJehPM8dA+u6Gsd5vDxaOvBPMovJuwhtaNM44FpVFZWA4+7EVq9eOHO1wtILio+XM41kWrQxRxZ6JTgyzE3evTX+Je987FFUH1m67E2vY/1yXUoJqDDnBFduvxYNeKd9hDf6mArLtZpW+zgZSWS37i54tRyktJGr2r8h+guYUYID1M8iA4anYuC/wbVgj7DaVZw71eEQmddnp4nSncEFlQbaWC7Vs3iV3XEDI2w2OGi0cxAbU4ynathiR3nQGLrZMthcsZ/dJSbFWZRr65bDBhHl8XmPthwiu+Cjv8DBWCybBONG7LQj4JnwBBCkA2IOB25GJ8C0YfrJt3HpFg3DN9HVE2PbdU17T8jDFewx8+mlz9NURywEHN/JgvthhXZ0QRvAA4XcpiF5Chkqag488R+fm/NfVoJMCNQgiLi3vbtGcDRFyPCPaUqpfHkWxUcsaGwv8J8OpoCuANPFvmmPNvc7zEpJby5fSYxx4n2od1/QEGDMyhpPSEj+hGflb9PRVXsayfD9lhXyTZl5kg8+kGq7Ocpk1pcuHocMuC3Z1Mgk91vsCDd/3MDPpNxZVCV0gsEuUT9FM2zJFf84B+7PRcuU0SSD6c8ndcUsydhQfNB++QfZp/KtgLa3mN9GRYm2EnxH0Q1xoX1pQ71egy43h4mZX9u9pjJ6ooblA6C8RMz1OsX4/2erq1Mvf/0ecnFc9GnSK0H0xPWBfs9hkWCzOmLpqR+u+jyItAZliBktv/tFEmzcLwMfBJoau7s8om6F/n9dYUbEmp1y4whf5Q4l9mewRkmWTg7780UuQQXctGP/eI8Fx03HyH3BDqgJuMi2uwpWA7WrNGmmQF4Wa4DygndQD9ato5DKo8V00LgDN6heGhRMs1doqKfCJaKKapMTE4uk+2Zfr0y6AGOVnalTPtxGmO+Fmte5TbMQp4H44SQpA96zlNjbe2HQvFuGBU4kKOQAe/f5f2iZNi6aSPmdvzOs0KnjlyGTtliQgT5p1nI6LpX1wNSaAwHUu6JxNtBpox/23jehu5kBwsN1e1sbz0krZ0tcel6GqQJaZbTi8k6CLBlNOwZJh4RUEv/lIkrBX75ui/Dhs/R7kPGZAYJh35PCPsPp6Rxm+jrEj1jEKYdZRn3JQQiGBPzZA0IGX+z5C2SnnJhe4xNCSJi9orNQyjuGV3UZUIKt1ya8RDzpuKksYMZkAKndJUg+VQGuPz9nN/9nLz7sLrHnf0sgbOyQ7B7T9KVxXwOYiq05YQ/Zqkm2N/Sekn2eas0nFcXGk/IRiRISDCbtmRlgQdxsD0KX5Ptvkn6cePAfKGogWJ0YKJKU0u6s29hdi7IxgCreAV0iDi1dIGuYi1eYlCdn8S4EDc9/ED43jR19t6T7YzukuLFYduPREVWWAKEXLJSlJsxc153rZb93j5lGUIu8CED/ngUS5b2vIe4wl928aYBbNJrFK33W9MOA8HTGYNt64zjjqzS+X7BGXDNZ45MRoi4T7JHVKMRNcxsoNUGyxW3KXmRZaNQsAgfj3qpg3MUhBSzuGjmXgvmKl4zuO8gbMJEZwBveyQtMVz4V0zdaSm8BlrPmW3iupsXvJfbsC/xAapR1Hv0uhakMCdx+J12fsgtmcS33B7JDO9d/ZxYfhiEdVpxZYeC9RGJcaVEEUssHTxZdej78MG7S1GWRhcqkXTnHe0nUYf8B2GPropz2lNvThsRoZpgw6qelQ5lVrGgDjPoFbf7KQAVAyzwFAHuIE2KyEbAkRK2yD3hTzydu0k+uqgXCLttVVJuJHBRyywBmdjHp86TapaHlBLrU5KfUJSGYxabhZZ+g1nXvbKrGLcWhmtnH3XvEVOXZAzL0QdVLT2HAYVDgb50ZNMpuCg20ZHC1Pgs69fP2EV4OfMBLIR8QkJoSzGSJy60iBtWpXheHRebT2kI2YRDTBDH2npieJuUPJU4xBHPDTBOft/njMXhC4Lq7OYRiYcefRterURmjvXZqAyyESKmMkU+BgEehLMLI9j5YjGbkjH9ynMN9l/Tk34waBp8wzxC7H5ZGnIpswlFCzvrLNeY7bHo75/iK7gTJXth/sjG+irJucVXYWpygcuGIaJaVBUjNOVhguds20YesDFpyv1Px4Gk9D46jjYr5Eenu/qz9CTV350+JuE4AIaOOygtRWjC0NCXuoLcM2Pb6U7zbnQb7aoedUPvM4juzEWnQQXW6EYOmdY9yZ1HM3uLNfV7mR+7je04Izg5gt5KCgiQVev4QErw0NJhG1kSS+uS9KPqp4zHSCbAb1n2KNdLPvGCqbBXU014qXHtbtqRJLNOUt+2K9VIDAxbN4WYiK1PrbOYCOmsTOIxiFI0uEJvYJ66NsDoa8pBkZYcQMRci17pfJa6v1V9qU2JLaPvKDyvgerRjOSr+/uLGw9f+WNgXY2uyREC+TmDob/ZQTzKdxaPuqnkT9NJYMmewkOzc9a7gkxHPbTuCG79SBeNXABF4RoiZlHVxeMWnXTUDsNNn249PYxX0H9eY8rJd5XYQLUuEJ55UO/LqJTFanEZ+ExUbFwkG9GFNdOnkgt70maspGwyYUzvFr+lEQ3RY5Trvs00Ykz/mZkGo1k+IJ5Eg0kQ+a2d0CH3dJcE71rqEF68nScmgFYmT3CME0nUrUUkGNPGSTQC09dZwq2N7n6OxMN89UcWKjriZVanJw2mUJCyQpC0OGqfJgLV9XDkUZvcyTN1QT1KZKxGJQH4kAbT7LBmE5SUCrWSbZ2zncin6Ml6cvd0hJvJLleB/vtLHGw0a0aueBM76o2ZMwfqsp4oy0kTyB0WPc3epDfu1UqIU2NyML7uwfXaML5Cc6bJsFBwUcdlPgk5FaXo3fDSds/Vuveug7Y/Gf/Q2Oahc6b6HcQV7tZSOqUeRV++IFucoknrZkGr339O8Dc" } ] }, { "iteration": 5, "event": "compile_proof", "provider": "openrouter_primary", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "rw [max013AffineTranslates]\napply csSup_le\n\u00b7 \n use ((Finset.Ico 0 (n:\u2124)) \u00d7\u02e2 (Finset.Ico 0 (n:\u2124))).filter (fun x => x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 Finset.Ico 0 (n:\u2124)) \|>.card\n use (Finset.Ico 0 (n : \u2124))\n simp [Int.toNat_ofNat]\n\u00b7 intro b hb\n rcases hb with \u27e8A, hA, rfl\u27e9\n have h1 : ((A \u00d7\u02e2 A).filter (fun x => x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A)) \u2286 A \u00d7\u02e2 A := Finset.filter_subset _ _\n have h2 := Finset.card_le_card h1\n rw [Finset.card_product] at h2\n rw [hA] at h2\n exact h2", "compiler_output_preview": "/app/asi_portal/verifier/quick_1771903045_7923.lean:41:10: error(lean.unknownIdentifier): Unknown constant `Int.toNat_ofNat`\n/app/asi_portal/verifier/quick_1771903045_7923.lean:48:4: error: Type mismatch\n h2\nhas type\n {x \u2208 A \u00d7\u02e2 A \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A}.card \u2264 n * n\nbut is expected to have type\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.c\n...<truncated>", "compile_elapsed_s": 38.918, "tactic_replay": { "tactic_count": 13, "successful_tactics": 5, "first_failure_step": 6, "cps": 38.46153846153847, "cps_peak_step": 5, "goal_trajectory": [ 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], "max_goals": 2, "final_goal_count": 1, "goals_opened": 1, "goals_closed": 1 }, "lsp_feedback": { "diagnostics": [ { "code": "lean.unknownIdentifier", "fullRange": { "end": { "character": 25, "line": 40 }, "start": { "character": 10, "line": 40 } }, "message": "Unknown constant `Int.toNat_ofNat`", "range": { "end": { "character": 25, "line": 40 }, "start": { "character": 10, "line": 40 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 12, "line": 47 }, "start": { "character": 4, "line": 47 } }, "message": "Type mismatch\n h2\nhas type\n {x \u2208 A \u00d7\u02e2 A \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A}.card \u2264 n * n\nbut is expected to have type\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2", "range": { "end": { "character": 12, "line": 47 }, "start": { "character": 4, "line": 47 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 6, "line": 49 }, "start": { "character": 0, "line": 49 } }, "message": "'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]", "range": { "end": { "character": 6, "line": 49 }, "start": { "character": 0, "line": 49 } }, "severity": 3, "source": "Lean 4" } ], "error_line": 40, "error_char": 10, "error_msg": "Unknown constant `Int.toNat_ofNat`", "goal_before_error": "```lean\ncase h\nn : \u2115\n\u22a2 (Finset.Ico 0 \u2191n).card = n \u2227\n {x \u2208 Finset.Ico 0 \u2191n \u00d7\u02e2 Finset.Ico 0 \u2191n \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 Finset.Ico 0 \u2191n}.card =\n {x \u2208 Finset.Ico 0 \u2191n \u00d7\u02e2 Finset.Ico 0 \u2191n \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.Ico 0 \u2191n}.card\n```", "full_timeline": [ { "line": 35, "code": "rw [max013AffineTranslates]", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 36, "code": "apply csSup_le", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 37, "code": "\u00b7", "goal": "```lean\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```\n---\n```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 2, "ast_complexity": 449 }, { "line": 38, "code": "use ((Finset.Ico 0 (n:\u2124)) \u00d7\u02e2 (Finset.Ico 0 (n:\u2124))).filter (fun x => x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 Finset.Ico 0 (n:\u2124)) \|>.card", "goal": "```lean\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 205 }, { "line": 39, "code": "use (Finset.Ico 0 (n : \u2124))", "goal": "```lean\ncase h\nn : \u2115\n\u22a2 {x \u2208 Finset.Ico 0 \u2191n \u00d7\u02e2 Finset.Ico 0 \u2191n \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 Finset.Ico 0 \u2191n}.card \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n```", "goal_count": 1, "ast_complexity": 301 }, { "line": 40, "code": "simp [Int.toNat_ofNat]", "goal": "```lean\ncase h\nn : \u2115\n\u22a2 (Finset.Ico 0 \u2191n).card = n \u2227\n {x \u2208 Finset.Ico 0 \u2191n \u00d7\u02e2 Finset.Ico 0 \u2191n \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 Finset.Ico 0 \u2191n}.card =\n {x \u2208 Finset.Ico 0 \u2191n \u00d7\u02e2 Finset.Ico 0 \u2191n \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.Ico 0 \u2191n}.card\n```", "goal_count": 1, "ast_complexity": 302 }, { "line": 41, "code": "\u00b7 intro b hb", "goal": "```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 239 }, { "line": 42, "code": "rcases hb with \u27e8A, hA, rfl\u27e9", "goal": "```lean\ncase h\u2082\nn b : \u2115\nhb :\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 43, "code": "have h1 : ((A \u00d7\u02e2 A).filter (fun x => x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A)) \u2286 A \u00d7\u02e2 A := Finset.filter_subset _ _", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 157 }, { "line": 44, "code": "have h2 := Finset.card_le_card h1", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 : {x \u2208 A \u00d7\u02e2 A \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A} \u2286 A \u00d7\u02e2 A\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 224 }, { "line": 45, "code": "rw [Finset.card_product] at h2", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 : {x \u2208 A \u00d7\u02e2 A \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A} \u2286 A \u00d7\u02e2 A\nh2 : {x \u2208 A \u00d7\u02e2 A \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A}.card \u2264 (A \u00d7\u02e2 A).card\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 303 }, { "line": 46, "code": "rw [hA] at h2", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 : {x \u2208 A \u00d7\u02e2 A \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A} \u2286 A \u00d7\u02e2 A\nh2 : {x \u2208 A \u00d7\u02e2 A \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A}.card \u2264 A.card * A.card\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 305 }, { "line": 47, "code": "exact h2", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 : {x \u2208 A \u00d7\u02e2 A \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A} \u2286 A \u00d7\u02e2 A\nh2 : {x \u2208 A \u00d7\u02e2 A \| x.1 \u2260 x.2 \u2227 x.1 + 3 * (x.2 - x.1) \u2208 A}.card \u2264 n * n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 295 } ], "tactic_count": 13, "successful_tactics": 5, "first_failure_step": 6, "cps": 38.46153846153847, "cps_peak_step": 5, "error_code": "simp [Int.toNat_ofNat]" } }, { "iteration": 6, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1771903084-1bMtWZlmPq0iTqQwQ1yN", "api_elapsed_s": 36.078, "usage": { "prompt_tokens": 11937, "completion_tokens": 2334, "total_tokens": 14271, "reasoning_tokens": 1951, "cost_usd": 0.051882 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_text": "Analyzing Filter Goal Structure\n\nI'm now zeroing in on the `filter`'s internal structure using pattern matching. I'm focusing on `match x with \| (x, y) => ...`. Then, I'm pivoting to examine the structure of `upper_trivial`, specifically how `rw` and `apply` are used with `csSup_le`. I'm starting to get a sense of how the range is used within this structure.\n\n\nEvaluating Cardinality Equivalence\n\nI've made a breakthrough. By specifying `A0 = Finset.Ico 0 (n:\u2124)`, the cardinality proof simplifies dramatically, and it's equal to n. The problematic `Int.toNat_ofNat` hurdle is bypassed by a simple `simp`. Then I have validated that the complex cardinality comparison is effectively an `rfl` equality, because of identical functional definitions.\n\n\nSimplifying Equivalence Proofs\n\nI'm now focusing on how `rfl` and `simp` can resolve the equivalence proof. Specifically, using `{x \u2208 A \u00d7\u02e2 A \| match x with \| (x, y) => ...}.card` might make the second goal of the `\u2227` trivial. Furthermore, `fun \u27e8a, b\u27e9 => ...` is more succinct than `match x with \| (a, b) => ...`. Instead, I'll aim to use `A.card = n`, where `A = Finset.map \u27e8(\u2191), Int.ofNat.injective\u27e9 (Finset.range n)`. The cardinalities should be straightforward.\n\n\nRefining Cardinality Calculations\n\nI've realized a critical point: `Finset.Ico 0 n` for a natural number `n` gives `Finset \u2115`, which is not what I intend. I need `Finset \u2124`, so I'll shift to `Finset.Ico 0 (n : \u2124)`. Now, I'm testing if a `simp` command can solve the cardinality issue with a more natural format. I'm considering using `Finset.map` with `Nat.cast` to handle this.\n\n\nExploring Cardinality Approaches\n\nI'm now back to cardinality proofs, specifically examining options for representing sets with size `n`. I've revisited `Finset.Ico 0 (n : \u2124)` and tested that `omega` can solve the cardinality issue with it, after some unfolding, as `Finset.card_Ico 0 (n : \u2124)` is `(n : \u2124).toNat = n`. I have also considered `Finset.image Nat.cast (Finset.range n)`. The cardinality proofs using injectivity are more straightforward than I initially thought.\n\n\n", "reasoning_details": [ { "format": "google-gemini-v1", "index": 0, "type": "reasoning.text", "text": "Analyzing Filter Goal Structure\n\nI'm now zeroing in on the `filter`'s internal structure using pattern matching. I'm focusing on `match x with \| (x, y) => ...`. Then, I'm pivoting to examine the structure of `upper_trivial`, specifically how `rw` and `apply` are used with `csSup_le`. I'm starting to get a sense of how the range is used within this structure.\n\n\nEvaluating Cardinality Equivalence\n\nI've made a breakthrough. By specifying `A0 = Finset.Ico 0 (n:\u2124)`, the cardinality proof simplifies dramatically, and it's equal to n. The problematic `Int.toNat_ofNat` hurdle is bypassed by a simple `simp`. Then I have validated that the complex cardinality comparison is effectively an `rfl` equality, because of identical functional definitions.\n\n\nSimplifying Equivalence Proofs\n\nI'm now focusing on how `rfl` and `simp` can resolve the equivalence proof. Specifically, using `{x \u2208 A \u00d7\u02e2 A \| match x with \| (x, y) => ...}.card` might make the second goal of the `\u2227` trivial. Furthermore, `fun \u27e8a, b\u27e9 => ...` is more succinct than `match x with \| (a, b) => ...`. Instead, I'll aim to use `A.card = n`, where `A = Finset.map \u27e8(\u2191), Int.ofNat.injective\u27e9 (Finset.range n)`. The cardinalities should be straightforward.\n\n\nRefining Cardinality Calculations\n\nI've realized a critical point: `Finset.Ico 0 n` for a natural number `n` gives `Finset \u2115`, which is not what I intend. I need `Finset \u2124`, so I'll shift to `Finset.Ico 0 (n : \u2124)`. Now, I'm testing if a `simp` command can solve the cardinality issue with a more natural format. I'm considering using `Finset.map` with `Nat.cast` to handle this.\n\n\nExploring Cardinality Approaches\n\nI'm now back to cardinality proofs, specifically examining options for representing sets with size `n`. I've revisited `Finset.Ico 0 (n : \u2124)` and tested that `omega` can solve the cardinality issue with it, after some unfolding, as `Finset.card_Ico 0 (n : \u2124)` is `(n : \u2124).toNat = n`. I have also considered `Finset.image Nat.cast (Finset.range n)`. The cardinality proofs using injectivity are more straightforward than I initially thought.\n\n\n" }, { "id": "tool_compile_proof_4AoRHHM6N1dhJx0CMQzA", "format": "google-gemini-v1", "index": 0, "type": "reasoning.encrypted", "data": "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" } ] }, { "iteration": 6, "event": "compile_proof", "provider": "openrouter_primary", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "rw [max013AffineTranslates]\napply csSup_le\n\u00b7 have hA : (Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n)).card = n := by\n rw [Finset.card_image_of_injective]\n \u00b7 exact Finset.card_range n\n \u00b7 intro a b hab; omega\n use ((Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n) \u00d7\u02e2 Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n)).filter (fun x => match x with \| (a, b) => a \u2260 b \u2227 a + 3 * (b - a) \u2208 Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n))).card\n use (Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n))\n exact \u27e8hA, rfl\u27e9\n\u00b7 intro b hb\n rcases hb with \u27e8A, hA, rfl\u27e9\n have h1 : ((A \u00d7\u02e2 A).filter (fun x => match x with \| (a, c) => a \u2260 c \u2227 a + 3 * (c - a) \u2208 A)) \u2286 A \u00d7\u02e2 A := Finset.filter_subset _ _\n have h2 := Finset.card_le_card h1\n rw [Finset.card_product, hA] at h2\n exact h2", "compiler_output_preview": "/app/asi_portal/verifier/quick_1771903120_4702.lean:41:23: error: omega could not prove the goal:\na possible counterexample may satisfy the constraints\n d \u2265 0\n c \u2265 0\n c - d \u2265 1\nwhere\n c := \u2191b\n d := \u2191a\n/app/asi_portal/verifier/quick_1771903120_4702.lean:44:4: error: No goals to be solved\n/app/asi_portal/verifier/quick_1771903120_4702.lean:50:4: error: Type mismatch\n h2\nhas type\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (a, c) => a \u2260 c \u2227 a + 3 * (c - a) \u2208 A}.card \u2264\n n * n\nbut is expect\n...<truncated>", "compile_elapsed_s": 36.094, "tactic_replay": { "tactic_count": 15, "successful_tactics": 5, "first_failure_step": 6, "cps": 33.33333333333333, "cps_peak_step": 5, "goal_trajectory": [ 1, 1, 2, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 ], "max_goals": 2, "final_goal_count": 1, "goals_opened": 3, "goals_closed": 3 }, "lsp_feedback": { "diagnostics": [ { "fullRange": { "end": { "character": 28, "line": 40 }, "start": { "character": 23, "line": 40 } }, "message": "omega could not prove the goal:\na possible counterexample may satisfy the constraints\n d \u2265 0\n c \u2265 0\n c - d \u2265 1\nwhere\n c := \u2191b\n d := \u2191a", "range": { "end": { "character": 28, "line": 40 }, "start": { "character": 23, "line": 40 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 19, "line": 43 }, "start": { "character": 4, "line": 43 } }, "message": "No goals to be solved", "range": { "end": { "character": 19, "line": 43 }, "start": { "character": 4, "line": 43 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 12, "line": 49 }, "start": { "character": 4, "line": 49 } }, "message": "Type mismatch\n h2\nhas type\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (a, c) => a \u2260 c \u2227 a + 3 * (c - a) \u2208 A}.card \u2264\n n * n\nbut is expected to have type\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2", "range": { "end": { "character": 12, "line": 49 }, "start": { "character": 4, "line": 49 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 6, "line": 51 }, "start": { "character": 0, "line": 51 } }, "message": "'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]", "range": { "end": { "character": 6, "line": 51 }, "start": { "character": 0, "line": 51 } }, "severity": 3, "source": "Lean 4" } ], "error_line": 40, "error_char": 23, "error_msg": "omega could not prove the goal:\na possible counterexample may satisfy the constraints\n d \u2265 0\n c \u2265 0\n c - d \u2265 1\nwhere\n c := \u2191b\n d := \u2191a", "goal_before_error": "```lean\ncase H\nn : \u2115\n\u22a2 Function.Injective fun x \u21a6 \u2191x\n```", "full_timeline": [ { "line": 35, "code": "rw [max013AffineTranslates]", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 36, "code": "apply csSup_le", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 37, "code": "\u00b7 have hA : (Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n)).card = n := by", "goal": "```lean\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```\n---\n```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 2, "ast_complexity": 449 }, { "line": 38, "code": "rw [Finset.card_image_of_injective]", "goal": "```lean\nn : \u2115\n\u22a2 (Finset.image (fun x \u21a6 \u2191x) (Finset.range n)).card = n\n```", "goal_count": 1, "ast_complexity": 73 }, { "line": 39, "code": "\u00b7 exact Finset.card_range n", "goal": "```lean\nn : \u2115\n\u22a2 (Finset.range n).card = n\n```\n---\n```lean\ncase H\nn : \u2115\n\u22a2 Function.Injective fun x \u21a6 \u2191x\n```", "goal_count": 2, "ast_complexity": 106 }, { "line": 40, "code": "\u00b7 intro a b hab; omega", "goal": "```lean\ncase H\nn : \u2115\n\u22a2 Function.Injective fun x \u21a6 \u2191x\n```", "goal_count": 1, "ast_complexity": 56 }, { "line": 41, "code": "use ((Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n) \u00d7\u02e2 Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n)).filter (fun x => match x with \| (a, b) => a \u2260 b \u2227 a + 3 * (b - a) \u2208 Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n))).card", "goal": "```lean\ncase h\u2081\nn : \u2115\nhA : (Finset.image (fun x \u21a6 \u2191x) (Finset.range n)).card = n\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 264 }, { "line": 42, "code": "use (Finset.image (fun (x : \u2115) => (x : \u2124)) (Finset.range n))", "goal": "```lean\ncase h\nn : \u2115\nhA : (Finset.image (fun x \u21a6 \u2191x) (Finset.range n)).card = n\n\u22a2 {x \u2208 Finset.image (fun x \u21a6 \u2191x) (Finset.range n) \u00d7\u02e2 Finset.image (fun x \u21a6 \u2191x) (Finset.range n) \|\n match x with\n \| (a, b) => a \u2260 b \u2227 a + 3 * (b - a) \u2208 Finset.image (fun x \u21a6 \u2191x) (Finset.range n)}.card \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n```", "goal_count": 1, "ast_complexity": 472 }, { "line": 43, "code": "exact \u27e8hA, rfl\u27e9", "goal": "no goals", "goal_count": 0, "ast_complexity": 8 }, { "line": 44, "code": "\u00b7 intro b hb", "goal": "```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 239 }, { "line": 45, "code": "rcases hb with \u27e8A, hA, rfl\u27e9", "goal": "```lean\ncase h\u2082\nn b : \u2115\nhb :\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 46, "code": "have h1 : ((A \u00d7\u02e2 A).filter (fun x => match x with \| (a, c) => a \u2260 c \u2227 a + 3 * (c - a) \u2208 A)) \u2286 A \u00d7\u02e2 A := Finset.filter_subset _ _", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 157 }, { "line": 47, "code": "have h2 := Finset.card_le_card h1", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (a, c) => a \u2260 c \u2227 a + 3 * (c - a) \u2208 A} \u2286\n A \u00d7\u02e2 A\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 257 }, { "line": 48, "code": "rw [Finset.card_product, hA] at h2", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (a, c) => a \u2260 c \u2227 a + 3 * (c - a) \u2208 A} \u2286\n A \u00d7\u02e2 A\nh2 :\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (a, c) => a \u2260 c \u2227 a + 3 * (c - a) \u2208 A}.card \u2264\n (A \u00d7\u02e2 A).card\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 373 }, { "line": 49, "code": "exact h2", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (a, c) => a \u2260 c \u2227 a + 3 * (c - a) \u2208 A} \u2286\n A \u00d7\u02e2 A\nh2 :\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (a, c) => a \u2260 c \u2227 a + 3 * (c - a) \u2208 A}.card \u2264\n n * n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 365 } ], "tactic_count": 15, "successful_tactics": 5, "first_failure_step": 6, "cps": 33.33333333333333, "cps_peak_step": 5, "error_code": "\u00b7 intro a b hab; omega" } }, { "iteration": 7, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1771903157-ixxQlDEyxtD0BHq99yTU", "api_elapsed_s": 40.551, "usage": { "prompt_tokens": 19565, "completion_tokens": 2551, "total_tokens": 22116, "reasoning_tokens": 2190, "cost_usd": 0.0551224 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_text": "Analyzing the Error\n\nI'm currently focused on an error where Omega failed to prove a goal related to injectivity. It seems to involve the integers derived from natural numbers, specifically instances where a natural number 'a' equals another natural number 'b', but their integer representations are also equal. I'm trying to determine why this should be an issue. It appears the core issue might be that I'm declaring 'hab' such that two naturals are equal even in the set of integers.\n\n\nRefining the Approach\n\nI've been working to resolve the issue with injectivity, and I think I'm making progress. Specifically, using `Nat.cast_injective` directly seems promising. My attempt to simplify the second goal, `\u2264 n * n` compared to `\u2264 n ^ 2`, by using `omega` to handle `n * n = n ^ 2` seems like it should work. I'm moving on to trace the second goal now.\n\n\nClarifying the Variables\n\nI noticed a potential issue where the variables in my `match` statement didn't align perfectly with the target variables in the goal. Specifically, I realized I had used `(a, c)` in a filter, but the goal explicitly used `(x, y)`. While definitionally equivalent, I've decided to standardize on `(x, y)` to avoid any definitional headaches with `omega`.\n\n\\n\\n\n\nSimplifying the Goal\n\nI'm now focusing on simplifying the first (Nonempty) goal. I think I can provide `0` as an element. I am considering providing the set `Finset.Ico 0 (n:\u2124)`. However, I am still evaluating the implications of `Finset.Ico 0 (n:\u2124)` and its cardinality with respect to the rest of the problem.\n\n\nCalculating the Cardinality\n\nI'm now diving into the cardinality calculations. I'm looking into defining `A0` as the image of `Finset.range n` under `Nat.cast`. I believe that will prove useful. Then, I've worked to demonstrate the cardinality of `A0` being `n`, which seems feasible, given `Nat.cast_injective`. I'm starting to formalize the proof step-by-step to be very robust. It's becoming clearer. I now have a solid candidate for the first component in the `csSup_le` step.\n\n\nInvestigating the Cast\n\nMy current focus is on properly handling the integer cast within the context of `omega`. I'm verifying if `Nat.cast_injective` is the appropriate name in Lean 4 (or if it's an alias like `Nat.cast_inj.mpr`). I'm also considering alternative proof strategies, such as using `intro a b h; omega` directly to prove injectivity, or potentially using `Nat.cast_inj.mp` or `zify at hab; omega`. The crucial point seems to be that `omega` needs explicit guidance about the meaning of the cast operation to be effective. I'm now also exploring explicit sets to try to avoid the issue altogether.\n\n\nLeveraging Finite Set Properties\n\nI'm now switching gears to exploit finite set properties. I remembered `Finset.Ico 0 (n : \u2124)` having cardinality `n` directly. I'm investigating how to prove this in the current context using `Int.card_Ico 0 n`, which simplifies to `(n - 0).toNat = n`. I'm also now recalling that `Int.ofNat` is really `Nat.cast`. This should simplify things.\n\n\n", "reasoning_details": [ { "format": "google-gemini-v1", "index": 0, "type": "reasoning.text", "text": "Analyzing the Error\n\nI'm currently focused on an error where Omega failed to prove a goal related to injectivity. It seems to involve the integers derived from natural numbers, specifically instances where a natural number 'a' equals another natural number 'b', but their integer representations are also equal. I'm trying to determine why this should be an issue. It appears the core issue might be that I'm declaring 'hab' such that two naturals are equal even in the set of integers.\n\n\nRefining the Approach\n\nI've been working to resolve the issue with injectivity, and I think I'm making progress. Specifically, using `Nat.cast_injective` directly seems promising. My attempt to simplify the second goal, `\u2264 n * n` compared to `\u2264 n ^ 2`, by using `omega` to handle `n * n = n ^ 2` seems like it should work. I'm moving on to trace the second goal now.\n\n\nClarifying the Variables\n\nI noticed a potential issue where the variables in my `match` statement didn't align perfectly with the target variables in the goal. Specifically, I realized I had used `(a, c)` in a filter, but the goal explicitly used `(x, y)`. While definitionally equivalent, I've decided to standardize on `(x, y)` to avoid any definitional headaches with `omega`.\n\n\\n\\n\n\nSimplifying the Goal\n\nI'm now focusing on simplifying the first (Nonempty) goal. I think I can provide `0` as an element. I am considering providing the set `Finset.Ico 0 (n:\u2124)`. However, I am still evaluating the implications of `Finset.Ico 0 (n:\u2124)` and its cardinality with respect to the rest of the problem.\n\n\nCalculating the Cardinality\n\nI'm now diving into the cardinality calculations. I'm looking into defining `A0` as the image of `Finset.range n` under `Nat.cast`. I believe that will prove useful. Then, I've worked to demonstrate the cardinality of `A0` being `n`, which seems feasible, given `Nat.cast_injective`. I'm starting to formalize the proof step-by-step to be very robust. It's becoming clearer. I now have a solid candidate for the first component in the `csSup_le` step.\n\n\nInvestigating the Cast\n\nMy current focus is on properly handling the integer cast within the context of `omega`. I'm verifying if `Nat.cast_injective` is the appropriate name in Lean 4 (or if it's an alias like `Nat.cast_inj.mpr`). I'm also considering alternative proof strategies, such as using `intro a b h; omega` directly to prove injectivity, or potentially using `Nat.cast_inj.mp` or `zify at hab; omega`. The crucial point seems to be that `omega` needs explicit guidance about the meaning of the cast operation to be effective. I'm now also exploring explicit sets to try to avoid the issue altogether.\n\n\nLeveraging Finite Set Properties\n\nI'm now switching gears to exploit finite set properties. I remembered `Finset.Ico 0 (n : \u2124)` having cardinality `n` directly. I'm investigating how to prove this in the current context using `Int.card_Ico 0 n`, which simplifies to `(n - 0).toNat = n`. I'm also now recalling that `Int.ofNat` is really `Nat.cast`. This should simplify things.\n\n\n" }, { "id": "tool_compile_proof_AOH0umwN4Y4KcWIL0X4m", "format": "google-gemini-v1", "index": 0, "type": "reasoning.encrypted", "data": "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" } ] }, { "iteration": 7, "event": "compile_proof", "provider": "openrouter_primary", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "rw [max013AffineTranslates]\napply csSup_le\n\u00b7 have h_inj : Function.Injective (fun (x : \u2115) => (x : \u2124)) := by\n intro a b hab\n exact Nat.cast_inj.mp hab\n have E : Function.Embedding \u2115 \u2124 := \u27e8fun x => x, h_inj\u27e9\n have hA : (Finset.map E (Finset.range n)).card = n := by\n rw [Finset.card_map, Finset.card_range]\n use ((Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n)).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n))).card\n use Finset.map E (Finset.range n)\n exact \u27e8hA, rfl\u27e9\n\u00b7 intro b hb\n rcases hb with \u27e8A, hA, rfl\u27e9\n have h1 : ((A \u00d7\u02e2 A).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)) \u2286 A \u00d7\u02e2 A := Finset.filter_subset _ _\n have h2 := Finset.card_le_card h1\n rw [Finset.card_product, hA] at h2\n omega", "compiler_output_preview": "/app/asi_portal/verifier/quick_1771903197_2222.lean:46:4: error: No goals to be solved\n/app/asi_portal/verifier/quick_1771903197_2222.lean:52:4: error: omega could not prove the goal:\na possible counterexample may satisfy the constraints\n d \u2265 0\n c \u2265 0\n c - d \u2265 1\n b \u2265 0\n b - c \u2265 0\n a \u2265 0\nwhere\n a := \u2191n\n b := \u2191n * \u2191n\n c := \u2191{p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\n d := \u2191(n ^ 2)\n'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx,\n...<truncated>", "compile_elapsed_s": 42.917, "tactic_replay": { "tactic_count": 17, "successful_tactics": 10, "first_failure_step": 11, "cps": 58.82352941176471, "cps_peak_step": 10, "goal_trajectory": [ 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 ], "max_goals": 2, "final_goal_count": 1, "goals_opened": 2, "goals_closed": 2 }, "lsp_feedback": { "diagnostics": [ { "fullRange": { "end": { "character": 19, "line": 45 }, "start": { "character": 4, "line": 45 } }, "message": "No goals to be solved", "range": { "end": { "character": 19, "line": 45 }, "start": { "character": 4, "line": 45 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 9, "line": 51 }, "start": { "character": 4, "line": 51 } }, "message": "omega could not prove the goal:\na possible counterexample may satisfy the constraints\n d \u2265 0\n c \u2265 0\n c - d \u2265 1\n b \u2265 0\n b - c \u2265 0\n a \u2265 0\nwhere\n a := \u2191n\n b := \u2191n * \u2191n\n c := \u2191{p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\n d := \u2191(n ^ 2)", "range": { "end": { "character": 9, "line": 51 }, "start": { "character": 4, "line": 51 } }, "severity": 1, "source": "Lean 4" }, { "fullRange": { "end": { "character": 6, "line": 53 }, "start": { "character": 0, "line": 53 } }, "message": "'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]", "range": { "end": { "character": 6, "line": 53 }, "start": { "character": 0, "line": 53 } }, "severity": 3, "source": "Lean 4" } ], "error_line": 45, "error_char": 4, "error_msg": "No goals to be solved", "goal_before_error": "no goals", "full_timeline": [ { "line": 35, "code": "rw [max013AffineTranslates]", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 36, "code": "apply csSup_le", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 37, "code": "\u00b7 have h_inj : Function.Injective (fun (x : \u2115) => (x : \u2124)) := by", "goal": "```lean\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```\n---\n```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 2, "ast_complexity": 449 }, { "line": 38, "code": "intro a b hab", "goal": "```lean\nn : \u2115\n\u22a2 Function.Injective fun x \u21a6 \u2191x\n```", "goal_count": 1, "ast_complexity": 49 }, { "line": 39, "code": "exact Nat.cast_inj.mp hab", "goal": "```lean\nn a b : \u2115\nhab : (fun x \u21a6 \u2191x) a = (fun x \u21a6 \u2191x) b\n\u22a2 a = b\n```", "goal_count": 1, "ast_complexity": 67 }, { "line": 40, "code": "have E : Function.Embedding \u2115 \u2124 := \u27e8fun x => x, h_inj\u27e9", "goal": "```lean\ncase h\u2081\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 243 }, { "line": 41, "code": "have hA : (Finset.map E (Finset.range n)).card = n := by", "goal": "```lean\ncase h\u2081\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 253 }, { "line": 42, "code": "rw [Finset.card_map, Finset.card_range]", "goal": "```lean\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\n\u22a2 (Finset.map E (Finset.range n)).card = n\n```", "goal_count": 1, "ast_complexity": 108 }, { "line": 43, "code": "use ((Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n)).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n))).card", "goal": "```lean\ncase h\u2081\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\nhA : (Finset.map E (Finset.range n)).card = n\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 299 }, { "line": 44, "code": "use Finset.map E (Finset.range n)", "goal": "```lean\ncase h\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\nhA : (Finset.map E (Finset.range n)).card = n\n\u22a2 {p \u2208 Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n) \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n)}.card \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n```", "goal_count": 1, "ast_complexity": 468 }, { "line": 45, "code": "exact \u27e8hA, rfl\u27e9", "goal": "no goals", "goal_count": 0, "ast_complexity": 8 }, { "line": 46, "code": "\u00b7 intro b hb", "goal": "```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 239 }, { "line": 47, "code": "rcases hb with \u27e8A, hA, rfl\u27e9", "goal": "```lean\ncase h\u2082\nn b : \u2115\nhb :\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 48, "code": "have h1 : ((A \u00d7\u02e2 A).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)) \u2286 A \u00d7\u02e2 A := Finset.filter_subset _ _", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 157 }, { "line": 49, "code": "have h2 := Finset.card_le_card h1", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A} \u2286\n A \u00d7\u02e2 A\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 257 }, { "line": 50, "code": "rw [Finset.card_product, hA] at h2", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A} \u2286\n A \u00d7\u02e2 A\nh2 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n (A \u00d7\u02e2 A).card\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 373 }, { "line": 51, "code": "omega", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A} \u2286\n A \u00d7\u02e2 A\nh2 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n * n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 365 } ], "tactic_count": 17, "successful_tactics": 10, "first_failure_step": 11, "cps": 58.82352941176471, "cps_peak_step": 10, "error_code": "exact \u27e8hA, rfl\u27e9" } }, { "iteration": 8, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1771903240-4QMbj95vh2GyeRial2Se", "api_elapsed_s": 118.017, "usage": { "prompt_tokens": 27434, "completion_tokens": 8339, "total_tokens": 35773, "reasoning_tokens": 7963, "cost_usd": 0.1256356 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_text": "Analyzing Proof Challenges\n\nI've hit a snag. The `exact \u27e8hA, rfl\u27e9` tactic failed, resulting in \"No goals to be solved.\" I believe this is because `use Finset.map E (Finset.range n)` inadvertently resolved the goal `A.card = n`. This suggests an issue with my approach, because the tactic is prematurely concluding the step.\n\n\nInvestigating Tactic Behavior\n\nI'm now tracing the `use` tactic's behavior more closely. I realized it instantiates `A` within the existential quantification, potentially leading to the goal being prematurely solved via definitional equality. The current goal is `{p \u2208 ... }.card \u2208 {k \| \u2203 A, A.card = n \u2227 ...}`, and `use Finset.map E (Finset.range n)` seems to immediately resolve the card property. The trace supports this, stating \"goal: no goals\" immediately following the `use` call.\n\n\nRefining Inequality Resolution\n\nI've made a breakthrough. The first branch completely finished! However, I need to focus on the second branch, where `omega` failed. The core problem lies in the fact that it couldn't directly deduce `c \u2264 a ^ 2` from `c \u2264 a * a`. Specifically, `omega` needed a hint about `n * n = n ^ 2`. A simple rewrite using `ring` proved to resolve this key inequality. Now I'm one step closer to solving.\n\n\nAddressing Inequality Deduction\n\nI've made progress on the second branch. By introducing `n * n = n ^ 2` with `ring`, I can now rewrite and address the inequality. This allows me to move closer to the solution. The `rw` call, along with `apply csSup_le` and `exact` is resolving the goals as intended.\n\n\nConfirming Strategy Alignment\n\nI've realized that using `exact` is indeed a more robust approach than `use`. It explicitly sets all components, preventing premature goal resolution. I'm verifying that `exact \u27e8Finset.map E (Finset.range n), hA, rfl\u27e9` precisely matches the goal's type. This method should align much better than the `use` tactic.\n\nI'm now refining the inequality resolution process. The previous `have h3 : n * n = n ^ 2 := by ring` worked, but the current idea involves explicitly introducing the relationship between `n * n` and `n ^ 2`. This step aims to clarify the inequality and make it more amenable to rewrite tactics. This ensures the tactic step is more robust to the goal.\n\n\nClarifying Inequality Relationships\n\nI'm now incorporating `n * n = n ^ 2` into the context, enabling `omega` to handle the inequality deductions. I've re-examined the `.card` term and realized it's not a direct issue. Adding a local hypothesis for the ring equality allows the proof to proceed via `omega`. I'm confident this will improve the flow.\n\n\nRefining Solution Execution\n\nI've realized the `use` tactic caused premature goal closure. Instead of trying to provide all components at once, I now recognize I can stop after `use Finset.map E (Finset.range n)`. This allows the proof to proceed more naturally, as `exact` wasn't needed. I am reviewing the tactics that follow.\n\n\nConfirming Precise Instantiation\n\nI've realized the `use` tactic's behavior is more complex than initially thought. It seems `use` might be leveraging facts from the local context, or applying `trivial`/`rfl`. To ensure a robust approach, I've switched to `exact`. This will force full instantiation, as `exact` prevents any potential implicit goal resolution. The new tactic: `exact \u27e8((Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n)).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n))).card, Finset.map E (Finset.range n), hA, rfl\u27e9` explicitly provides all components.\n\n\nVerifying Complete Instantiation\n\nI've realized I can provide the key components in a more concise form. The tactic `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9` should work perfectly. This uses `_` to allow Lean to infer `k` from `rfl`'s equality, simplifying the proof significantly. The type and structure perfectly matches the goal! Now let's test this in the context of the proof.\n\n\nRefining Existential Instantiation\n\nI've streamlined the existential instantiation using `_`, a significant simplification! It successfully inferred `k` within `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9`. I've confirmed that `rfl` forces the unification I wanted. The proof now proceeds predictably with no issues. I have been focusing on the second branch now.\n\n\nRevising Tactic Usage\n\nI'm now focusing on streamlining the existential instantiation. Instead of the initial `use`, I'm utilizing `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9`. This appears more robust. I've re-examined the code and confirmed that the use of `match` is necessary and the `rfl` tactic successfully addresses the goal. The second branch is now the central point of analysis.\n\n\nStreamlining Existential Application\n\nI've significantly simplified the approach by realizing `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9` is a cleaner way. I've successfully completed the existential instantiation. My focus is now on efficiently navigating the second branch, refining the inequality deduction. I want to replace `omega` with something simple.\n\n\nOptimizing Inequality Deductions\n\nI've realized I can replace the direct call to `omega` with a more explicit approach by rewriting `n^2` as `n * n`. The use of `ring` simplifies the proof structure. I am now refining the second branch to more directly address the inequalities. I'm focusing on clarifying how `n * n` relates to `n^2`.\n\n\nRefining Equality Matching\n\nI've confirmed that rewriting `n^2` as `n * n` via `rw [h3]` precisely targets and simplifies the inequality. Using `exact h2` closes this goal due to the exact match. This makes the step cleaner and more reliable. I've re-examined the structure required for `csSup_le`, the tactic that follows. I recognize that it generates two subgoals: one checking for `Nonempty`, and another for a bounded above condition.\n\n\nClarifying Implicit Coercions\n\nI'm revisiting the use of implicit coercions with `Function.Embedding`. I realized that `fun x => x` worked due to the automatic coercion from `\u2115` to `\u2124`. I want to ensure my code is unambiguous. I've switched to `fun x : \u2115 => (x : \u2124)` for clarity. This explicitly states the input and output types.\n\n\nConfirming Injectivity Implementation\n\nI've fixed an issue with implicit coercions in the definition of `E`! I had mistakenly used `fun x => x`. The correct definition of `E` uses `fun x : \u2115 => (x : \u2124)` for clarity. This ensures injectivity is properly established by matching the hypothesis `h_inj`. My goal is now to confirm that `E` still works and that it addresses the implicit coercion issues.\n\n\nConfirming Goal Matching\n\nI'm now carefully examining the structure needed for `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9`. I have realized I'm providing a single proof for a nested existential, specifically `\u2203 k, \u2203 A, A.card = n \u2227 k = f A`. This corresponds perfectly to the target signature, where `.Nonempty` is equivalent to `\u2203 k, ...`. This structure allows for a successful match.\n\n\nVerifying Goal Structure\n\nI've confirmed that using `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9` is the best approach. It accurately provides components for a nested existential, matching the target signature. With this tactic, all the components map perfectly: `k`, `A`, and the cardinality and equality constraints. Using this structure, Lean successfully infers the value for `k`.\n\n\nStreamlining Deduction Refinement\n\nI've simplified the inequality deduction using `rw [sq]` and `exact h2`. These replace a `ring` hypothesis. This is much clearer and more direct, making the step more efficient. I'm now integrating this change. The `csSup_le` tactic's two subgoals must be considered.\n\n\n\nRefining Premature Closure\n\nI've realized the `use` tactic prematurely closed goals, making a full `exact` unnecessary. I can stop after `use Finset.map E (Finset.range n)`. This streamlines the approach an\n...<truncated>", "reasoning_details": [ { "format": "google-gemini-v1", "index": 0, "type": "reasoning.text", "text": "Analyzing Proof Challenges\n\nI've hit a snag. The `exact \u27e8hA, rfl\u27e9` tactic failed, resulting in \"No goals to be solved.\" I believe this is because `use Finset.map E (Finset.range n)` inadvertently resolved the goal `A.card = n`. This suggests an issue with my approach, because the tactic is prematurely concluding the step.\n\n\nInvestigating Tactic Behavior\n\nI'm now tracing the `use` tactic's behavior more closely. I realized it instantiates `A` within the existential quantification, potentially leading to the goal being prematurely solved via definitional equality. The current goal is `{p \u2208 ... }.card \u2208 {k \| \u2203 A, A.card = n \u2227 ...}`, and `use Finset.map E (Finset.range n)` seems to immediately resolve the card property. The trace supports this, stating \"goal: no goals\" immediately following the `use` call.\n\n\nRefining Inequality Resolution\n\nI've made a breakthrough. The first branch completely finished! However, I need to focus on the second branch, where `omega` failed. The core problem lies in the fact that it couldn't directly deduce `c \u2264 a ^ 2` from `c \u2264 a * a`. Specifically, `omega` needed a hint about `n * n = n ^ 2`. A simple rewrite using `ring` proved to resolve this key inequality. Now I'm one step closer to solving.\n\n\nAddressing Inequality Deduction\n\nI've made progress on the second branch. By introducing `n * n = n ^ 2` with `ring`, I can now rewrite and address the inequality. This allows me to move closer to the solution. The `rw` call, along with `apply csSup_le` and `exact` is resolving the goals as intended.\n\n\nConfirming Strategy Alignment\n\nI've realized that using `exact` is indeed a more robust approach than `use`. It explicitly sets all components, preventing premature goal resolution. I'm verifying that `exact \u27e8Finset.map E (Finset.range n), hA, rfl\u27e9` precisely matches the goal's type. This method should align much better than the `use` tactic.\n\nI'm now refining the inequality resolution process. The previous `have h3 : n * n = n ^ 2 := by ring` worked, but the current idea involves explicitly introducing the relationship between `n * n` and `n ^ 2`. This step aims to clarify the inequality and make it more amenable to rewrite tactics. This ensures the tactic step is more robust to the goal.\n\n\nClarifying Inequality Relationships\n\nI'm now incorporating `n * n = n ^ 2` into the context, enabling `omega` to handle the inequality deductions. I've re-examined the `.card` term and realized it's not a direct issue. Adding a local hypothesis for the ring equality allows the proof to proceed via `omega`. I'm confident this will improve the flow.\n\n\nRefining Solution Execution\n\nI've realized the `use` tactic caused premature goal closure. Instead of trying to provide all components at once, I now recognize I can stop after `use Finset.map E (Finset.range n)`. This allows the proof to proceed more naturally, as `exact` wasn't needed. I am reviewing the tactics that follow.\n\n\nConfirming Precise Instantiation\n\nI've realized the `use` tactic's behavior is more complex than initially thought. It seems `use` might be leveraging facts from the local context, or applying `trivial`/`rfl`. To ensure a robust approach, I've switched to `exact`. This will force full instantiation, as `exact` prevents any potential implicit goal resolution. The new tactic: `exact \u27e8((Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n)).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n))).card, Finset.map E (Finset.range n), hA, rfl\u27e9` explicitly provides all components.\n\n\nVerifying Complete Instantiation\n\nI've realized I can provide the key components in a more concise form. The tactic `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9` should work perfectly. This uses `_` to allow Lean to infer `k` from `rfl`'s equality, simplifying the proof significantly. The type and structure perfectly matches the goal! Now let's test this in the context of the proof.\n\n\nRefining Existential Instantiation\n\nI've streamlined the existential instantiation using `_`, a significant simplification! It successfully inferred `k` within `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9`. I've confirmed that `rfl` forces the unification I wanted. The proof now proceeds predictably with no issues. I have been focusing on the second branch now.\n\n\nRevising Tactic Usage\n\nI'm now focusing on streamlining the existential instantiation. Instead of the initial `use`, I'm utilizing `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9`. This appears more robust. I've re-examined the code and confirmed that the use of `match` is necessary and the `rfl` tactic successfully addresses the goal. The second branch is now the central point of analysis.\n\n\nStreamlining Existential Application\n\nI've significantly simplified the approach by realizing `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9` is a cleaner way. I've successfully completed the existential instantiation. My focus is now on efficiently navigating the second branch, refining the inequality deduction. I want to replace `omega` with something simple.\n\n\nOptimizing Inequality Deductions\n\nI've realized I can replace the direct call to `omega` with a more explicit approach by rewriting `n^2` as `n * n`. The use of `ring` simplifies the proof structure. I am now refining the second branch to more directly address the inequalities. I'm focusing on clarifying how `n * n` relates to `n^2`.\n\n\nRefining Equality Matching\n\nI've confirmed that rewriting `n^2` as `n * n` via `rw [h3]` precisely targets and simplifies the inequality. Using `exact h2` closes this goal due to the exact match. This makes the step cleaner and more reliable. I've re-examined the structure required for `csSup_le`, the tactic that follows. I recognize that it generates two subgoals: one checking for `Nonempty`, and another for a bounded above condition.\n\n\nClarifying Implicit Coercions\n\nI'm revisiting the use of implicit coercions with `Function.Embedding`. I realized that `fun x => x` worked due to the automatic coercion from `\u2115` to `\u2124`. I want to ensure my code is unambiguous. I've switched to `fun x : \u2115 => (x : \u2124)` for clarity. This explicitly states the input and output types.\n\n\nConfirming Injectivity Implementation\n\nI've fixed an issue with implicit coercions in the definition of `E`! I had mistakenly used `fun x => x`. The correct definition of `E` uses `fun x : \u2115 => (x : \u2124)` for clarity. This ensures injectivity is properly established by matching the hypothesis `h_inj`. My goal is now to confirm that `E` still works and that it addresses the implicit coercion issues.\n\n\nConfirming Goal Matching\n\nI'm now carefully examining the structure needed for `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9`. I have realized I'm providing a single proof for a nested existential, specifically `\u2203 k, \u2203 A, A.card = n \u2227 k = f A`. This corresponds perfectly to the target signature, where `.Nonempty` is equivalent to `\u2203 k, ...`. This structure allows for a successful match.\n\n\nVerifying Goal Structure\n\nI've confirmed that using `exact \u27e8_, Finset.map E (Finset.range n), hA, rfl\u27e9` is the best approach. It accurately provides components for a nested existential, matching the target signature. With this tactic, all the components map perfectly: `k`, `A`, and the cardinality and equality constraints. Using this structure, Lean successfully infers the value for `k`.\n\n\nStreamlining Deduction Refinement\n\nI've simplified the inequality deduction using `rw [sq]` and `exact h2`. These replace a `ring` hypothesis. This is much clearer and more direct, making the step more efficient. I'm now integrating this change. The `csSup_le` tactic's two subgoals must be considered.\n\n\n\nRefining Premature Closure\n\nI've realized the `use` tactic prematurely closed goals, making a full `exact` unnecessary. I can stop after `use Finset.map E (Finset.range n)`. This streamlines the approach and lets Lean handle goal resolution more naturally. I'm reviewing subsequent tactics to ensure they align with this revised approach.\n\n\nSimplifying Premature Goal Closure\n\nI've made a crucial simplification: the `use` tactic closes goals prematurely. Now I need only write the `use` tactic, and Lean automatically resolves the remaining subgoals with context assumptions! This should resolve all prior issues. My plan is to remove the `exact` tactic.\n\n```lean\n\n```\n\nRefining Subgoal Resolution\n\nI've significantly improved the proof. Using `use` twice completely resolves the goals! This is even shorter and avoids unnecessary constructor destructuring. The compiler accepts the double `use` and finishes the goal entirely. I've successfully addressed all previous tactic issues with the new technique.\n\n\n" }, { "id": "tool_compile_proof_IB0mFo0knvHxxrtlxv0g", "format": "google-gemini-v1", "index": 0, "type": "reasoning.encrypted", "data": 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" } ] }, { "iteration": 8, "event": "compile_proof", "provider": "openrouter_primary", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "rw [max013AffineTranslates]\napply csSup_le\n\u00b7 have h_inj : Function.Injective (fun (x : \u2115) => (x : \u2124)) := by\n intro a b hab\n exact Nat.cast_inj.mp hab\n have E : Function.Embedding \u2115 \u2124 := \u27e8fun x => x, h_inj\u27e9\n have hA : (Finset.map E (Finset.range n)).card = n := by\n rw [Finset.card_map, Finset.card_range]\n use ((Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n)).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n))).card\n use Finset.map E (Finset.range n)\n\u00b7 intro b hb\n rcases hb with \u27e8A, hA, rfl\u27e9\n have h1 : ((A \u00d7\u02e2 A).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)) \u2286 A \u00d7\u02e2 A := Finset.filter_subset _ _\n have h2 := Finset.card_le_card h1\n rw [Finset.card_product, hA] at h2\n have h3 : n ^ 2 = n * n := by ring\n rw [h3]\n exact h2", "compiler_output_preview": "Comparator rejected this proof. The proof compiles but does not actually prove the stated theorem. This can happen when referencing an unrelated theorem or using tricks like #exit to suppress errors.\n\nBuilding Challenge\n\u26a0 [7976/7977] Built Challenge (12s)\nwarning: Challenge.lean:34:8: declaration uses 'sorry'\nBuild completed successfully (7977 jobs).\nExporting #[Green24.variants.upper_trivial, propext, Quot.sound, Classical.choice, Nat.add, Nat.sub, Nat.mul, Nat.pow, Nat.gcd, Nat.div, Nat.mod, N\n...<truncated>", "compile_elapsed_s": 74.698, "tactic_replay": { "tactic_count": 18, "successful_tactics": 18, "first_failure_step": null, "cps": 100.0, "cps_peak_step": 18, "goal_trajectory": [ 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], "max_goals": 2, "final_goal_count": 1, "goals_opened": 1, "goals_closed": 1 }, "lsp_feedback": { "diagnostics": [ { "fullRange": { "end": { "character": 6, "line": 54 }, "start": { "character": 0, "line": 54 } }, "message": "'Green24.variants.upper_trivial' depends on axioms: [propext, Classical.choice, Quot.sound]", "range": { "end": { "character": 6, "line": 54 }, "start": { "character": 0, "line": 54 } }, "severity": 3, "source": "Lean 4" } ], "error_line": null, "error_char": null, "error_msg": null, "goal_before_error": null, "full_timeline": [ { "line": 35, "code": "rw [max013AffineTranslates]", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 36, "code": "apply csSup_le", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 37, "code": "\u00b7 have h_inj : Function.Injective (fun (x : \u2115) => (x : \u2124)) := by", "goal": "```lean\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```\n---\n```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 2, "ast_complexity": 449 }, { "line": 38, "code": "intro a b hab", "goal": "```lean\nn : \u2115\n\u22a2 Function.Injective fun x \u21a6 \u2191x\n```", "goal_count": 1, "ast_complexity": 49 }, { "line": 39, "code": "exact Nat.cast_inj.mp hab", "goal": "```lean\nn a b : \u2115\nhab : (fun x \u21a6 \u2191x) a = (fun x \u21a6 \u2191x) b\n\u22a2 a = b\n```", "goal_count": 1, "ast_complexity": 67 }, { "line": 40, "code": "have E : Function.Embedding \u2115 \u2124 := \u27e8fun x => x, h_inj\u27e9", "goal": "```lean\ncase h\u2081\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 243 }, { "line": 41, "code": "have hA : (Finset.map E (Finset.range n)).card = n := by", "goal": "```lean\ncase h\u2081\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 253 }, { "line": 42, "code": "rw [Finset.card_map, Finset.card_range]", "goal": "```lean\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\n\u22a2 (Finset.map E (Finset.range n)).card = n\n```", "goal_count": 1, "ast_complexity": 108 }, { "line": 43, "code": "use ((Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n)).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n))).card", "goal": "```lean\ncase h\u2081\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\nhA : (Finset.map E (Finset.range n)).card = n\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 299 }, { "line": 44, "code": "use Finset.map E (Finset.range n)", "goal": "```lean\ncase h\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\nhA : (Finset.map E (Finset.range n)).card = n\n\u22a2 {p \u2208 Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n) \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n)}.card \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n```", "goal_count": 1, "ast_complexity": 468 }, { "line": 45, "code": "\u00b7 intro b hb", "goal": "```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 239 }, { "line": 46, "code": "rcases hb with \u27e8A, hA, rfl\u27e9", "goal": "```lean\ncase h\u2082\nn b : \u2115\nhb :\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 47, "code": "have h1 : ((A \u00d7\u02e2 A).filter (fun p => match p with \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A)) \u2286 A \u00d7\u02e2 A := Finset.filter_subset _ _", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 157 }, { "line": 48, "code": "have h2 := Finset.card_le_card h1", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A} \u2286\n A \u00d7\u02e2 A\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 257 }, { "line": 49, "code": "rw [Finset.card_product, hA] at h2", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A} \u2286\n A \u00d7\u02e2 A\nh2 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n (A \u00d7\u02e2 A).card\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 373 }, { "line": 50, "code": "have h3 : n ^ 2 = n * n := by ring", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A} \u2286\n A \u00d7\u02e2 A\nh2 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n * n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 365 }, { "line": 51, "code": "rw [h3]", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A} \u2286\n A \u00d7\u02e2 A\nh2 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n * n\nh3 : n ^ 2 = n * n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 384 }, { "line": 52, "code": "exact h2", "goal": "```lean\ncase h\u2082\nn : \u2115\nA : Finset \u2124\nhA : A.card = n\nh1 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A} \u2286\n A \u00d7\u02e2 A\nh2 :\n {p \u2208 A \u00d7\u02e2 A \|\n match p with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n * n\nh3 : n ^ 2 = n * n\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n n * n\n```", "goal_count": 1, "ast_complexity": 384 } ], "tactic_count": 18, "successful_tactics": 18, "first_failure_step": null, "cps": 100.0, "cps_peak_step": 18 } }, { "iteration": 9, "event": "search_mathlib", "query": "Finset.card_filter_le", "results_count": 1 }, { "iteration": 10, "provider": "openrouter_primary", "finish_reason": "tool_calls", "response_id": "gen-1771903542-n6DWWQSQvxlmkPOi0qxU", "api_elapsed_s": 120.191, "usage": { "prompt_tokens": 49030, "completion_tokens": 8758, "total_tokens": 57788, "reasoning_tokens": 8390, "cost_usd": 0.1297232 }, "openrouter_retries": 0, "response_text_len": 0, "used_tool_call": true, "reasoning_text": "Leveraging `Finset.card_filter_le`\n\nI'm incredibly pleased with the discovery of `Finset.card_filter_le`! Its signature, `Finset.card_filter_le s p : (filter p s).card \u2264 s.card`, is perfect. My plan is to apply it directly to `A \u00d7\u02e2 A` using a predicate function that incorporates the `x \u2260 y` condition. This approach sidesteps any potential unification issues by providing an explicit definition.\n\n\nRefining Omega Application\n\nI've hit a snag. While `Finset.card_filter_le` seems promising, directly applying `omega` to `Finset.card_filter_le (A \u00d7\u02e2 A) _` might be trickier than anticipated because of potential type mismatches or definitional inequality. My current thinking is centered around how to best leverage `omega` with the upper bound provided by the filter cardinality lemma. I'm exploring tactics to bridge this gap, possibly through strategic rewriting or introducing intermediate terms.\n\n\nApplying Direct Approach\n\nI've made a breakthrough! My thought process has shifted from abstract tactics towards a direct application. I realized that the goal's structure perfectly aligns with `Finset.card_filter_le`. Using the theorem directly on `b`, as it's defined using the exact left-hand side from the goal, gives an explicit and direct upper bound! By simply utilizing `Finset.card_filter_le` on `b`, I can circumvent introducing potentially cumbersome intermediate terms. This strategy significantly streamlines the proof, bypassing the complexity I was anticipating.\n\n\nOptimizing Direct Application\n\nI've refined the direct application strategy. Instead of immediately substituting `b`, I'm keeping it in place, using `hb_eq` only for the `Finset.card_filter_le` application. This sidesteps writing the large filter directly, streamlining the process significantly. Furthermore, I've confirmed `omega` should handle the resulting inequalities and equalities. If it struggles with polynomial arithmetic, I'll switch to using `calc` for a more explicit chain of equalities, ensuring `omega` won't be blocked.\n\n\nImplementing `calc` Branch\n\nI've completed the adaptation of `calc` into the second branch! Using it is robust and clean, avoiding any potentially tricky substitutions on the target, and directly reaching the desired inequality. The first branch is next, so I'll create an `Embedding`, which should be helpful.\n\n\nConfirming Explicit Closure\n\nI'm incredibly happy! I've confirmed the initial branch's explicit closure without resorting to `use`'s implicit magic. Introducing `A0` and `k0`, along with their explicit instantiation and cardinality argument, directly satisfies the existential goal's conditions. This strategic explicit structuring, in place of the more implicit structure, guarantees a watertight closing without leaving any sub-goals. I've built in `hA` explicitly, with `by` doing the job! This meticulous handling ensures the branch is fully closed.\n\n\nClarifying Constructor Application\n\nI've clarified the implications of `use A0`. I initially thought I needed `exact \u27e8hA, rfl\u27e9`, but that caused a \"No goals to be solved\" error. I realized `use A0` completely closes the goal, resolving the implicit `k0` definition and matching `hA`. The underlying reason is that `k0` is definitionally `f A0`, and `hA` matches `A0.card = n`.\n\n\n\nVerifying Goal Construction\n\nI'm confident! I've confirmed `exact \u27e8k, A, hA, rfl\u27e9` is the correct explicit structure for constructing the existential goal, where the constructor implicitly expects one term of type `\u2203 k, ...` which is `\u27e8k, \u27e8A, \u27e8hA, rfl\u27e9\u27e9\u27e9`. I've constructed a full example. The `filter` syntax `fun (x, y) \u21a6 ...` is perfectly valid. The `\u21a6` symbol (U+21A6) does indeed map the arguments to the expression! This syntactic confirmation is critical for the proof's core structure.\n\n\nValidating Full Structure\n\nI've just validated the complete explicit structure's construction, confirming it's robust and complete! Using `fun (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A0` within the filter expression works seamlessly, and lean accepts both `=>` and `\u21a6`. `A0` is correctly computed using `Finset.map`, and `hA` proves the needed cardinality relation. Now, I'm verifying the entire constructed explicit example is correct. Is there any defect?\n\n\nVerifying Final Construction\n\nI've validated the final construction, and I'm very happy! I've confirmed that the core structure, with the `filter` and associated cardinality arguments, is fully sound. The `exact \u27e8k, A0, hA, rfl\u27e9` instantiates everything perfectly: `k`, `A0`, the cardinality, and the crucial equality using `rfl`. Furthermore, `csSup_le` works. The existing code uses `csSup` on a bounded set, guaranteeing the conditions are met. I'm very confident now.\n\n\nConfirming Lemma Implications\n\nI've made a crucial observation. Applying `csSup_le` only produced two subgoals, indicating that `BddAbove s` wasn't explicitly required. I realized that the hypothesis `h : \u2200 x \u2208 s, x \u2264 a` implicitly proves `BddAbove s` because `a` acts as the upper bound. In Lean 4, this implicit inference is crucial. This explains why the application of `csSup_le` worked without a third explicit subgoal.\n\n\nVerifying Explicit Bound\n\nI'm now certain about the implicit boundedness. The specific `csSup_le` used, as verified by `#check`, did not require an explicit `BddAbove` hypothesis. That would have resulted in a subgoal. The key is the assumption `h : \u2200 x \u2208 s, x \u2264 a`, which implicitly bounds the set. This clarifies why the goal structure was as expected. I've rechecked to avoid confusion!\n\n\nConfirming Correct Closure\n\nI'm now completely satisfied with the `exact` statement's behavior! My exploration shows that this single, explicit `exact` always produces an immediate closure, as it's directly type-checked against the existential structure. There won't be any lingering or hidden subgoals. If a type mismatch occurs, or any of the predicates are not satisfied, Lean will throw an error immediately during this call. This directly eliminates the danger of unstated assumptions.\n\n\nRefining Variable Closure\n\nI'm revisiting the use of `A0` as a `let` binding. While the current structure works, I want to confirm that `A0`'s definition is fully expanded. To confirm, I'll rewrite the code to explicitly inline the definition of `A0` within the existential constructor and the cardinality proof to eliminate any potential issues. This will help confirm that no \"delayed\" evaluation is interfering with the result. I also want to make sure I'm following common conventions.\n\n\nEliminating In-line Definitions\n\nI'm verifying that avoiding `let` and directly inlining `A0`'s definition within the existential constructor and cardinality proof is the safest approach. This avoids potential issues from deferred evaluation. I've re-examined previous tests, and they confirm that such inlining works perfectly, providing a completely explicit and transparent structure.\n\n\nValidating Code Execution\n\nI'm now focusing on validating the execution flow of branch 2 with detailed type analysis. The detailed analysis confirmed the validity of each step, and particularly `Finset.card_filter_le`'s application. I was sure to carefully track the variables, including how `hb_eq` is used with the filter and the application of `Finset.card_filter_le`. Everything now aligns perfectly.\n\n\nConfirming Variable Replacement\n\nI'm verifying the variable replacement strategy is sound. Specifically, I'm confirming that replacing `(A \u00d7\u02e2 A).card` with `n ^ 2` in `h1` results in `b \u2264 n ^ 2`. The rewrite strategy `rw [h2] at h1` achieves this by substituting the right-hand side of `h2` into `h1`. This is valid, and the goal is complete.\n\n\nValidating Product Replacement\n\nI'm now completely assured. The application of `rw` with `Finset.card_product` and `hA` functions as expected. I've re-examined the process, confirming that `Finset.card_product` correctly transforms `(A \u00d7\u02e2 A).card` to `A.card * A.card`. Then, `r\n...<truncated>", "reasoning_details": [ { "format": "google-gemini-v1", "index": 0, "type": "reasoning.text", "text": "Leveraging `Finset.card_filter_le`\n\nI'm incredibly pleased with the discovery of `Finset.card_filter_le`! Its signature, `Finset.card_filter_le s p : (filter p s).card \u2264 s.card`, is perfect. My plan is to apply it directly to `A \u00d7\u02e2 A` using a predicate function that incorporates the `x \u2260 y` condition. This approach sidesteps any potential unification issues by providing an explicit definition.\n\n\nRefining Omega Application\n\nI've hit a snag. While `Finset.card_filter_le` seems promising, directly applying `omega` to `Finset.card_filter_le (A \u00d7\u02e2 A) _` might be trickier than anticipated because of potential type mismatches or definitional inequality. My current thinking is centered around how to best leverage `omega` with the upper bound provided by the filter cardinality lemma. I'm exploring tactics to bridge this gap, possibly through strategic rewriting or introducing intermediate terms.\n\n\nApplying Direct Approach\n\nI've made a breakthrough! My thought process has shifted from abstract tactics towards a direct application. I realized that the goal's structure perfectly aligns with `Finset.card_filter_le`. Using the theorem directly on `b`, as it's defined using the exact left-hand side from the goal, gives an explicit and direct upper bound! By simply utilizing `Finset.card_filter_le` on `b`, I can circumvent introducing potentially cumbersome intermediate terms. This strategy significantly streamlines the proof, bypassing the complexity I was anticipating.\n\n\nOptimizing Direct Application\n\nI've refined the direct application strategy. Instead of immediately substituting `b`, I'm keeping it in place, using `hb_eq` only for the `Finset.card_filter_le` application. This sidesteps writing the large filter directly, streamlining the process significantly. Furthermore, I've confirmed `omega` should handle the resulting inequalities and equalities. If it struggles with polynomial arithmetic, I'll switch to using `calc` for a more explicit chain of equalities, ensuring `omega` won't be blocked.\n\n\nImplementing `calc` Branch\n\nI've completed the adaptation of `calc` into the second branch! Using it is robust and clean, avoiding any potentially tricky substitutions on the target, and directly reaching the desired inequality. The first branch is next, so I'll create an `Embedding`, which should be helpful.\n\n\nConfirming Explicit Closure\n\nI'm incredibly happy! I've confirmed the initial branch's explicit closure without resorting to `use`'s implicit magic. Introducing `A0` and `k0`, along with their explicit instantiation and cardinality argument, directly satisfies the existential goal's conditions. This strategic explicit structuring, in place of the more implicit structure, guarantees a watertight closing without leaving any sub-goals. I've built in `hA` explicitly, with `by` doing the job! This meticulous handling ensures the branch is fully closed.\n\n\nClarifying Constructor Application\n\nI've clarified the implications of `use A0`. I initially thought I needed `exact \u27e8hA, rfl\u27e9`, but that caused a \"No goals to be solved\" error. I realized `use A0` completely closes the goal, resolving the implicit `k0` definition and matching `hA`. The underlying reason is that `k0` is definitionally `f A0`, and `hA` matches `A0.card = n`.\n\n\n\nVerifying Goal Construction\n\nI'm confident! I've confirmed `exact \u27e8k, A, hA, rfl\u27e9` is the correct explicit structure for constructing the existential goal, where the constructor implicitly expects one term of type `\u2203 k, ...` which is `\u27e8k, \u27e8A, \u27e8hA, rfl\u27e9\u27e9\u27e9`. I've constructed a full example. The `filter` syntax `fun (x, y) \u21a6 ...` is perfectly valid. The `\u21a6` symbol (U+21A6) does indeed map the arguments to the expression! This syntactic confirmation is critical for the proof's core structure.\n\n\nValidating Full Structure\n\nI've just validated the complete explicit structure's construction, confirming it's robust and complete! Using `fun (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A0` within the filter expression works seamlessly, and lean accepts both `=>` and `\u21a6`. `A0` is correctly computed using `Finset.map`, and `hA` proves the needed cardinality relation. Now, I'm verifying the entire constructed explicit example is correct. Is there any defect?\n\n\nVerifying Final Construction\n\nI've validated the final construction, and I'm very happy! I've confirmed that the core structure, with the `filter` and associated cardinality arguments, is fully sound. The `exact \u27e8k, A0, hA, rfl\u27e9` instantiates everything perfectly: `k`, `A0`, the cardinality, and the crucial equality using `rfl`. Furthermore, `csSup_le` works. The existing code uses `csSup` on a bounded set, guaranteeing the conditions are met. I'm very confident now.\n\n\nConfirming Lemma Implications\n\nI've made a crucial observation. Applying `csSup_le` only produced two subgoals, indicating that `BddAbove s` wasn't explicitly required. I realized that the hypothesis `h : \u2200 x \u2208 s, x \u2264 a` implicitly proves `BddAbove s` because `a` acts as the upper bound. In Lean 4, this implicit inference is crucial. This explains why the application of `csSup_le` worked without a third explicit subgoal.\n\n\nVerifying Explicit Bound\n\nI'm now certain about the implicit boundedness. The specific `csSup_le` used, as verified by `#check`, did not require an explicit `BddAbove` hypothesis. That would have resulted in a subgoal. The key is the assumption `h : \u2200 x \u2208 s, x \u2264 a`, which implicitly bounds the set. This clarifies why the goal structure was as expected. I've rechecked to avoid confusion!\n\n\nConfirming Correct Closure\n\nI'm now completely satisfied with the `exact` statement's behavior! My exploration shows that this single, explicit `exact` always produces an immediate closure, as it's directly type-checked against the existential structure. There won't be any lingering or hidden subgoals. If a type mismatch occurs, or any of the predicates are not satisfied, Lean will throw an error immediately during this call. This directly eliminates the danger of unstated assumptions.\n\n\nRefining Variable Closure\n\nI'm revisiting the use of `A0` as a `let` binding. While the current structure works, I want to confirm that `A0`'s definition is fully expanded. To confirm, I'll rewrite the code to explicitly inline the definition of `A0` within the existential constructor and the cardinality proof to eliminate any potential issues. This will help confirm that no \"delayed\" evaluation is interfering with the result. I also want to make sure I'm following common conventions.\n\n\nEliminating In-line Definitions\n\nI'm verifying that avoiding `let` and directly inlining `A0`'s definition within the existential constructor and cardinality proof is the safest approach. This avoids potential issues from deferred evaluation. I've re-examined previous tests, and they confirm that such inlining works perfectly, providing a completely explicit and transparent structure.\n\n\nValidating Code Execution\n\nI'm now focusing on validating the execution flow of branch 2 with detailed type analysis. The detailed analysis confirmed the validity of each step, and particularly `Finset.card_filter_le`'s application. I was sure to carefully track the variables, including how `hb_eq` is used with the filter and the application of `Finset.card_filter_le`. Everything now aligns perfectly.\n\n\nConfirming Variable Replacement\n\nI'm verifying the variable replacement strategy is sound. Specifically, I'm confirming that replacing `(A \u00d7\u02e2 A).card` with `n ^ 2` in `h1` results in `b \u2264 n ^ 2`. The rewrite strategy `rw [h2] at h1` achieves this by substituting the right-hand side of `h2` into `h1`. This is valid, and the goal is complete.\n\n\nValidating Product Replacement\n\nI'm now completely assured. The application of `rw` with `Finset.card_product` and `hA` functions as expected. I've re-examined the process, confirming that `Finset.card_product` correctly transforms `(A \u00d7\u02e2 A).card` to `A.card * A.card`. Then, `rw [hA]` precisely substitutes all instances of `A.card` with `n`, simplifying the goal to a trivially true equality via `rfl`. This ensures the equality rewriting, a crucial step, is robust.\n\n\nConfirming Final Checks\n\nI'm now incredibly confident. I've re-examined the core logic and verified that `rw [hb_eq]` with `Finset.card_filter_le` closes the goal directly. This streamlined approach eliminates any need for `omega` or polynomial arithmetic, significantly simplifying the proof. The goal's explicit structure, with all variables scoped and fully defined, guarantees a robust and reliable result. I've also double-checked for potential edge cases, including `n = 0`, confirming that the code functions correctly under all conditions. Further, `csSup_le` works. Now, no subgoals should exist, since I explicitly set all goals.\n\n\nConfirming Definitively Flawless Approach\n\nI'm incredibly pleased. The \"unsolved goals\" warning definitively pinpoints `exact`'s failure due to definitional inequality, specifically the variable mismatch. The explicit extraction of the filter from `hb_eq` using `rw` guarantees 100% definitional equality. I've completely bypassed explicit filter definition. This now ensures flawless unification! No more `exact` unification problems or potential hidden sub-goals!\n\n\nVerifying Implicit Variables\n\nI'm now checking if `Finset.card_filter_le` introduces any implicit variables that could cause unification problems. It requires a `Finset \u03b1` and a predicate `\u03b1 \u2192 Prop`. I'm confident that the explicit construction `(A \u00d7\u02e2 A).filter (fun (x, y) => ...)` addresses this. The explicit pairing in the filter ensures no issues. The key is that `A \u00d7\u02e2 A` provides the `Finset \u03b1`, and the function matches. The problem is now completely eliminated.\n\n\nRefining Explicit Placement\n\nI'm now zeroing in on the cleanest way to express the application of `Finset.card_filter_le`. My aim is to be completely explicit and eliminate any potential type inference issues. My latest idea is to use `exact Finset.card_filter_le (A \u00d7\u02e2 A) _`. This will tell the Lean to infer the `p` argument from the goal itself. My goal is to maximize the level of explicitness, so the code is clear.\n\n\nFinalizing Application Strategy\n\nI'm now certain about my final approach, the explicit instantiation strategy using `exact`. I've analyzed the \"unsolved goal\" error. The explicit instantiation is key; no implicit variable inference or subtle type mismatches will arise because all dependencies are fully defined. This approach guarantees full definitional equality, as I've explicitly set every variable's type. This eliminates every source of error. I've completely bypassed explicit filter definition. The resulting code will be incredibly robust.\n\n\n" }, { "id": "tool_compile_proof_M2sW0pWHppPy2V5kM2m3", "format": "google-gemini-v1", "index": 0, "type": "reasoning.encrypted", "data": 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" } ] }, { "iteration": 10, "event": "compile_proof", "provider": "openrouter_primary", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "rw [max013AffineTranslates]\napply csSup_le\n\u00b7 have h_inj : Function.Injective (fun (x : \u2115) => (x : \u2124)) := by\n intro a b hab\n exact Nat.cast_inj.mp hab\n have E : Function.Embedding \u2115 \u2124 := \u27e8fun x => x, h_inj\u27e9\n have hA : (Finset.map E (Finset.range n)).card = n := by\n rw [Finset.card_map, Finset.card_range]\n exact \u27e8((Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n)).filter (fun (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n))).card, Finset.map E (Finset.range n), hA, rfl\u27e9\n\u00b7 intro b hb\n rcases hb with \u27e8A, hA, hb_eq\u27e9\n have h1 : b \u2264 (A \u00d7\u02e2 A).card := by\n rw [hb_eq]\n exact Finset.card_filter_le _ _\n have h2 : (A \u00d7\u02e2 A).card = n * n := by\n rw [Finset.card_product, hA]\n have h3 : n * n = n ^ 2 := by ring\n rw [h3] at h2\n rw [h2] at h1\n exact h1", "compiler_output_preview": "Comparator rejected this proof. The proof compiles but does not actually prove the stated theorem. This can happen when referencing an unrelated theorem or using tricks like #exit to suppress errors.\n\nBuilding Challenge\n\u26a0 [7976/7977] Replayed Challenge\nwarning: Challenge.lean:34:8: declaration uses 'sorry'\nBuild completed successfully (7977 jobs).\nExporting #[Green24.variants.upper_trivial, propext, Quot.sound, Classical.choice, Nat.add, Nat.sub, Nat.mul, Nat.pow, Nat.gcd, Nat.div, Nat.mod, Nat.\n...<truncated>", "compile_elapsed_s": 55.991, "tactic_replay": { "tactic_count": 20, "successful_tactics": 20, "first_failure_step": null, "cps": 100.0, "cps_peak_step": 20, "goal_trajectory": [ 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], "max_goals": 2, "final_goal_count": 1, "goals_opened": 1, "goals_closed": 1 }, "lsp_feedback": { "diagnostics": [ { "fullRange": { "end": { "character": 6, "line": 56 }, "start": { "character": 0, "line": 56 } }, "message": "'Green24.variants.upper_trivial' depends on axioms: [propext, Classical.choice, Quot.sound]", "range": { "end": { "character": 6, "line": 56 }, "start": { "character": 0, "line": 56 } }, "severity": 3, "source": "Lean 4" } ], "error_line": null, "error_char": null, "error_msg": null, "goal_before_error": null, "full_timeline": [ { "line": 35, "code": "rw [max013AffineTranslates]", "goal": "```lean\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 52 }, { "line": 36, "code": "apply csSup_le", "goal": "```lean\nn : \u2115\n\u22a2 sSup\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 37, "code": "\u00b7 have h_inj : Function.Injective (fun (x : \u2115) => (x : \u2124)) := by", "goal": "```lean\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```\n---\n```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 2, "ast_complexity": 449 }, { "line": 38, "code": "intro a b hab", "goal": "```lean\nn : \u2115\n\u22a2 Function.Injective fun x \u21a6 \u2191x\n```", "goal_count": 1, "ast_complexity": 49 }, { "line": 39, "code": "exact Nat.cast_inj.mp hab", "goal": "```lean\nn a b : \u2115\nhab : (fun x \u21a6 \u2191x) a = (fun x \u21a6 \u2191x) b\n\u22a2 a = b\n```", "goal_count": 1, "ast_complexity": 67 }, { "line": 40, "code": "have E : Function.Embedding \u2115 \u2124 := \u27e8fun x => x, h_inj\u27e9", "goal": "```lean\ncase h\u2081\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 243 }, { "line": 41, "code": "have hA : (Finset.map E (Finset.range n)).card = n := by", "goal": "```lean\ncase h\u2081\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 253 }, { "line": 42, "code": "rw [Finset.card_map, Finset.card_range]", "goal": "```lean\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\n\u22a2 (Finset.map E (Finset.range n)).card = n\n```", "goal_count": 1, "ast_complexity": 108 }, { "line": 43, "code": "exact \u27e8((Finset.map E (Finset.range n) \u00d7\u02e2 Finset.map E (Finset.range n)).filter (fun (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 Finset.map E (Finset.range n))).card, Finset.map E (Finset.range n), hA, rfl\u27e9", "goal": "```lean\ncase h\u2081\nn : \u2115\nh_inj : Function.Injective fun x \u21a6 \u2191x\nE : \u2115 \u21aa \u2124\nhA : (Finset.map E (Finset.range n)).card = n\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n```", "goal_count": 1, "ast_complexity": 299 }, { "line": 44, "code": "\u00b7 intro b hb", "goal": "```lean\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 239 }, { "line": 45, "code": "rcases hb with \u27e8A, hA, hb_eq\u27e9", "goal": "```lean\ncase h\u2082\nn b : \u2115\nhb :\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 223 }, { "line": 46, "code": "have h1 : b \u2264 (A \u00d7\u02e2 A).card := by", "goal": "```lean\ncase h\u2082\nn b : \u2115\nA : Finset \u2124\nhA : A.card = n\nhb_eq :\n b =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 175 }, { "line": 47, "code": "rw [hb_eq]", "goal": "```lean\nn b : \u2115\nA : Finset \u2124\nhA : A.card = n\nhb_eq :\n b =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\n\u22a2 b \u2264 (A \u00d7\u02e2 A).card\n```", "goal_count": 1, "ast_complexity": 175 }, { "line": 48, "code": "exact Finset.card_filter_le _ _", "goal": "```lean\nn b : \u2115\nA : Finset \u2124\nhA : A.card = n\nhb_eq :\n b =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\n\u22a2 {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card \u2264\n (A \u00d7\u02e2 A).card\n```", "goal_count": 1, "ast_complexity": 266 }, { "line": 49, "code": "have h2 : (A \u00d7\u02e2 A).card = n * n := by", "goal": "```lean\ncase h\u2082\nn b : \u2115\nA : Finset \u2124\nhA : A.card = n\nhb_eq :\n b =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\nh1 : b \u2264 (A \u00d7\u02e2 A).card\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 198 }, { "line": 50, "code": "rw [Finset.card_product, hA]", "goal": "```lean\nn b : \u2115\nA : Finset \u2124\nhA : A.card = n\nhb_eq :\n b =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\nh1 : b \u2264 (A \u00d7\u02e2 A).card\n\u22a2 (A \u00d7\u02e2 A).card = n * n\n```", "goal_count": 1, "ast_complexity": 202 }, { "line": 51, "code": "have h3 : n * n = n ^ 2 := by ring", "goal": "```lean\ncase h\u2082\nn b : \u2115\nA : Finset \u2124\nhA : A.card = n\nhb_eq :\n b =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\nh1 : b \u2264 (A \u00d7\u02e2 A).card\nh2 : (A \u00d7\u02e2 A).card = n * n\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 225 }, { "line": 52, "code": "rw [h3] at h2", "goal": "```lean\ncase h\u2082\nn b : \u2115\nA : Finset \u2124\nhA : A.card = n\nhb_eq :\n b =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\nh1 : b \u2264 (A \u00d7\u02e2 A).card\nh2 : (A \u00d7\u02e2 A).card = n * n\nh3 : n * n = n ^ 2\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 244 }, { "line": 53, "code": "rw [h2] at h1", "goal": "```lean\ncase h\u2082\nn b : \u2115\nA : Finset \u2124\nhA : A.card = n\nhb_eq :\n b =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\nh1 : b \u2264 (A \u00d7\u02e2 A).card\nh2 : (A \u00d7\u02e2 A).card = n ^ 2\nh3 : n * n = n ^ 2\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 244 }, { "line": 54, "code": "exact h1", "goal": "```lean\ncase h\u2082\nn b : \u2115\nA : Finset \u2124\nhA : A.card = n\nhb_eq :\n b =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card\nh1 : b \u2264 n ^ 2\nh2 : (A \u00d7\u02e2 A).card = n ^ 2\nh3 : n * n = n ^ 2\n\u22a2 b \u2264 n ^ 2\n```", "goal_count": 1, "ast_complexity": 236 } ], "tactic_count": 20, "successful_tactics": 20, "first_failure_step": null, "cps": 100.0, "cps_peak_step": 20 } } ], "verification": { "banned_tokens": [], "command": [ "[path] "lean", "/app/asi_portal/verifier/quick_1771903718_1296.lean" ], "comparator_passed": null, "duration_s": 12.650257110595703, "goals_remaining": null, "is_verified": false, "lean_file_path": "", "problem": { "axioms_check_name": "upper_trivial", "axioms_check_skipped": false, "benchmark_id": "GreensOpenProblems__24__upper_trivial", "category": "research_open", "endings": [ "end variants", "end Green24" ], "imports": [ "FormalConjectures.Util.ProblemImports" ], "preamble_chars": 950, "source_file": "FormalConjectures/GreensOpenProblems/24.lean", "title": "upper_trivial" }, "returncode": 1, "status": "failed", "stderr": "", "stdout": "/app/asi_portal/verifier/quick_1771903718_1296.lean:38:3: error: unexpected token '?'; expected '{' or tactic\n/app/asi_portal/verifier/quick_1771903718_1296.lean:38:2: error: unsolved goals\ncase h\u2081\nn : \u2115\n\u22a2 {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card}.Nonempty\n/app/asi_portal/verifier/quick_1771903718_1296.lean:35:68: error: unsolved goals\ncase h\u2082\nn : \u2115\n\u22a2 \u2200\n b \u2208\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card},\n b \u2264 n ^ 2\n'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n", "timed_out": false } }
Gemini 3 Flash Agent v8	Verified	0	Feb 17, 2026	unfold max013AffineTranslates apply csSup_le · obtain ⟨A, hA⟩ : ∃ A : Finset ℤ, A.card = n := ⟨(Finset.range n).image Int.ofNat, by rw [Finset.card_image_of_injective _ Int.ofNat_injective, Finset.card_range]⟩ exact ⟨_, ⟨A, hA, rfl⟩⟩ · rintro k ⟨A, hA, rfl⟩ apply le_trans (Finset.card_filter_le _ _) rw [Finset.card_product, hA, pow_two]	View { "agent_telemetry": { "compilations": 3, "iterations": 3, "duration_s": 178.46, "best_status": "verified", "best_goals": 0, "max_iterations": 8, "model": "gemini-3-flash-preview", "error": null, "provider": "gemini_direct", "finish_reason": "STOP", "usage": { "prompt_tokens": 48895, "completion_tokens": 345, "total_tokens": 74988, "reasoning_tokens": 25748, "cost_usd": 0.0 }, "reasoning_effort": "", "response_trace_count": 6 }, "response_trace": [ { "iteration": 1, "provider": "gemini_direct", "finish_reason": "STOP", "api_elapsed_s": 108.306, "usage": { "prompt_tokens": 2025, "completion_tokens": 93, "total_tokens": 19867, "reasoning_tokens": 17749, "cost_usd": 0.0 }, "thoughts_tokens": 17749, "gemini_retries": 0, "has_function_call": true, "response_text_len": 0, "model_text_preview": "" }, { "iteration": 1, "event": "compile_proof", "provider": "gemini_direct", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "unfold max013AffineTranslates\napply Nat.sSup_le\nintro k hk\nrcases hk with \u27e8A, hA, rfl\u27e9\napply le_trans (Finset.card_filter_le _ _)\nrw [Finset.card_product, hA, pow_two]\nexact le_refl _", "compiler_output_preview": "[path]:37:8: error: unknown constant 'Nat.sSup_le'\n[path]:38:2: error: no goals to be solved\n'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n" }, { "iteration": 2, "provider": "gemini_direct", "finish_reason": "STOP", "api_elapsed_s": 38.958, "usage": { "prompt_tokens": 20071, "completion_tokens": 80, "total_tokens": 26388, "reasoning_tokens": 6237, "cost_usd": 0.0 }, "thoughts_tokens": 6237, "gemini_retries": 0, "has_function_call": true, "response_text_len": 0, "model_text_preview": "" }, { "iteration": 2, "event": "compile_proof", "provider": "gemini_direct", "status": "failed", "goals_remaining": null, "timed_out": false, "banned_tokens": [], "proof_code": "unfold max013AffineTranslates\napply sSup_le\nrintro k \u27e8A, hA, rfl\u27e9\napply le_trans (Finset.card_filter_le _ _)\nrw [Finset.card_product, hA, pow_two]", "compiler_output_preview": "[path]:37:2: error: tactic 'apply' failed, could not unify the conclusion of `@sSup_le`\n @sSup ?\u03b1 CompleteSemilatticeSup.toSupSet ?s \u2264 ?a\nwith the goal\n @sSup \u2115 Nat.instSupSet\n {k \|\n \u2203 A,\n A.card = n \u2227\n k =\n {x \u2208 A \u00d7\u02e2 A \|\n match x with\n \| (x, y) => x \u2260 y \u2227 x + 3 * (y - x) \u2208 A}.card} \u2264\n n ^ 2\n\nNote: The fu\n...<truncated>" }, { "iteration": 3, "provider": "gemini_direct", "finish_reason": "STOP", "api_elapsed_s": 12.869, "usage": { "prompt_tokens": 26799, "completion_tokens": 172, "total_tokens": 28733, "reasoning_tokens": 1762, "cost_usd": 0.0 }, "thoughts_tokens": 1762, "gemini_retries": 0, "has_function_call": true, "response_text_len": 0, "model_text_preview": "" }, { "iteration": 3, "event": "compile_proof", "provider": "gemini_direct", "status": "verified", "goals_remaining": 0, "timed_out": false, "banned_tokens": [], "proof_code": "unfold max013AffineTranslates\napply csSup_le\n\u00b7 obtain \u27e8A, hA\u27e9 : \u2203 A : Finset \u2124, A.card = n := \u27e8(Finset.range n).image Int.ofNat, by rw [Finset.card_image_of_injective _ Int.ofNat_injective, Finset.card_range]\u27e9\n exact \u27e8_, \u27e8A, hA, rfl\u27e9\u27e9\n\u00b7 rintro k \u27e8A, hA, rfl\u27e9\n apply le_trans (Finset.card_filter_le _ _)\n rw [Finset.card_product, hA, pow_two]", "compiler_output_preview": "'Green24.variants.upper_trivial' depends on axioms: [propext, Classical.choice, Quot.sound]\n\n" } ], "verification": { "banned_tokens": [], "command": [ "[path] "lean", "[path] ], "duration_s": 6.236074686050415, "goals_remaining": 0, "is_verified": true, "lean_file_path": "[path] "problem": { "axioms_check_name": "upper_trivial", "axioms_check_skipped": false, "benchmark_id": "GreensOpenProblems__24__upper_trivial", "category": "research_open", "endings": [ "end variants", "end Green24" ], "imports": [ "FormalConjectures.Util.ProblemImports" ], "preamble_chars": 950, "source_file": "FormalConjectures/GreensOpenProblems/24.lean", "title": "upper_trivial" }, "returncode": 0, "status": "verified", "stderr": "", "stdout": "'Green24.variants.upper_trivial' depends on axioms: [propext, Classical.choice, Quot.sound]\n", "timed_out": false } }
Gemini 3 Flash Agent v7	Failed	--	Feb 16, 2026	unfold max013AffineTranslates apply Nat.sSup_le intros k hk obtain ⟨A, hA_card, rfl⟩ := hk apply (Finset.card_filter_le (A ×ˢ A) _).trans rw [Finset.card_product, hA_card, pow_two]	View { "agent_telemetry": { "compilations": 0, "iterations": 1, "duration_s": 45.87, "best_status": "pending", "best_goals": null, "max_iterations": 8, "model": "gemini-3-flash-preview", "error": null, "provider": "gemini_direct", "finish_reason": "MAX_TOKENS", "usage": { "prompt_tokens": 2025, "completion_tokens": 326, "total_tokens": 10213, "reasoning_tokens": 7862, "cost_usd": 0.0 }, "reasoning_effort": "", "response_trace_count": 1 }, "response_trace": [ { "iteration": 1, "provider": "gemini_direct", "finish_reason": "MAX_TOKENS", "api_elapsed_s": 45.873, "usage": { "prompt_tokens": 2025, "completion_tokens": 326, "total_tokens": 10213, "reasoning_tokens": 7862, "cost_usd": 0.0 }, "has_function_call": false, "response_text_len": 799 } ], "verification": { "banned_tokens": [], "command": [ "[path] "lean", "[path] ], "duration_s": 6.739211559295654, "goals_remaining": null, "is_verified": false, "lean_file_path": "[path] "problem": { "axioms_check_name": "upper_trivial", "axioms_check_skipped": false, "benchmark_id": "GreensOpenProblems__24__upper_trivial", "category": "research_open", "endings": [ "end variants", "end Green24" ], "imports": [ "FormalConjectures.Util.ProblemImports" ], "preamble_chars": 950, "source_file": "FormalConjectures/GreensOpenProblems/24.lean", "title": "upper_trivial" }, "returncode": 1, "status": "failed", "stderr": "", "stdout": "[path]:37:8: error: unknown constant 'Nat.sSup_le'\n[path]:38:2: error: no goals to be solved\n'Green24.variants.upper_trivial' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n", "timed_out": false } }
Gemini 3 Flash Preview	Failed	--	Feb 14, 2026	apply Finset.filter_subset have h_card_le := Finset.card_le_card h_subset rw [Finset.card_product, hA_card] at h_card_le exact h_card_le	View { "generation": { "attempts": 1, "banned_tokens_found": [], "elapsed_s": 3.99, "finish_reason": "stop", "max_tokens": 8192, "model_id": "google/gemini-3-flash-preview", "proof_code_length": 142, "raw_content_length": 1194, "reasoning_effort": "", "structured_output": true, "temperature": 0.0, "usage": { "completion_tokens": 450, "completion_tokens_details": { "image_tokens": 0, "reasoning_tokens": 0 }, "cost": 0.0022815, "cost_details": { "upstream_inference_completions_cost": 0.00135, "upstream_inference_cost": 0.0022815, "upstream_inference_prompt_cost": 0.0009315 }, "is_byok": false, "prompt_tokens": 1863, "prompt_tokens_details": { "audio_tokens": 0, "cache_write_tokens": 0, "cached_tokens": 0, "video_tokens": 0 }, "total_tokens": 2313 } }, "verification": { "banned_tokens": [], "command": [ "[path] "lean", "[path] ], "duration_s": 7.749795436859131, "goals_remaining": null, "is_verified": false, "lean_file_path": "[path] "problem": { "benchmark_id": "GreensOpenProblems__24__upper_trivial", "category": "research_open", "endings": [ "end variants", "end Green24" ], "imports": [ "FormalConjectures.Util.ProblemImports" ], "preamble_chars": 950, "source_file": "FormalConjectures/GreensOpenProblems/24.lean", "title": "upper_trivial" }, "returncode": 1, "status": "failed", "stderr": "", "stdout": "[path]:36:2: error: tactic 'apply' failed, could not unify the conclusion of `@Finset.filter_subset`\n ?a \u2208 ?s\nwith the goal\n max013AffineTranslates n \u2264 n ^ 2\n\nNote: The full type of `@Finset.filter_subset` is\n \u2200 {\u03b1 : Type ?u.1407} (p : \u03b1 \u2192 Prop) [inst : DecidablePred p] (s : Finset \u03b1), Finset.filter p s \u2286 s\nn : \u2115\n\u22a2 max013AffineTranslates n \u2264 n ^ 2\n", "timed_out": false } }

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