Back to Problems
KotherConjecture.variants.le_KotherRadical
Specification
The **Köthe conjecture**: every left nil radical is contained in the Köthe radical.
Lean 4 Statement
theorem KotherConjecture.variants.le_KotherRadical {I : Ideal R} (hI : IsNil I) :
(I : Set R) ⊆ KotheRadical R
Browse
All Problems
Explore all 300 unsolved conjectures.
View problems →
Docs
Verification Pipeline
How zero-trust verification works.
Read docs →
Evaluation Results
Recent Submissions
| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Partial | 1 | Feb 25, 2026 |
intro x hx
simp [KotheRadical]
|
View{
"agent_telemetry": {
"compilations": 3,
"iterations": 10,
"duration_s": 310.36,
"best_status": "partial",
"best_goals": 1,
"max_iterations": 10,
"model": "openai/gpt-5.3-codex",
"error": null,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"usage": {
"prompt_tokens": 73087,
"completion_tokens": 8306,
"total_tokens": 81393,
"reasoning_tokens": 7981,
"cost_usd": 0.16072385
},
"reasoning_effort": "high",
"response_trace_count": 13
},
"response_trace": [
{
"iteration": 1,
"event": "search_mathlib",
"query": "Ideal.toTwoSided",
"results_count": 1
},
{
"iteration": 2,
"event": "search_mathlib",
"query": "le_sSup",
"results_count": 1
},
{
"iteration": 3,
"event": "search_mathlib",
"query": "mem_toTwoSided",
"results_count": 1
},
{
"iteration": 4,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772022297-meIxNNHSHJrW0PNCy98Q",
"api_elapsed_s": 10.545,
"usage": {
"prompt_tokens": 2570,
"completion_tokens": 519,
"total_tokens": 3089,
"reasoning_tokens": 425,
"cost_usd": 0.0081347
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Structuring proof skeleton with holes**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Structuring proof skeleton with holes**"
},
{
"id": "rs_02d87ccecc950c2201699eea19a1c08194809daaf5ede3b286",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "gAAAAABpnuoi8pur5hqR65_r7nLabUbOFhbA3oOBnFbwA-cn1qB8NIGs97EUPZW8N3eo_OtQYnKfwgaPWymdfPVAw2n20oIfp3GDFBSF0VAoBJmFX21I4IhmnUQCminujeKy8G6SiJszFxzDf60tYjlv-hb-k1LaNST4jlbJzoYtZI4WeCwDBy8GYpNXDT3UMS-8ZcqDPmfgr8AYzHu3tcbnlJYwgJd2x8t4GMGAAZh5KaXH7qzJbr7WuwOll7BYiedqBWqlW-iiH-tq7oFS300-3P0NwA98Yy9HqYIfBEDw7B98qHElZGC1PZJHDjh2gmAFiavTP3Lxw_fC84uf-dffvo_2qS5u1-ikZx8FvptK1fucloWOd4QN1pQFpjotkEm1QPD_MEzhOAat5VTZn3zc9hZxndB4fq4RTEYJStWKZq84vL2kdXcsJvBAJEP-kGVgGeVvT1ucPv0mA4jZPhksFHpHie6_HsEZ3Eq8GnAdFjJ7dGcu72tG7n8tOU9E27nObmggFq1773wHswECKXXME2VsenMOPY_99it9umx0f3IYn81c8D6sF_zaktaeeWzQD-UK4dQ6oqSIM1ABu57iGpyK7g7Pnv_HlcjkDroR_kZS97aUtfabzMzt1lzmbHPr8I_FZw0gwzxpGiT9PuJUd2HMV81n1kJz_gKDV9xwdzngiAHKnantA5yaqiz9WBW9ds33nemQTcgGpVohCbW7V-uXeqvf5iTg_fjYJDN2lUhXZ7M7NYxny4pvSCyD_VBBByYR4HEASK8sXu5Gn7G5jLEmTSbrVD67GXwxfHDwMnA5k73KFUvS0oXm22kZ14zQEDe74S6wGEOav8VU1TdTGKalHN0mvFNZHuv0wKbGlz2nR8GiQQBvdd_LfjYOx_LVZ_5tgKkmBpiNqk1Crw3Xmo0SspGWnZX1IQeYWu3pbcV_kag1GQ_s2nHv_BhBJvyHvysatotRYAAJxdmnSfrbH7bc9BYKIhMnR831EhAb7fRIrdPsqFsZje5GlcJTj1YoF53m2lJtKnuBK8EiJcfgrnpcGTzWWypOYpilQwN6n5DCY489Cim7HtJXOk21ZIRl1jDm6nX75jfuV404kora6r5S9bV78OuM-tPt0RFXzu8f5DFK8yGz7_juvoHv9UbyRrdtV82LpQtqxWDmsb72sumMjJG9mxRPAuXciBgowI9iPbvHAczdYPlgCT-ud6u9ZF06z96at6jMrlDjLC4-9n8UW6i-sbAGXsCQARcozyKf41zIBTNNizZuTf__Un6GKZVN_GjYx21u4e4aHZKf3QavaixSwj3OAwfvIljBKEOEACsx3bdZ5KfwIBgddNAVooD4suhTHGQEBDr5zLinXpStnQYnWiFGI5JI7bvmnP8yEyZF2-pQUJDzB13roGtXW_qY6z5oFx0sZsa0z6HUMzYK6B2iyoaaTHWeXQoErOY_WjsMexiYvBZTwfQUOkRsLWgnCsSr2A8aCfKhQDQZOkbt1__6i13VZNuSbEdhRaYsemABwrXX_rb-l-da5eG_OGX9qxXGjOfkKWq_a5a1rwELXM7P67uzaJyErbEyQPN6ePEb_hWqHh8b4H9-GXvLays4gIxOtZdk81pdRyYrb5IWw3J1QcEdLaZ73QXBGt-HCix099dg7n5Ghs2KqI0BEZeKXVR9Z2s1EpYIKZuRR11_3La6HVE2x2xTabY1D4TjhsHfN6o9cVA0bw_nbL08IrUO4Esy35kYi3guLzjnRuE9yH8VjlhWjArImlFvFur6I8BkcoMd8caDxGJNrVxypBNEGbGfcGAyOiZJn6K13RZ0LaiCxy-sCEvo94Gl-BJoJ6nBDDvd9R67FIxaS8dqe5vn7T9zsIyqjnwuWtU5O1Oh5jidkQvAihRAWXxH3bmycLHgbz-oTtpZMenb7EUPUnqjOWkrX1yyGX7HZZmAdQmnBOCi1eY-aDQhJE6smOcQIblRPhv74PwJbePJO0zeDAhKhn54yeI3PiED9EYHOtgrmmCyH3cmHFMtADdeUg1XNT-hjNVuVxta5xWT3RTXWVG8ytMWW60tyJjuvNFFcWiy4uo6eOmpr9op5X3IzHlGnDxhvYH_kuqOPvtc2Ci_FsnSXPRsKczpP6Pr49bD9cedf7dBqxsB633fKhZFRkLhTADiWKtO2c4vwS5qlbkllDfbiQjkY9qG1fW426WLEbJxRne_kIR4XmFsVastuEPipNwEmF7OwoNgfpPDenve-0-2_oEKOwybj_JzrFCyq7YOwYwxU3xN8o8hjQAmvVgUOWYSaVSW7bTnDN2q-3a2P95Ob8uhxF65-uvEcD6ufLPB13msE4GyDR6iQ6d8mD-q1ASPw6RPEe59JBhwhZjfVwiWC3xnY9ZkxNGI4PfP8kE9ldbA0MBoeJVRUtyJmcRYSltdAW5Sfa_wBFwylDREscBGGlIxwSHgBF4IKfhNYJtRLmTQC8P9eqXc_p4Mxl3etSMnYO9mZsmHemtPtnISkWSvsVAk1YFNVqOsI4VjFcmkkpq5raP1TgfJrBg3RfuLVp2crQrjzKTjAi84Yq2UVQAY_7KoRuM5Six32i4nEQtJyI-bgxFwK_DEM02tv1PjLX_Lhg6F1WC72uM9LlKNpoPsZB_2R-dwoCZzIXqIWQ1Q88dufif9EoNJDNOMRZF43xxJ5f2Q6gKjStt4fnwRcL9uTvkh9wm9Mz4sFIvTGhT-oaq9O-GU9V3HWp2dhEqxgFp4hq2BIlWjyL7I6neIJzbOBSWRoCjN4qhQ60LBVHQAbql0QPd2CGuQPX96s1k8B3Fkhfg="
}
]
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 2,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro x hx\nhave hI' : IsNil (Ideal.toTwoSided I) := ?_\nhave hle : Ideal.toTwoSided I \u2264 KotheRadical R := by\n exact le_sSup hI'\nhave hx' : x \u2208 Ideal.toTwoSided I := ?_\nexact hle hx'",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772022307_2141.lean:39:20: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n I.IsTwoSided\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.\n/app/asi_portal/verifier/quick_1772022307_2141.lean:40:13: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n I.IsTwoSided\n\nHint: Type class instance resolution failures can be inspected with the \n...<truncated>",
"compile_elapsed_s": 56.085,
"tactic_replay": {
"tactic_count": 6,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1,
2,
1,
2,
3
],
"max_goals": 3,
"final_goal_count": 3,
"goals_opened": 3,
"goals_closed": 1
},
"lsp_feedback": {
"diagnostics": [
{
"code": "lean.synthInstanceFailed",
"fullRange": {
"end": {
"character": 38,
"line": 38
},
"start": {
"character": 20,
"line": 38
}
},
"message": "failed to synthesize instance of type class\n I.IsTwoSided\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.",
"range": {
"end": {
"character": 38,
"line": 38
},
"start": {
"character": 20,
"line": 38
}
},
"severity": 1,
"source": "Lean 4"
},
{
"code": "lean.synthInstanceFailed",
"fullRange": {
"end": {
"character": 31,
"line": 39
},
"start": {
"character": 13,
"line": 39
}
},
"message": "failed to synthesize instance of type class\n I.IsTwoSided\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.",
"range": {
"end": {
"character": 31,
"line": 39
},
"start": {
"character": 13,
"line": 39
}
},
"severity": 1,
"source": "Lean 4"
},
{
"code": "lean.synthInstanceFailed",
"fullRange": {
"end": {
"character": 35,
"line": 41
},
"start": {
"character": 17,
"line": 41
}
},
"message": "failed to synthesize instance of type class\n I.IsTwoSided\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.",
"range": {
"end": {
"character": 35,
"line": 41
},
"start": {
"character": 17,
"line": 41
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 15,
"line": 42
},
"start": {
"character": 36,
"line": 36
}
},
"leanTags": [
1
],
"message": "unsolved goals\ncase refine_2.refine_1\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\nhI' : IsNil sorry\nhle : sorry \u2264 Nil* R\n\u22a2 x \u2208 sorry\n\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 IsNil sorry",
"range": {
"end": {
"character": 0,
"line": 37
},
"start": {
"character": 36,
"line": 36
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 44
},
"start": {
"character": 0,
"line": 44
}
},
"message": "'Koethe.KotherConjecture.variants.le_KotherRadical' 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": 36,
"error_char": 36,
"error_msg": "unsolved goals\ncase refine_2.refine_1\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\nhI' : IsNil sorry\nhle : sorry \u2264 Nil* R\n\u22a2 x \u2208 sorry\n\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 IsNil sorry",
"goal_before_error": null,
"full_timeline": [
{
"line": 37,
"code": "intro x hx",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\n\u22a2 \u2191I \u2286 \u2191(Nil* R)\n```",
"goal_count": 1,
"ast_complexity": 81
},
{
"line": 38,
"code": "have hI' : IsNil (Ideal.toTwoSided I) := ?_",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 x \u2208 \u2191(Nil* R)\n```",
"goal_count": 1,
"ast_complexity": 98
},
{
"line": 39,
"code": "have hle : Ideal.toTwoSided I \u2264 KotheRadical R := by",
"goal": "```lean\ncase refine_2\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\nhI' : IsNil sorry\n\u22a2 x \u2208 \u2191(Nil* R)\n```\n---\n```lean\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 IsNil sorry\n```",
"goal_count": 2,
"ast_complexity": 245
},
{
"line": 40,
"code": "exact le_sSup hI'",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\nhI' : IsNil sorry\n\u22a2 sorry \u2264 Nil* R\n```",
"goal_count": 1,
"ast_complexity": 117
},
{
"line": 41,
"code": "have hx' : x \u2208 Ideal.toTwoSided I := ?_",
"goal": "```lean\ncase refine_2\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\nhI' : IsNil sorry\nhle : sorry \u2264 Nil* R\n\u22a2 x \u2208 \u2191(Nil* R)\n```\n---\n```lean\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 IsNil sorry\n```",
"goal_count": 2,
"ast_complexity": 266
},
{
"line": 42,
"code": "exact hle hx'",
"goal": "```lean\ncase refine_2.refine_2\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\nhI' : IsNil sorry\nhle : sorry \u2264 Nil* R\nhx' : x \u2208 sorry\n\u22a2 x \u2208 \u2191(Nil* R)\n```\n---\n```lean\ncase refine_2.refine_1\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\nhI' : IsNil sorry\nhle : sorry \u2264 Nil* R\n\u22a2 x \u2208 sorry\n```\n---\n```lean\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 IsNil sorry\n```",
"goal_count": 3,
"ast_complexity": 452
}
],
"tactic_count": 6,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "(I : Set R) \u2286 KotheRadical R := by"
}
},
{
"iteration": 5,
"event": "search_mathlib",
"query": "Ideal.twoSided",
"results_count": 8
},
{
"iteration": 6,
"event": "search_mathlib",
"query": "TwoSidedIdeal.span",
"results_count": 1
},
{
"iteration": 7,
"event": "search_mathlib",
"query": "IsNilpotent span",
"results_count": 0
},
{
"iteration": 8,
"event": "search_mathlib",
"query": "Ideal.mul_mem_right",
"results_count": 1
},
{
"iteration": 9,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772022479-aln4EvWUDr4da7TcXaZX",
"api_elapsed_s": 20.065,
"usage": {
"prompt_tokens": 12613,
"completion_tokens": 803,
"total_tokens": 13416,
"reasoning_tokens": 771,
"cost_usd": 0.03170195
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Assessing nil ideal two-sidedness**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Assessing nil ideal two-sidedness**"
},
{
"id": "rs_02d87ccecc950c2201699eead012e48194992a42177c703d91",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "gAAAAABpnurjaxEdzs37yDbAeinz4L7UWlIO_Bkh7mrzD2mz09nPIZ7kteVPXS-lZiofmrT2WlxokhlLNGOtQIJpb_FqpNGR5fVgLBgg7lo-xGhm1tN8r43xAiU64Z2yJZBd_QjL6e2rwSS8txAtBLnD_2xzbXaKc3lnGc30lezRdX3Of4zw7KZgaeIctjRTTl0Y0ObTZov1z-trapOf76CG7tRyYVN78cZzy3CDrPRHKrFvs8WHA3YT0KF9Q7VpK_URP7Z3CPpg6u2s2AL8GbR1Good8PcZQQsosGkyFNm-ey4RP_nhjG3HMpl-66OU3SHJ7m5L86I7Dw4gniu5u9bQQ0Ekw4mSLbaLWICN14kJEWP19qUHCPfcN542VbVO0HxwYyXxPLHPjKqcn-t6_pEHmTXVSoK_0J_9fGrrEQoFROV_vx1M4Ufha4vFTomdxD5ZW_bCVBXhTQFtEieMBk4Oz2Xa5ZuPmhVOCUtec_N4BBZJJ2Ebx7sqiBcedjP7eGKib4aI1YQcw9vc-1TNTGkAoAJn76GC0JtywYrS1nKrsnRkEp9OIPwwu4v4utmgH-wZvqKDtnRzWu_zBid8IaweP0XGXaPT25SfLrb3E9bVgq7YaeVHZsBtfgVgDr2t8q2Rm5GpW50PVVutumClM8DKN1awl6iNEGkt10BGyvRYpxUhINT2nqjOHDoTMWGjWN1nje09lNtrz49kE16wQrD3M_pEhqr8Tn9KAU9uwijnWogLD5jCLRjnmhiuje2aJLBa32cTVLU-bYbqTmy2cQP416B2ukRYpxMoASuSVQ4jFm8M2Dh3iu1WfltDojiVvo9RQYVR7tMJ6_LKEgcWM9ff2f_fQNsJLh8Ngv1hpZBry0O0lO2uHLvFdfBI_sau6bGDUV6jVoGSQ-0ndX0ZQZkoieH4DxbHEQ6lZbHlmeHbB226bIvDhkPGu67r1NJpahZzZfHoOs-NtrGFC6iaug3SgvghSu9GubeKRdDU6ts30n3CLmxTdiPS7gq-44d6wVsEu-lxJfzd8eX3Bjll6Wp68IT7duOraD376ObD00vs3SSlDjN-z__WFlzBB6E2Hwdfu5YM6DJgT3pfLy4k-sj5RQ3PPE87xhladU5jFqqnDI9tr_BIFBlNXlZi3g6DHMbMSnRyAsnImyF-aD0GSP5HISUXam83N5QUFH_iZyKy-ZRZ2wTfmGnua7kpTF7OefPoEu0QcBsNf1rv48Hb34uNFw4N4pdBuCVnKSR9aDbTIwGfXBCeJf5DRQiSVJvWPipN10OnUYogg37XE9E4RYZdqSbcWT1M8TTEKsr-9tjZkSnVp-IU3nhkknKeipcr0razVVg_EGglXfejcyrGTDOPSqA6subWHM_JTpmZuEVIrcpqLaipprG2XLTS5xQ7CSQMEhe3JDZNr_lMnP--oUVqiq0qW2wdA6OU2fwWWkS5Bv0JBe391G4qZx26eHPmhJPw92U-fph_agSxyFlGOug-CmzYxyob_eqvVHt5oomRN1HBybM6cZsItKFuWFksC6vrCpQyVCsCaUEguG1a5u82j4mYr012cBXPVEoJ_ysj0gCLh6SN7ua-G1jqgTRZPQVbExHWW6UpwJd-tJnUlc0uhlXQBg9gR8C9Zyy2Oy-aQ8yNZO3ImLbB0g-Us5it99FBPu0VRwxliyUng9x1D914OQpj7oIe4lj0Nmiz82TJ31xCPtIrYKQTULLonmHx-r8iY-q31wTg8-L7ebIQQV5qKI04owHz3gCqvS0V1LXhI_eHn3nHHeuh-qpOxq-vfA47YI1PmgVHKtx3gdDlXH0Syaxak-HWJQ9dkzESDhBPDqTkj06Zr2SlpyydFS6x2vEq5KFG9x667aIQ0n88D2jfKBWrz3XgxfEg6rlDybVOWJRwoMmFww7tmzPBp8XyJlyoCV99Nnvvh4GmpoogHm1QoJyAoXFW4cba8omCjhE_QMWdaskNs2bq2j2cPNRM09o-L-toH3AOHaF6gnDiXdq8mPjaSJzQo4vifES_8y6W8Sju-mFOvwU_Wt-MctMSUugTXuI66HG8cYCJ8Kp38Cls7jB8ehPzJQoXhNjPv-IaKC75AJK2Of5UV42y31cEV8Ve8Ux8s0GdxzwIvLN68UXwdn6jUs-OS-RzPsAJV0f6-O_sWj8z8AlbFozlNdgTRbvTxBZidAwEouBemzdQDy3wYB6mLMyVBiK3-s6IcVQSQ_Qj2BGN7VUUj1FjFPuAEF2vDHID2lLlyUvwTcrhJjuEPFKAvzm-iaU0lySmkV0tmr9VWjl4mZNVGVwU6vEpsgriy6YgFhnE3ho1Rz53xy6piOJ2EjOaD8leP-lc3lnVyPCAKizCeROznDQ6TFcxcedU2chmUHKx4hb8kkV_kPDpHHbDniaORLpZXKrVVkjccjVA4KnetoRe_616pznYtaJ9uPiWKedmYM1nubeN0uIU1LYrqyM85d_IkORRs-QnnSIh5oNA1OMlmpHaQ9D_a3fKcDEV883Z0KW2tB3COdIi58j_auE8nriaJAIaczIaObFKYPi77RowIrYwbibmmouZSNqx_Jcv0pfPXwYWtP24yN2xasW9pYVu7nQBmMqwQB_aAjBDapiUIQYM-4suZE6kZl5n8tNFHB_V2wQQitba4K8cHiXvNKf9xGEr1t1W_PXNrzG1TEq5Z1gIh3RsGVmIB_EQMmltyGMx-2ccuuFOkLZfXwUFJ4zQq4JWZkeabVEICEfW3vTdv-IBnXsmvzBAT_cH8ACYST5IfWYC34wCJQm2Nmre1UgggzL4ydKYVdAj3toR4caLeJ5Twek2OvWaneEy1HlduAESF8-GHReszyLx5ABv6CdKr5C1KfzDX2bpUYn_Gx1NPbMVmGX0IIWM7QN6y8sW0cQFL6QzDe8t838Ge_XRoxF7md_FWZhEca51AJnwwT2pdVDVrlfvZFamdYlEqSo7ZUMsQpLY3_egiCt7MGxzO6qGYVCHvmwKI-ECjRchBH8BgegwSJfZf40sf9x6sAcCMbEA2fqERMngxGNUQDw1N31cLENKsF2rjvH86GSES1QCJ_9fzgsqrkQWtg9ygi8WSavsQVEz8_EYaEs8B3LtMDjXxxP3nPGA4Sgrsyr3NZdXzTIvZnCbitHGQLKbSqSj5xZqLaK18hzsbT5-bYpcmSWkRq_39Qhkao7scKw8Qg-ypRNwWgBOjdklc5WlhZAKKWm5VnyCruQl8J9U7yIvkNwIqxlL90zA2NS44uEwSzyLTsjOodoa7Lb5rp_9fC2wrnmcfZubitH51peVerY1nlG5pxa_iF1LIElG68z2_RA8EcB3sKX2ee3bpVV0A2zMf-eKe1MwLq7DNKoR3ORBLDgRAb7_pA55ewlttiXDWU7oPJKByENPQG35GAuA7J9MNS7Ys0h3COu4rCHyLAdOwRdFq3siOTBGlLsch2NYTsJLPOjLRZdSKNTGs6Jw3YyTE5UmJpw7feHp3POdRieV45jur6yvOOJPLW99Kddlq2n4OasdZj1knQMMAWx-fKEJigDGa53SQajZO7qBXmm0xX0dvsWu99TQOqsPhDgIMg2IgBwqYdPvmc5gHqyg7HJTYNgI1jaTe_fKaeJTX9ejlNEqsznHTJP7NkgPNCBeIkxyg2GjnZSh1gQEwSoTrb7iDLXdhLIhX9oWc9lFxNp8sxp63yVTjSW-hqgoD3oiUKrUELGr5daGvz4gzx8AHT1Kg7VUSDZDL5xNTOkp6qRhnmUrZpPSDdETnGQcV3QdVvc6XmA66QK9YOB9vX-4JYraHnQ421fWMaBJUdyz_5KD8NCfxll8hc5P32E6hx_dQM6wxkQksLkQbgQSwDpD-PGRoYSVIgNWrxzRgyVf1aygVK8orrBscYAo02yxSTxf2eL9F93snJA24Uj1ASTZ_azbwaGM1BhhGzTSKDQo3JG6qlWrhBch2x12WKA9Yy_bCAGPmFQWlrC9jZV6w6jBbBVlfdGNuBIfxysgDSD3S4S_TwGUtoRp6Qju9xkpjq5MV5XsLH19hbAd8wi2t68uQrXsqh5W-ANcgrn8ZlSGma7Fz_wYjXY4MdlTzlzeBN11AEDmSoeub4ZkL60pZDrkQQtUYAa58BCVK1mUDPGj4JDDzvmXcnp46pqJlHOcCb3pyOx9Ndgc_37G3maG_Qceu-hSiJoRj4_8fgunHWR93C35HZ30g8T1OYHmIofzFly9LR6jXAHB2ezw2PhSXog08cNYulmImN57-AjR4VrikaYZkq0KBfHL_RElvD1bFP0YKIkdUiZ2nItU_uvjVDxInO6tXnv-Wlk4q2Fhj5Y2k9RsA4yij_7LX22euc0wrQ21zwCSXDCPbWy0xLt5CgeJdh83UsNEDRJ0HekSl4CjKZH0qjK2kLJ-qYlT2VWUw4jioh6ZT4wD5JaYIqnuPA1BMlkXUBcJmRQNmXbbgWSy-Gc5xgg018jVO4uY2X4Pg1EXPlNLCdvvBJ-eHGHsgV55xWOwCB0dUO26XIk5UEfux2kYAwxWl9eg1oM9vpb3eRf5cUsBh_rhXTSOQHn7qQadW1reDqO5aZD7CAinqf_z_7C3I2Iw9GIsHALR3H6Vy_LQxAKg8mfMW8kDApk23a9MWGOP6h-QLm93gnWHzpEA1_nxKkn0c3mLs91n_BS1rJ_-tMfkz0b7_VQ1XjEROZaVXISDbDfLlCXutou0snmtM_M61YIv6dkzla1PdybGNmDm6erLG-gDMVp1yzngdKtrWRvHqT_Pu7VjeyEVUZOnsRdBeGAi7WRUtoIpu-KmrUAI4nNGrF9jJZPIcyPtlfukiX5b1Bztlbfk5D_om6FJLcdbyv8ZDpnXX36SZ0aPkGqpUyuDbu0rv4ZH6gIF876KvC8nJ0l1BeomE_H15WPEAIn4qi3KSH3EWhf-RiPP2vkZQgQ05F8jn2_kuz8ZlmUvsH6BhgwBCLZHfBCuS55bIZBU3hzFZQW4-VJAq4_J-cXoQCdmN8TVjhMZPZhdUfaiuCUYfiPdEMayEQ0SuL3tVKUFFP0s6koQgTi3cg5aPZosV5OlaRF-HNrmYmnEAr-XOstBsTgKrLIiJAEoQ1JAIlHR_3mcYLgqNWFHnxss30gMv71li0or-pgmR107mTobkEH-8eYHsJUPR1hP5wI8oY1QoURUPwmfxTcUvlxSlLFiX0F-c95sdcfOIuY3oDDhP3ofc5ASODi7bo2ZpSsY8UBSA2dd65cbiueKzqVtF5Y_Bbi2UGVcKq7SPlXoZJbWdKqVNVNHfcOeTOQuTe1J2-jApZdpIoau0Fay8d362WJtX7MSx5UzFZ7JJLeQhE98eEADCMF_cE4W_Zg0Iqy8fXIG1Ee6bqE7udW5rfCjuDxtYtFvWefT-ACYHBQ8qo8oH-5sAHf9Jj106t3EhDa4XlA1OA=="
}
]
},
{
"iteration": 9,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro x hx\nsimp [KotheRadical]",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772022499_5978.lean:37:36: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 x \u2208 sSup {I | IsNil I}\n'Koethe.KotherConjecture.variants.le_KotherRadical' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 42.983,
"tactic_replay": {
"tactic_count": 2,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 21,
"line": 38
},
"start": {
"character": 36,
"line": 36
}
},
"leanTags": [
1
],
"message": "unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 x \u2208 sSup {I | IsNil I}",
"range": {
"end": {
"character": 0,
"line": 37
},
"start": {
"character": 36,
"line": 36
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 40
},
"start": {
"character": 0,
"line": 40
}
},
"message": "'Koethe.KotherConjecture.variants.le_KotherRadical' 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": 36,
"error_char": 36,
"error_msg": "unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 x \u2208 sSup {I | IsNil I}",
"goal_before_error": null,
"full_timeline": [
{
"line": 37,
"code": "intro x hx",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\n\u22a2 \u2191I \u2286 \u2191(Nil* R)\n```",
"goal_count": 1,
"ast_complexity": 81
},
{
"line": 38,
"code": "simp [KotheRadical]",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 x \u2208 \u2191(Nil* R)\n```",
"goal_count": 1,
"ast_complexity": 98
}
],
"tactic_count": 2,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "(I : Set R) \u2286 KotheRadical R := by"
}
},
{
"iteration": 10,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772022542-WbQL6TwvYYtMRiy3KRT8",
"api_elapsed_s": 15.921,
"usage": {
"prompt_tokens": 14893,
"completion_tokens": 523,
"total_tokens": 15416,
"reasoning_tokens": 495,
"cost_usd": 0.01382955
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Assessing compile budget and leveraging sorry axioms**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Assessing compile budget and leveraging sorry axioms**"
},
{
"id": "rs_02d87ccecc950c2201699eeb0ef84c8194a3bfdfc66896daf6",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "gAAAAABpnuseg4c39gGgqAioW8WD7EtMn5K3Fl3qmFF1NV2gKWpbZPdtS6-iKCQ-MKi8KO7wl3AaJN8zau-zqah1_YlWAIclf0t1gguy7OIGepSS4pvtZ6kAHxVvk45xDYdcPWx1_WrD78MhwEEVMLxNs8q8kUZf-kg02FFurf0y3ddNwKHQhH1Rdw72Jht5m4PfyeEwUJAzjp3Js0Us2tvqsxGKOHXo28eaW_Nm5fT0Xu5IKc0l2R3aUeNI1D6Wn0XMiIZMkXHNbDOmD_GlIS156VmDuesS_Om9jbeiJUUrQ73D19puzvlymCT-Pd600q9FBLtxO0NqXHfeTNT0ylXo3GiIDB-NioJOtwX6R4XbizayDkpOwijxRKgzrywPDTF5v3J1Z5beN_Z_MG1PPQ8xz0_N9-23C0b5-Rvt0Ra9njAuR4p2oSfAYQZx0EIEk52RWBtHUaWhYIX4kqsVRyXFMKvulE4YEMqGhERKzhGrfUW7nUJZZyjYI_lvQjFxq5YHNXqfQCanvSJRWj07Fg2Y3V05dZYmd8DmAUR8zwln7Go0rac0VoWF5iJJboLdS94PiaZ26w-NmcsiUkL4Y4wfscCZK1LRW1mHmSMpJgjQfe4MscOLe_8hIxxxY0Cuw2MvLZ-j3SVKIniJHhPWJjgodoD2gAFgc3zXcecC3-_X6e1f02YTHLRoKJQycETOZMQIqbaR2Y50wWnCLQHzyxLqU_aGmq2mJ-nxZzi1FfbhJqDAlm-WOzwTxoozGlWWia2pHN6Hk5XdccZI4V2TLLPrC1L4TNLoWqvlF8Sjm96V_SuVRa0oAJ6VW-WlxVFWFciajdG3dqjqLxtAeA0c1-FmxEoXVGeO1iyO_msG0nKHbQzULUFzH-xDzsFSdA08G19AuR4lLXSFfLdfoIPBBwP95AbOUJsxIDoooOOjcQ0WyppPt_wcFthSLWSoUxrXMpF143jc7WYdSYmyrpG6BM8xewJviqNOQVMNPcrlxklCaR1_O7sLpqIJ5rTwGzTZtPqqouN8_RfYk2QYD2jJEtiHP48V7Y_3cjF5koHux3qck8J_M8g_WGEt6mceDfxpYwD9Y8KyUN5xUZuu18WHMN32BKyzTg4--rn_2IwpgPX_1os23YdrOiaQVuoX6J7bKLQjvikQ7xmvcDyM0eycNH3C2uoePrjZW4NVa52WcQ-gIhglFqrZj5YEpNsxpe8iofmwjQEQP7HKvF0sV7W2XePogEW1WHe99qpx9Kl32gJSEYBwNk7OvD6WVAAjiRbT0GsLFHVcJVflsZ44VuHTDKsztk04ocNEYthypTBfa0Hrdx1sZGoW4JvWxxw94Vb7bc2lSohi_zYpQjF_LErnzBoCBoKO2xabNWxGQlmyrQR9D0fDvrIBoIznR3Mr0Qp2oWF3zZkz8uedCo44hqjcvItGdI2P5MzvZz5JWwGHOAb7eokL1Stbn1C1B6sDHejqoqxvSqTti8l0T_ynA2tV2ayRqkzAlrTXs0OhPxvGZafYAa6CF74VaXicpSZDoWMAGDCwTDJnOB22fdX_dwEfhXqSwoTmfCIVLkT9agMnIeEhmmeR-IpNnDkcSNSkP7vCWpTM4sr9KmIaVLZ0c7RkGNxPDbQ1gjeH2Q0lx5ohNAe8-uuaRDriQ3I9O9i74LQEftxAhLgyziUs5HNSsKQYjidPwXlMGeEi2iHglYoZZ2KaaLfh-O2KCFcFN39t2S2tVuB5tudFyIp-qJjAeAzVJ0lHJ8qBaKCyOK1TVgLayaeAXCph2qOMoEB22NQTflxtfnzvJPXsA58h_dHg3CCtkMqLzTCiGufSZNsfwHUx8dUQGjTGmx_sxFmnDOwSpOCMgl2m0wOJ_gCiwtCoEvSDCJw-kumV-GxSwQhSwU89JVtNEvpfQ4cQqXmncOd920YwVvfH15MuOhzzKEiAai4AzCFPU_-RoofRiHGhJ_p_PeTPS0k--0TEW8kZwBMGkImrLI9sMaG6FSQy8GXxpFaGhmtEDwp7qN0sIIwVAnTcujuq66kLCaLR9AW3LBDYohk-JnmVqdXSOKE9wpQBWVU0eq98Yrz7cRVQkmI9Yw3Uj0xNNv1NqcqJ8Nr2HsoaFLp_ULtWUGNSnQnBoEJ95ffkBJlhRO0NiMDWMSHa44qzklnaqhxVolNfWHHbGITL1ULJQSg3-NH7R-_HsAfLuxw0-gJjgjypGjzsTqGveY4S8Hk4U_SF_o7WYjqj8ER1EbcifXdiGnE4MBjEIy_gui42NLlfvSS3njCeCYj_tXNRVvuSjpEEEP2dnBiMHKUIfo4Rztb8jDkmGZ2gJigx1mD9P01kifcl7BXgV-wY_LvBfPyRZK_AMiJk29DyONZ6-j4Yfmc5Q4hjL0yic_a88gz3r6MH5FGYYiFQdFP23f04qzClt0m2ETNMLOX21Lfez2db5B9SEjhhhmJ_R9Oml7Pj03uitAPWvN1FdwtblExclUFwM4hqgCPgCMMugK86bEMnlFDEIh_2qGEYZN90xTWuAcDkd8ZHis6FrRoFpJj8Ic3cTMvTFJX8wsBjRu2amhS8sBEHhItVe1VtQXUB1RFNYGOa89LEJp7pfqtjMrEu33gE6GqJdtQIQKl74ZdSW7Oi5etZtzBvNC2A5ioS5xmOWpgVvz2rv3YnQrVQTb3AfwwERxb_qA2mPcz-he8TEkj9vlWBnSBx4luoJcgKV6boQ7ifKud2ZYCUvsj-e1NuvSXGPTlQDfFlK5Di9eVOabv7Zw4xvalN3_v6hIWXTWeoke9QfClIKWMKmiGn0S55z5ziQPDhCrKH5WGNjX4FldfADbZuNYca8emivl025R7H7lpBoHXRgojN0SC4n5PCKM3-QVp8jWk_KmxR59TpYACb1DlghtWk-l1CTCLJu_P4n5gvFD3EzhGuQHgINi6r2ApzCji54qNy-6YyFXFAmi_PuNYWbMNP7eT4bwaOokkt8uvDHm577RMQlP8wsLn-PR0OYh56Wz2nP2KKK1gd2gEONqwEnMinJ4xs04ba4m80zabc8t8T-f1YM4-LtOxpDoS98bgFFAQjECi04QFTgPaTPCqlDar-r-23pOLSzyi07ORTUacstKvRdc_ltvAAl0NXqL1wjYaNbwo1EOw_8IvqV-Vge4WUqmOfqTcPm6UYVqo07QWwvZ3HoYaY03WNwPhocoxdAzUjAqFacTM7LVIIi8HPVASu79JhbaQIjzipLzYA3KatQdVM2iXyP6HUT98v6MqsJ45_5FEuX6hGVzuKvcMdQlPEgWtSKLuwo68RdVICYhQ5SNtkzzneulvDUAgkNjzJhc4NNVu997pU8hlOQnmCjWbyT2ZidxgAEIGAOoEb-9GuayMGuHUsEQL8PxbGA9bmxRTs1waKGMB71Yt45P_4v9u0nA4S7-8Kk88miDWnnpUMLDoB4XFIFVNRiSt_4bjMv_0r3Xb2jQe_CQW7xImVPh8fc5ZrR7NqInmxsfCVwB_htY2mu6QXr7Nz5oB40g6zInGjs6s3veAWA5MyfbDKbeKcP3NDy3a-6BSKiOmX2ygd6jPa2Rfxq_L9e9-_2h54383eAz3xS1hlS3Eflka6P682VNVWVmYR-KRmlLyiFXUPdICRbVwvvBjSpWqK71lQ6dslq0VIy4Fy2cMoptywXUm3E3CBtPyta0ujTbcz3C44CPrE9cuA3acao6tnTYkWIHIh-eDkIPMLbKkvxuTHVsKkm2j1fg74BUaTJxwHeEmi_ozefmXFNhZeuAH7T5uZ5HtN0MVkgkbVU570vFbMrvKv3urAToQ5cMr_ATxKFXJrQNvvtIBJd3eVWJApAEMfkudhJehuiBAio2c-fOQ9MXoZlhuRN_WrFj8S9tZIM2bvb0Ud-Xu59m94O2bjjs_pg5xEQWdcLryoSCCw7hqVKYAvH1fUffnoap-xjNiSJGoQweyMHA=="
}
]
},
{
"iteration": 10,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro x hx\naesop",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772022558_3442.lean:39:2: error: aesop: failed to prove the goal after exhaustive search.\n/app/asi_portal/verifier/quick_1772022558_3442.lean:37:36: error: unsolved goals\nR : Type u_1\ninst : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 I\n\u22a2 x \u2208 Nil* R\n'Koethe.KotherConjecture.variants.le_KotherRadical' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 31.833,
"tactic_replay": {
"tactic_count": 2,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 7,
"line": 38
},
"start": {
"character": 2,
"line": 38
}
},
"message": "aesop: failed to prove the goal after exhaustive search.",
"range": {
"end": {
"character": 7,
"line": 38
},
"start": {
"character": 2,
"line": 38
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 7,
"line": 38
},
"start": {
"character": 36,
"line": 36
}
},
"leanTags": [
1
],
"message": "unsolved goals\nR : Type u_1\ninst : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 I\n\u22a2 x \u2208 Nil* R",
"range": {
"end": {
"character": 0,
"line": 37
},
"start": {
"character": 36,
"line": 36
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 40
},
"start": {
"character": 0,
"line": 40
}
},
"message": "'Koethe.KotherConjecture.variants.le_KotherRadical' 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": 36,
"error_char": 36,
"error_msg": "unsolved goals\nR : Type u_1\ninst : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 I\n\u22a2 x \u2208 Nil* R",
"goal_before_error": null,
"full_timeline": [
{
"line": 37,
"code": "intro x hx",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\n\u22a2 \u2191I \u2286 \u2191(Nil* R)\n```",
"goal_count": 1,
"ast_complexity": 81
},
{
"line": 38,
"code": "aesop",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 x \u2208 \u2191(Nil* R)\n```",
"goal_count": 1,
"ast_complexity": 98
}
],
"tactic_count": 2,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "(I : Set R) \u2286 KotheRadical R := by"
}
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772022590_5092.lean"
],
"comparator_passed": null,
"duration_s": 12.974811315536499,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "KotherConjecture.variants.le_KotherRadical",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Koethe__KotherConjecture.variants.le_KotherRadical",
"category": "research_open",
"endings": [
"end Koethe"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 755,
"source_file": "FormalConjectures/Wikipedia/Koethe.lean",
"title": "KotherConjecture.variants.le_KotherRadical"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772022590_5092.lean:37:36: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 x \u2208 sSup {I | IsNil I}\n'Koethe.KotherConjecture.variants.le_KotherRadical' 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 |
View{
"agent_telemetry": {
"compilations": 0,
"iterations": 0,
"duration_s": 0.05,
"best_status": "pending",
"best_goals": null,
"max_iterations": 10,
"model": "google/gemini-3.1-pro-preview",
"error": "OpenRouter API error 402: {\"error\":{\"message\":\"This request requires more credits, or fewer max_tokens. You requested up to 65536 tokens, but can only afford 58661. To increase, visit https://openrouter.ai/settings/keys and create a key with a higher total limit\",\"code\":402,\"metadata\":{\"provider_name\":null}},\"user_id\":\"user_2zsm3L1zzBL2yxyrvmrXwgeTimc\"}",
"provider": "openrouter_primary",
"finish_reason": "",
"usage": {
"prompt_tokens": 0,
"completion_tokens": 0,
"total_tokens": 0,
"reasoning_tokens": null,
"cost_usd": 0.0
},
"reasoning_effort": "high",
"response_trace_count": 0
},
"response_trace": []
}
|
|
| Gemini 3 Flash Agent v8 | Partial | 1 | Feb 17, 2026 |
intro x hx
have : ∃ J : TwoSidedIdeal R, IsNil J ∧ x ∈ J := by
refine ?_
obtain ⟨J, hJ, hxJ⟩ := this
have h : J ≤ KotheRadical R := le_sSup hJ
exact h hxJ
|
View{
"agent_telemetry": {
"compilations": 6,
"iterations": 6,
"duration_s": 136.17,
"best_status": "partial",
"best_goals": 1,
"max_iterations": 8,
"model": "gemini-3-flash-preview",
"error": "Gemini API error 429: {\n \"error\": {\n \"code\": 429,\n \"message\": \"Resource has been exhausted (e.g. check quota).\",\n \"status\": \"RESOURCE_EXHAUSTED\"\n }\n}\n",
"provider": "gemini_direct",
"finish_reason": "STOP",
"usage": {
"prompt_tokens": 51784,
"completion_tokens": 681,
"total_tokens": 62237,
"reasoning_tokens": 9772,
"cost_usd": 0.0
},
"reasoning_effort": "",
"response_trace_count": 12
},
"response_trace": [
{
"iteration": 1,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 47.36,
"usage": {
"prompt_tokens": 1910,
"completion_tokens": 84,
"total_tokens": 6380,
"reasoning_tokens": 4386,
"cost_usd": 0.0
},
"thoughts_tokens": 4386,
"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": [
"sorry"
],
"proof_code": "intro x hx\napply mem_sSup.2\nintro J hJ\n/- The goal is now x \u2208 J if J is an upper bound of all nil two-sided ideals. -/\n/- This is not the right direction. `sSup` in a complete lattice of ideals. -/\nsorry",
"compiler_output_preview": "\nBanned tokens found: sorry"
},
{
"iteration": 2,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 12.737,
"usage": {
"prompt_tokens": 6471,
"completion_tokens": 126,
"total_tokens": 8479,
"reasoning_tokens": 1882,
"cost_usd": 0.0
},
"thoughts_tokens": 1882,
"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": "intro x hx\nrw [KotheRadical]\n/- The goal is now x \u2208 sSup {I : TwoSidedIdeal R | IsNil I} -/\n/- We want to show that if I is a nil left ideal, it's contained in the sum of all nil two-sided ideals. -/\n/- This is exactly the K\u00f6the conjecture. -/\nrefine @sSup_le _ _ {J : TwoSidedIdeal R | IsNil J} ?_ ?_",
"compiler_output_preview": "[path]:43:2: error: type mismatch\n sSup_le ?m.4842\nhas type\n sSup {J | IsNil J} \u2264 ?m.4841 : Prop\nbut is expected to have type\n x \u2208 \u2191(sSup {I | IsNil I}) : Prop\n'Koethe.KotherConjecture.variants.le_KotherRadical' depends on axioms: [propext, sorryAx, Quot.sound]\n\n"
},
{
"iteration": 3,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 8.176,
"usage": {
"prompt_tokens": 8685,
"completion_tokens": 181,
"total_tokens": 9820,
"reasoning_tokens": 954,
"cost_usd": 0.0
},
"thoughts_tokens": 954,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 3,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro x hx\n/- We need to show that x is in the Kothe radical.\n The Kothe radical is the sum of all nil two-sided ideals.\n If I is a nil left ideal, it is a well-known (but unproven)\n conjecture that I is contained in the Kothe radical. -/\nobtain \u27e8J, hJ, hxJ\u27e9 : \u2203 J : TwoSidedIdeal R, IsNil J \u2227 x \u2208 J := by\n /- This existence is exactly the K\u00f6the conjecture. -/\n refine ?_\n/- Once we have such a J, the result follows by the definition of KotheRadical. -/\napply le_sSup hJ\nexact hxJ",
"compiler_output_preview": "[path]:43:66: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 \u2203 J, IsNil J \u2227 x \u2208 J\n[path]:47:2: error: tactic 'apply' failed, could not unify the type of `le_sSup ?m.4923`\n ?m.4895 \u2264 s\n...<truncated>"
},
{
"iteration": 4,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 5.999,
"usage": {
"prompt_tokens": 10210,
"completion_tokens": 75,
"total_tokens": 11035,
"reasoning_tokens": 750,
"cost_usd": 0.0
},
"thoughts_tokens": 750,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro x hx\nobtain \u27e8J, hJ, hxJ\u27e9 : \u2203 J : TwoSidedIdeal R, IsNil J \u2227 x \u2208 J := by\n refine ?_\nexact (le_sSup hJ) hxJ",
"compiler_output_preview": "[path]:39:66: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 \u2203 J, IsNil J \u2227 x \u2208 J\n[path]:41:8: error: Function expected at\n le_sSup ?m.4923\nbut this term has type\n ?m.4895 \u2264 sSup ?m.\n...<truncated>"
},
{
"iteration": 5,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 9.123,
"usage": {
"prompt_tokens": 11476,
"completion_tokens": 90,
"total_tokens": 12813,
"reasoning_tokens": 1247,
"cost_usd": 0.0
},
"thoughts_tokens": 1247,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 5,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro x hx\nhave : \u2203 J : TwoSidedIdeal R, IsNil J \u2227 x \u2208 J := by\n refine ?_\nobtain \u27e8J, hJ, hxJ\u27e9 := this\nhave h : J \u2264 KotheRadical R := le_sSup hJ\nexact h hxJ",
"compiler_output_preview": "[path]:39:51: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 \u2203 J, IsNil J \u2227 x \u2208 J\n'Koethe.KotherConjecture.variants.le_KotherRadical' depends on axioms: [propext, sorryAx, Quot.sound]\n\n"
},
{
"iteration": 6,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 4.952,
"usage": {
"prompt_tokens": 13032,
"completion_tokens": 125,
"total_tokens": 13710,
"reasoning_tokens": 553,
"cost_usd": 0.0
},
"thoughts_tokens": 553,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 6,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "partial",
"goals_remaining": 2,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro x hx\nhave : \u2203 J : TwoSidedIdeal R, IsNil J \u2227 x \u2208 J := by\n let J := TwoSidedIdeal.span {x}\n use J\n constructor\n \u00b7 refine ?_ -- IsNil J\n \u00b7 refine ?_ -- x \u2208 J\nobtain \u27e8J, hJ, hxJ\u27e9 := this\nhave h : J \u2264 KotheRadical R := le_sSup hJ\nexact h hxJ",
"compiler_output_preview": "[path]:43:4: error: unsolved goals\ncase h.left\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\nJ : TwoSidedIdeal R := TwoSidedIdeal.span {x}\n\u22a2 IsNil J\n[path]:44:4: error: unsolved goals\ncase h.right\nR : Type u\n...<truncated>"
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"[path]
],
"duration_s": 6.55593466758728,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"axioms_check_name": "KotherConjecture.variants.le_KotherRadical",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Koethe__KotherConjecture.variants.le_KotherRadical",
"category": "research_open",
"endings": [
"end Koethe"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 755,
"source_file": "FormalConjectures/Wikipedia/Koethe.lean",
"title": "KotherConjecture.variants.le_KotherRadical"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:39:51: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI : Ideal R\nhI : IsNil I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 \u2203 J, IsNil J \u2227 x \u2208 J\n'Koethe.KotherConjecture.variants.le_KotherRadical' depends on axioms: [propext, sorryAx, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
intro y hy
obtain ⟨r, hr⟩ := Ideal.mem_span_singleton.1 hy
rw [hr]
exact IsNilpotent.mul_left (hI x hx) r
apply Set.mem_sSup.2
exists (TwoSidedIdeal.span {x})
constructor
· intro y hy
obtain ⟨a, b, r, hr⟩ := TwoSidedIdeal.mem_span_singleton.1 hy
rw [hr]
apply IsNilpotent.mul_left
apply IsNilpotent.mul_right
exact hI x hx
· exact TwoSidedIdeal.subset_span (Set.mem_singleton x)
|
View{
"generation": {
"model_id": "google/gemini-3-flash-preview",
"raw_content_length": 1465,
"proof_code_length": 394,
"finish_reason": "stop",
"usage": {
"prompt_tokens": 1962,
"completion_tokens": 426,
"total_tokens": 2388,
"cost": 0.002259,
"is_byok": false,
"prompt_tokens_details": {
"cached_tokens": 0,
"cache_write_tokens": 0,
"audio_tokens": 0,
"video_tokens": 0
},
"cost_details": {
"upstream_inference_cost": 0.002259,
"upstream_inference_prompt_cost": 0.000981,
"upstream_inference_completions_cost": 0.001278
},
"completion_tokens_details": {
"reasoning_tokens": 0,
"image_tokens": 0
}
},
"elapsed_s": 3.59,
"temperature": 0.0,
"max_tokens": 8192,
"attempts": 2,
"banned_tokens_found": [],
"reasoning_effort": "",
"structured_output": true
}
}
|