intro x hx
have hmain : IsNilpotent x := ?_
exact hmain
View
{
"agent_telemetry": {
"compilations": 3,
"iterations": 10,
"duration_s": 322.21,
"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": 80378,
"completion_tokens": 8175,
"total_tokens": 88553,
"reasoning_tokens": 7904,
"cost_usd": 0.1666091
},
"reasoning_effort": "high",
"response_trace_count": 13
},
"response_trace": [
{
"iteration": 1,
"event": "search_mathlib",
"query": "KotheConjecture",
"results_count": 0
},
{
"iteration": 2,
"event": "search_mathlib",
"query": "IsNil",
"results_count": 2
},
{
"iteration": 3,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772022203-ZwVxI3PW30qsiGF1kFeX",
"api_elapsed_s": 4.292,
"usage": {
"prompt_tokens": 5087,
"completion_tokens": 124,
"total_tokens": 5211,
"reasoning_tokens": 83,
"cost_usd": 0.00297745
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Assessing goal shape and lemmas**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Assessing goal shape and lemmas**"
},
{
"id": "rs_0973f977e8b29db701699ee9bca0c48196b520a28df504037e",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "gAAAAABpnum_B7gfxlEWy2WhVJslS4U0QaK6gWOUfAW4_hjFTKDMOeqJIwtHruVczCsvyKMtM-yQL8tYCyR5mSQgTJoFUqCqWsZlOH7nzjd31RcIeX7Cl5YZif_yQYgQpYU0G0ZzX9gnk3cDC3P5HVlzLhYvHdFumi1MFiGlh6347KGRT1xcw413VY0df8bMqpHrjYTAc2ILMdlrRdPChmHgJJ9X6k5zvKJJOiIePSlWVU286NiT88PMjHeA06r7ziogWt0NKC-MzJ2RPzz6NZdvLinAHBxz4vSaGzpn7La-jVppPh-fIEULWyzZ-gFbmOJBO_gwUiHZE5tsQkrluZf_I1Ce1dGFPkTkRhMNzXrHIW5moeiHSe-cIHhaJa_7b11FmM4X1UL2mvsG0OgTWq7y1OeaoE5LDoH-MBaHDxSz0OAoUUBY1N05348mwETfdKFEiC2WdOCUpYDrnWZRmC9S7PNrMdBmrQIaY3qT9pLiLNEGQMdwFJr_0CI32SRbbP_1dGquXMM2s2krTEIjFwHvjCoN7x32yrlK6-1PsRik4DOEBDPRGWUl-pm_9Xk3use3YPKhebci2HE6bARMEXCVfuz8zvb6TFXkJOoyCe4ZezBMR_YGiFKL5KUne319B5TXFSlUS2CyEByRuphjNJ6J-MeM0G26y_sKXSH3VrYeoCu0TJ2LFUQJ8JQb8tukpWgpJiukSyewFX81LXLqFAudEWB0u1U9Gl19f0G3AY5jG6ASeAp5aLi1YfxTbwbLLYnHlG8DhMJ2-Pxem-QzwAjxW2_WlvOFPbRBHCxuB8obUKnOfrO77fHEsyUkx3sD2wXK1ndrwzdOtO9x0X1HkfNchIe5TvNwKQlx-rNqDDI4xoXtC-dIiF8KuBLSQmz_D7xh848n-jcwOlM456HELp_tggGWsTCu2aDlamm-K5E756j0bCtrREi88SfjGTgfLMjQzkqZ0B3eBQMxorniRoI1pNwQjv6NpBnh_feE2fqb0sLFxo1oMCR4O83OyNIqB1IAAAJE4vUwDoTfJMMfIxtsCoSEzvTRLZL-PYZbrq4yDRzsy7C0fa_YK7H7MeQYtBr3ixhCwPfdQ35NSQOqWTdePgM248KtxZdccCmpPxvNvzfHldI_n_Qo8QvYNc1S55L-lyHzhfRkVyjJNuFUsGcOKoN_fepj-u_vQEX5MWub0WznaxnBL6scvawxqGs3ozmL015CDXNn_k1WjoaeIp3fa5i9LBtK5lwXcyINZR926GYe1X2PZvxy5UIuJTdH98oRN0xwnl3-FbEoWIct5Do432jC0JAQVfWcqJWW-tDIXY8DDbrOfIQDaUb1CkFibZ06TL0W8jS9Irs-ERKBqeMq_f83T-oxXY0zgSqb34UCRzDULud0LTQzXlYJf-o6FxaV57Arhnwn"
}
]
},
{
"iteration": 3,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro x hx\nhave hmain : IsNilpotent x := ?_\nexact hmain",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772022207_4057.lean:36:89: error: unsolved goals\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x\n'Koethe.KotheConjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 30.506,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1,
2
],
"max_goals": 2,
"final_goal_count": 2,
"goals_opened": 1,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 13,
"line": 38
},
"start": {
"character": 89,
"line": 35
}
},
"leanTags": [
1
],
"message": "unsolved goals\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x",
"range": {
"end": {
"character": 0,
"line": 36
},
"start": {
"character": 89,
"line": 35
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 40
},
"start": {
"character": 0,
"line": 40
}
},
"message": "'Koethe.KotheConjecture' 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": 35,
"error_char": 89,
"error_msg": "unsolved goals\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x",
"goal_before_error": null,
"full_timeline": [
{
"line": 36,
"code": "intro x hx",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\n\u22a2 IsNil (I + J)\n```",
"goal_count": 1,
"ast_complexity": 95
},
{
"line": 37,
"code": "have hmain : IsNilpotent x := ?_",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x\n```",
"goal_count": 1,
"ast_complexity": 116
},
{
"line": 38,
"code": "exact hmain",
"goal": "```lean\ncase refine_2\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\nhmain : IsNilpotent x\n\u22a2 IsNilpotent x\n```\n---\n```lean\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x\n```",
"goal_count": 2,
"ast_complexity": 287
}
],
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "theorem KotheConjecture (I J : Ideal R) (hI : IsNil I) (hJ : IsNil J) : IsNil (I + J) := by"
}
},
{
"iteration": 4,
"event": "search_mathlib",
"query": "IsNilpotent.add",
"results_count": 6
},
{
"iteration": 5,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772022254-E6dbqqepyI7bCA2Ye3Xa",
"api_elapsed_s": 13.214,
"usage": {
"prompt_tokens": 7761,
"completion_tokens": 533,
"total_tokens": 8294,
"reasoning_tokens": 505,
"cost_usd": 0.00935095
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Assessing nilpotent sum strategies**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Assessing nilpotent sum strategies**"
},
{
"id": "rs_0973f977e8b29db701699ee9efbc3c8196aeca757f7d470e01",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "gAAAAABpnun7E6BKUtelfPVrzpCJB8wXvXUzvd9rSnuS7hUN2PCIsFVDt29m96coRjengprFqaz94hnco0o-BY1Xc5MN-nhCy1-lrJx3MJQ-ZvRu9Pr3gkI8DRyURqxgI-EIkiDoQZDqwHzVcle1-JxVGttOloTiufVdcj0neAL6W2hTbAJmWaUCrSAZw0YzueiAijsi5HGbZMSmev11QfXVHj2-811xu-Y_qyhET-5EC7k6L2r1_mLVzWBL8cgk3A17TVMqFG3J4DY42ZtNB2a2kcjj_b5BfN1SjL-2OjPM5AtFgvaX2jGWgtOu_q_27Yp2yANWTS3mRJDwUDnGhSZbzeTq00BqnWVn9-pDmX1nPuL2OSQCX7m_dNVv8eQ1OQj_aPKOFRvGpDBuHdCZtLHemHpmDs9FRqT18z78H9-HYqywNZqtLwMaa3YvAOBUPzd1_KzF4j0KcBpzK_JZS5e4lW-VIfKdzV3dY7-ccC_rrkklSp1VVy3iA1Xndgvve9BkpMv4VAq80WF7efBeTUi9FGqgHKIIkyLuLPnkVR-XW2pesVRJI72VoOXp-WlNk9IkluScJKqVR9ikWq-HmljR3NbnMNqYNnD26R8qM8HwXNqa3tkduZ8PwufAOfkj6uu9AUqqFAViwQA7L64s1S4GTKmOhk0FylBFGWKQQ0DBnVbDCDFQ3aK_73uliw1mplzdfgwBujpNuiekSVU82hN17jdb5MNF803DseYGyPIeRiy4D3JMtM_sDbiMhkI-N5qQrOetA8fvvRR0HrE4Ux72HQRjMfhqIMMouzAmm-OtJ1hDimGRAIKiVaHUTcxO9pXNQsHFxeOAz7qCErDLc1WxwrQkr_l5C0gkkzUFVsXdiMun5IF70Or54VhvocLQeJ8BWCjDu6qEa2XCUSdhp8r1cB2COWJkqAEjwSlY0NP5ukslzTm_o-QVbx4ciNBQWpmkZZ-O4yah1ySJKAs6hK0rRNffytuEEseIHadI_Oh4hzfUFOT1ggmOemPhzjVyGhwVR0RfPxxiBaHizGGJcbtXxzgxjsZlalL44bfmsfytQoQYRTvrVvUPAGrhGLIAxzz5zVBIs62MCw5XRCSGTo_ec6EuBXToG546RPbw3SwhOdtXcTTJEvVrykeqsgyhu4Odjx7EcJkf-KmimVa8R520_ZftrfffUbqdihPdBurISS7TLHdnNC0WwECDNL3G_t01qFAeApBb8oVlZN6_KhDlmfc300cVzdKc_-3yVqENFBTwjo-vWz2v16edKbPfAM9pdBdtfIpOsqN9BO2DQkNXLCMNJpAHRArxnLf8K1gYBwtMnfyo2VeDY-JCA1cQ8T0zPC4KFLtFEq1w_GpUJGSYXxcEFs3zZIxqGclazByXWkMQ5OlG6XX-LVnh37CgRrwHGWGlERJELh_FmvxhKIysk_R6GAAeybcBfhaTR5-vHnGMwDJkVFlpklIKwse9Yd5yjlj9dHFY4CQZZj8iOe-aPS4uy7e9GfToqSWcJJVhB1F2hHnMz9SCGtP7OMgh-4aSwXaEv5-5Mhw9cOn68rH8y9Sa261aBR1f2BMA6JRFFaEvpgGMb_Lpk6Q0GtCrfJR-ZruurkN7KJRp1e3DrdPbEp09mQND2UFsHVgLJG32juyfutcrV8Wao0OLCKBuUSDgLrSKYQ3OC02v5lPlzKxyKO8R6lXZUUvqRiAeDvsWiniU6pCWLSyJeQtZ73iSDBA2szos_C_BcImWa7nXxXRbgTq-DantRtAsFNLSEdpwVYFc0vq4MNIwAffB3t5f5D3DWltosf2aE0h_qnA9AEj2w3hrd2btvz2-Dm_kSIIQdOIkjz8ZyundxVapEN-u0miyv_76K-YhIoT1ccCzxO5IWH_4CXHB9ihnM_XDSu85aLSWSJ5M7OgqBZppdoHWMLcv4Es7LX5nHhNsvSHn9F1yiTbJE5rWCw58XuF3WFTNUIE2eP6Sz6A5shnR6kJgSk5XoWB8X4D4L3pp5tz4VgI48BZTEe8j-dcFswdw7-jPPerFyWXQaYOy9knxT-uU5rCLayB9NdOxyA3qrssaCse9jHRy6HBNBxU1-sRU8hdrazr1eLsuPCYCbHoGYHXaFZRH-jUgEobj7_nYpNIosPApBZZm7tygz1TcZxk0rzIgI87xnVsBM5rEIgAZdfRKNdGrIVwc6Mqp7SFzGuBKiPHTQCxprUfPAMwXZKsIWl2pRTZ0Jb7U3m472cKm_QmjOa9bZVc--wahNtAYCIG_vLR-UOJN21hNNHOXeFgcleyIm-9bjfTlu-UBPkfalWIzrptZmH8t2opbmAvAWb4UEdRF-nblsscMx3Fl5gR3RdR7SKpRNEHoCUyJuFN8IGJQ08TFR2r_A4yfmmldQNx_ezXjK0kHS0_9wLS5OURNReimRbzy7MePP7woMPVZSi6JBPh7mY2s5glsUDyYw1wNJugalGFkqMYOjmgvkUb4i39YOCgzNmZ3bYK_bf0cc4Qs1Vukug_-rG4cqWvQeCiK6DnwpCVwf4rmy5V7UQaTn8TabOYqldT9fkEkPhURcX43PrOI_e68O4HXSvyb9HfhV2L77NJAMbN7NrH6E8rqIYQ4bHv3PUDJlCqjBi-dgteWi7_AraTFQBfXT-XXzGcEG_pNpa_z-XBdRQwCiA9uzwBZMem7LOvabRseD9fGvvUkE7SU4HCXMvMKKpSyg_SqulS0fgCLcwwKILnPxwwa9FoSCtdyhE1Kl9XKYgBmiWn0xVn6_DLHr73vMQrP1YwrA4St3hNYJatueSCIea_dufzInk6gGtX_a0v3jcV2PGGzcFf7BuXBvYvWunREhMAp0Cgd-txmGZS-Cct3UnSWcWyiLT79OB4HYiG_ag4zHvUdZI71ymeMKU68jfeA2BQXO8F7FePy7BNn4Sw_E2NoNK-fpG1NRpbd1Oqxd7WbX61Z902VF4CdMMcsOzcTsP3L_3OkYwT39lYXKMh9UfT2tbhKwLag-g_pd_BXwq5Zq7vblJsFVZmmk5p0daRNUCevhf6oD9m6QvUnW-5OiOvop7r44mnRcRYXHMGQG9RYbrr-k4bzskZdAa43Malu2BH1ioBYZGjkYqss1hfmgIBLO6h_z4HN4xU-y8DmA1ux1phIbQd5NJT8PFDxc4eKwLpRuynPp2tMXIAXK1lNhRo8-207_BzLjX181sxTVFmasO-vpFkVMt3arHBSQpJc5sybl4L1Abv1YxjfmqnQqOpzALarCi3Z8THgigHHdQNPO978GydKiDG9JhxwcnjKRlH8sNJqIuogNR6z9xvzA8vY5TSjB7RoQGsN-ec3q3ZNCzGQ6iKVc1mhWkAv-UklqotyQIr4IbVg01QYF3JVb8kX27Indjj--7tyBQSPgd6uvFju6qMKGMc3VAg-BVqX1n2mCxL9kR9lUIaPl4NeIfjKcxYGIS8YQ9rVZld_vaasdstOwdU_RXIZTIc0DNJOeuPi-QAFgq5f3tIBTGI6dj1leN0uMEdaqbnLae7xo3bGYeJEFLzNBo-dEWaoU5hT34EfDQEI9aL7XT2WOQT3MefpQIdwj4kM3N4xay8="
}
]
},
{
"iteration": 5,
"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_1772022267_8684.lean:36:0: error: (deterministic) timeout at `whnf`, maximum number of heartbeats (200000) has been reached\n\nNote: Use `set_option maxHeartbeats <num>` to set the limit.\n\nHint: Additional diagnostic information may be available using the `set_option diagnostics true` command.\n'Koethe.KotheConjecture' does not depend on any axioms\n\n\n",
"compile_elapsed_s": 76.548,
"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": 37
},
"start": {
"character": 0,
"line": 35
}
},
"message": "(deterministic) timeout at `whnf`, maximum number of heartbeats (200000) has been reached\n\nNote: Use `set_option maxHeartbeats <num>` to set the limit.\n\nHint: Additional diagnostic information may be available using the `set_option diagnostics true` command.",
"range": {
"end": {
"character": 0,
"line": 36
},
"start": {
"character": 0,
"line": 35
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 39
},
"start": {
"character": 0,
"line": 39
}
},
"message": "'Koethe.KotheConjecture' does not depend on any axioms",
"range": {
"end": {
"character": 6,
"line": 39
},
"start": {
"character": 0,
"line": 39
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 35,
"error_char": 0,
"error_msg": "(deterministic) timeout at `whnf`, maximum number of heartbeats (200000) has been reached\n\nNote: Use `set_option maxHeartbeats <num>` to set the limit.\n\nHint: Additional diagnostic information may be available using the `set_option diagnostics true` command.",
"goal_before_error": null,
"full_timeline": [
{
"line": 36,
"code": "intro x hx",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\n\u22a2 IsNil (I + J)\n```",
"goal_count": 1,
"ast_complexity": 95
},
{
"line": 37,
"code": "aesop",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x\n```",
"goal_count": 1,
"ast_complexity": 116
}
],
"tactic_count": 2,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "theorem KotheConjecture (I J : Ideal R) (hI : IsNil I) (hJ : IsNil J) : IsNil (I + J) := by"
}
},
{
"iteration": 6,
"event": "search_mathlib",
"query": "Koethe",
"results_count": 0
},
{
"iteration": 7,
"event": "search_mathlib",
"query": "IsNil (I + J)",
"results_count": 0
},
{
"iteration": 8,
"event": "search_mathlib",
"query": "Ideal IsNilpotent mem_sup",
"results_count": 0
},
{
"iteration": 9,
"event": "search_mathlib",
"query": "nilpotent add of mem ideal",
"results_count": 0
},
{
"iteration": 10,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772022375-YAuMMreQ9pV0LSNTTfMu",
"api_elapsed_s": 45.77,
"usage": {
"prompt_tokens": 11357,
"completion_tokens": 2092,
"total_tokens": 13449,
"reasoning_tokens": 2065,
"cost_usd": 0.03202675
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Assessing nilpotent ideal interactions**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Assessing nilpotent ideal interactions**"
},
{
"id": "rs_0973f977e8b29db701699eea68717081969d05d98cffe069aa",
"format": "openai-responses-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": "intro x hx\ncases hx",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772022421_5969.lean:38:2: error: Tactic `cases` failed: major premise type is not an inductive type\n \u2200 t \u2208 Set.range fun s \u21a6 \u22c2 (_ : s \u2208 {x | I \u2264 x \u2227 J \u2264 x}), \u2191s, x \u2208 t\n\nExplanation: the `cases` tactic is for constructor-based reasoning as well as for applying custom cases principles with a 'using' clause or a registered '@[cases_eliminator]' theorem. The above type neither is an inductive type nor has a registered theorem.\n\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal\n...<truncated>",
"compile_elapsed_s": 27.975,
"tactic_replay": {
"tactic_count": 2,
"successful_tactics": 1,
"first_failure_step": 2,
"goal_trajectory": [
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 10,
"line": 37
},
"start": {
"character": 2,
"line": 37
}
},
"message": "Tactic `cases` failed: major premise type is not an inductive type\n \u2200 t \u2208 Set.range fun s \u21a6 \u22c2 (_ : s \u2208 {x | I \u2264 x \u2227 J \u2264 x}), \u2191s, x \u2208 t\n\nExplanation: the `cases` tactic is for constructor-based reasoning as well as for applying custom cases principles with a 'using' clause or a registered '@[cases_eliminator]' theorem. The above type neither is an inductive type nor has a registered theorem.\n\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x",
"range": {
"end": {
"character": 10,
"line": 37
},
"start": {
"character": 2,
"line": 37
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 39
},
"start": {
"character": 0,
"line": 39
}
},
"message": "'Koethe.KotheConjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 39
},
"start": {
"character": 0,
"line": 39
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 37,
"error_char": 2,
"error_msg": "Tactic `cases` failed: major premise type is not an inductive type\n \u2200 t \u2208 Set.range fun s \u21a6 \u22c2 (_ : s \u2208 {x | I \u2264 x \u2227 J \u2264 x}), \u2191s, x \u2208 t\n\nExplanation: the `cases` tactic is for constructor-based reasoning as well as for applying custom cases principles with a 'using' clause or a registered '@[cases_eliminator]' theorem. The above type neither is an inductive type nor has a registered theorem.\n\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x",
"goal_before_error": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x\n```",
"full_timeline": [
{
"line": 36,
"code": "intro x hx",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\n\u22a2 IsNil (I + J)\n```",
"goal_count": 1,
"ast_complexity": 95
},
{
"line": 37,
"code": "cases hx",
"goal": "```lean\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x\n```",
"goal_count": 1,
"ast_complexity": 116
}
],
"tactic_count": 2,
"successful_tactics": 1,
"first_failure_step": 2,
"cps": 50.0,
"cps_peak_step": 1,
"error_code": "cases hx"
}
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772022449_3269.lean"
],
"comparator_passed": null,
"duration_s": 12.863720417022705,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "KotheConjecture",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Koethe__KotheConjecture",
"category": "research_open",
"endings": [
"end Koethe"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 755,
"source_file": "FormalConjectures/Wikipedia/Koethe.lean",
"title": "KotheConjecture"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772022449_3269.lean:36:89: error: unsolved goals\ncase refine_1\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\nx : R\nhx : x \u2208 I + J\n\u22a2 IsNilpotent x\n'Koethe.KotheConjecture' 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": 0.11,
"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
3
Feb 17, 2026
intro i hi
obtain ⟨a, ha, b, hb, hab⟩ := Submodule.mem_sup.1 hi
have h_ra_nil (r : R) : IsNilpotent (r * a) := hI (r • a) (I.smul_mem r ha)
have h_rb_nil (r : R) : IsNilpotent (r * b) := hJ (r • b) (J.smul_mem r hb)
have h_ar_nil (r : R) : IsNilpotent (a * r) := by
obtain ⟨n, hn_pow⟩ := h_ra_nil r
use n + 1
have h_pow (k : ℕ) : (a * r) ^ k * a = a * (r * a) ^ k := by
induction k with
| zero => simp
| succ k ih => simp [pow_succ, ← mul_assoc, ih, mul_assoc]
rw [pow_succ, ← mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]
have h_br_nil (r : R) : IsNilpotent (b * r) := by
obtain ⟨n, hn_pow⟩ := h_rb_nil r
use n + 1
have h_pow (k : ℕ) : (b * r) ^ k * b = b * (r * b) ^ k := by
induction k with
| zero => simp
| succ k ih => simp [pow_succ, ← mul_assoc, ih, mul_assoc]
rw [pow_succ, ← mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]
rw [← hab]
show IsNilpotent (a + b)
View
{
"agent_telemetry": {
"compilations": 7,
"iterations": 7,
"duration_s": 270.89,
"best_status": "partial",
"best_goals": 3,
"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": 206200,
"completion_tokens": 4386,
"total_tokens": 241198,
"reasoning_tokens": 30612,
"cost_usd": 0.0
},
"reasoning_effort": "",
"response_trace_count": 14
},
"response_trace": [
{
"iteration": 1,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 114.621,
"usage": {
"prompt_tokens": 1904,
"completion_tokens": 520,
"total_tokens": 22044,
"reasoning_tokens": 19620,
"cost_usd": 0.0
},
"thoughts_tokens": 19620,
"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": "intro i hi\nobtain \u27e8a, ha, b, hb, hab\u27e9 := Submodule.mem_sup.1 hi\nhave h_ra_nil (r : R) : IsNilpotent (r * a) := by\n have h := hI (r \u2022 a) (Submodule.smul_mem I r ha)\n rwa [Submodule.smul_def] at h\nhave h_rb_nil (r : R) : IsNilpotent (r * b) := by\n have h := hJ (r \u2022 b) (Submodule.smul_mem J r hb)\n rwa [Submodule.smul_def] at h\nhave h_ar_nil (r : R) : IsNilpotent (a * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_ra_nil r\n use n + 1\n have h_pow : \u2200 k, (a * r) ^ k * a = a * (r * a) ^ k := by\n intro k\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih => rw [pow_succ, \u2190 mul_assoc, ih, mul_assoc, pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow, hn_pow, mul_zero, zero_mul]\nhave h_br_nil (r : R) : IsNilpotent (b * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_rb_nil r\n use n + 1\n have h_pow : \u2200 k, (b * r) ^ k * b = b * (r * b) ^ k := by\n intro k\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih => rw [pow_succ, \u2190 mul_assoc, ih, mul_assoc, pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow, hn_pow, mul_zero, zero_mul]\nrw [\u2190 hab]\nshow IsNilpotent (a + b)",
"compiler_output_preview": "[path]:41:9: error: tactic 'rewrite' failed, did not find instance of the pattern in the target expression\n ?a \u2022 ?S\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nr : R\nh : IsNilpotent (r \u2022 a)\n\u22a2 IsNilpotent (r * a)\n[path]
},
{
"iteration": 2,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 33.533,
"usage": {
"prompt_tokens": 23023,
"completion_tokens": 432,
"total_tokens": 28860,
"reasoning_tokens": 5405,
"cost_usd": 0.0
},
"thoughts_tokens": 5405,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 2,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "partial",
"goals_remaining": 3,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro i hi\nobtain \u27e8a, ha, b, hb, hab\u27e9 := Submodule.mem_sup.1 hi\nhave h_ra_nil (r : R) : IsNilpotent (r * a) := hI (r \u2022 a) (I.smul_mem r ha)\nhave h_rb_nil (r : R) : IsNilpotent (r * b) := hJ (r \u2022 b) (J.smul_mem r hb)\nhave h_ar_nil (r : R) : IsNilpotent (a * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_ra_nil r\n use n + 1\n have h_pow (k : \u2115) : (a * r) ^ k * a = a * (r * a) ^ k := by\n induction k with\n | zero => simp\n | succ k ih => simp [pow_succ, \u2190 mul_assoc, ih, mul_assoc]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nhave h_br_nil (r : R) : IsNilpotent (b * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_rb_nil r\n use n + 1\n have h_pow (k : \u2115) : (b * r) ^ k * b = b * (r * b) ^ k := by\n induction k with\n | zero => simp\n | succ k ih => simp [pow_succ, \u2190 mul_assoc, ih, mul_assoc]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nrw [\u2190 hab]\nshow IsNilpotent (a + b)",
"compiler_output_preview": "[path]:47:18: error: unsolved goals\ncase succ\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nh_ra_nil : \u2200 (r : R), IsNilpotent (r * a)\nh_rb_nil : \u2200 (r : R), IsNilpotent (r * b)\nr : R\nn : \u2115\nhn_pow : (r * a) ^ n = 0\nk : \u2115\nih : (a * r) ^ k * a = a * (r * a) ^ k\n\u22a2 (a * r) ^ k * (a * (r * a)) = a * ((r * \n...<truncated>"
},
{
"iteration": 3,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 13.313,
"usage": {
"prompt_tokens": 29869,
"completion_tokens": 442,
"total_tokens": 32150,
"reasoning_tokens": 1839,
"cost_usd": 0.0
},
"thoughts_tokens": 1839,
"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 i hi\nobtain \u27e8a, ha, b, hb, hab\u27e9 := Submodule.mem_sup.1 hi\nhave h_ra_nil (r : R) : IsNilpotent (r * a) := hI (r \u2022 a) (I.smul_mem r ha)\nhave h_rb_nil (r : R) : IsNilpotent (r * b) := hJ (r \u2022 b) (J.smul_mem r hb)\nhave h_ar_nil (r : R) : IsNilpotent (a * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_ra_nil r\n use n + 1\n have h_pow (k : \u2115) : (a * r) ^ k * a = a * (r * a) ^ k := by\n induction k with\n | zero => simp\n | succ k ih => rw [pow_succ, mul_assoc, ih, \u2190 mul_assoc, \u2190 pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nhave h_br_nil (r : R) : IsNilpotent (b * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_rb_nil r\n use n + 1\n have h_pow (k : \u2115) : (b * r) ^ k * b = b * (r * b) ^ k := by\n induction k with\n | zero => simp\n | succ k ih => rw [pow_succ, mul_assoc, ih, \u2190 mul_assoc, \u2190 pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nrw [\u2190 hab]\nshow IsNilpotent (a + b)",
"compiler_output_preview": "[path]:47:46: error: tactic 'rewrite' failed, did not find instance of the pattern in the target expression\n (a * r) ^ k * a\ncase succ\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nh_ra_nil : \u2200 (r : R), IsNilpotent (r * a)\nh_rb_nil : \u2200 (r : R), IsNilpotent (r * b)\nr : R\nn : \u2115\nhn_pow : (r * a) ^ n =\n...<truncated>"
},
{
"iteration": 4,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 18.193,
"usage": {
"prompt_tokens": 33070,
"completion_tokens": 734,
"total_tokens": 36339,
"reasoning_tokens": 2535,
"cost_usd": 0.0
},
"thoughts_tokens": 2535,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "partial",
"goals_remaining": 3,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro i hi\nobtain \u27e8a, ha, b, hb, hab\u27e9 := Submodule.mem_sup.1 hi\nhave h_ra_nil (r : R) : IsNilpotent (r * a) := hI (r \u2022 a) (Submodule.smul_mem I r ha)\nhave h_rb_nil (r : R) : IsNilpotent (r * b) := hJ (r \u2022 b) (Submodule.smul_mem J r hb)\nhave h_ar_nil (r : R) : IsNilpotent (a * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_ra_nil r\n use n + 1\n have h_pow (k : \u2115) : (a * r) ^ k * a = a * (r * a) ^ k := by\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih =>\n calc (a * r) ^ (k + 1) * a\n _ = (a * r) ^ k * (a * r) * a := by rw [pow_succ]\n _ = (a * r) ^ k * a * (r * a) := by rw [mul_assoc]\n _ = a * (r * a) ^ k * (r * a) := by rw [ih]\n _ = a * ((r * a) ^ k * (r * a)) := by rw [mul_assoc]\n _ = a * (r * a) ^ (k + 1) := by rw [pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nhave h_br_nil (r : R) : IsNilpotent (b * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_rb_nil r\n use n + 1\n have h_pow (k : \u2115) : (b * r) ^ k * b = b * (r * b) ^ k := by\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih =>\n calc (b * r) ^ (k + 1) * b\n _ = (b * r) ^ k * (b * r) * b := by rw [pow_succ]\n _ = (b * r) ^ k * b * (r * b) := by rw [mul_assoc]\n _ = b * (r * b) ^ k * (r * b) := by rw [ih]\n _ = b * ((r * b) ^ k * (r * b)) := by rw [mul_assoc]\n _ = b * (r * b) ^ (k + 1) := by rw [pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nrw [\u2190 hab]\nshow IsNilpotent (a + b)",
"compiler_output_preview": "[path]:50:43: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nh_ra_nil : \u2200 (r : R), IsNilpotent (r * a)\nh_rb_nil : \u2200 (r : R), IsNilpotent (r * b)\nr : R\nn : \u2115\nhn_pow : (r * a) ^ n = 0\nk : \u2115\nih : (a * r) ^ k * a = a * (r * a) ^ k\n\u22a2 (a * r) ^ k * (a * r * a) = (a * r) ^ k * a * (r *\n...<truncated>"
},
{
"iteration": 5,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 9.55,
"usage": {
"prompt_tokens": 37203,
"completion_tokens": 790,
"total_tokens": 38718,
"reasoning_tokens": 725,
"cost_usd": 0.0
},
"thoughts_tokens": 725,
"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": 3,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro i hi\nobtain \u27e8a, ha, b, hb, hab\u27e9 := Submodule.mem_sup.1 hi\nhave h_ra_nil (r : R) : IsNilpotent (r * a) := hI (r \u2022 a) (Submodule.smul_mem I r ha)\nhave h_rb_nil (r : R) : IsNilpotent (r * b) := hJ (r \u2022 b) (Submodule.smul_mem J r hb)\nhave h_ar_nil (r : R) : IsNilpotent (a * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_ra_nil r\n use n + 1\n have h_pow (k : \u2115) : (a * r) ^ k * a = a * (r * a) ^ k := by\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih =>\n calc (a * r) ^ (k + 1) * a\n _ = (a * r) ^ k * (a * r) * a := by rw [pow_succ]\n _ = (a * r) ^ k * (a * r * a) := by rw [mul_assoc]\n _ = (a * r) ^ k * a * (r * a) := by rw [\u2190 mul_assoc]\n _ = a * (r * a) ^ k * (r * a) := by rw [ih]\n _ = a * ((r * a) ^ k * (r * a)) := by rw [mul_assoc]\n _ = a * (r * a) ^ (k + 1) := by rw [pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nhave h_br_nil (r : R) : IsNilpotent (b * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_rb_nil r\n use n + 1\n have h_pow (k : \u2115) : (b * r) ^ k * b = b * (r * b) ^ k := by\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih =>\n calc (b * r) ^ (k + 1) * b\n _ = (b * r) ^ k * (b * r) * b := by rw [pow_succ]\n _ = (b * r) ^ k * (b * r * b) := by rw [mul_assoc]\n _ = (b * r) ^ k * b * (r * b) := by rw [\u2190 mul_assoc]\n _ = b * (r * b) ^ k * (r * b) := by rw [ih]\n _ = b * ((r * b) ^ k * (r * b)) := by rw [mul_assoc]\n _ = b * (r * b) ^ (k + 1) := by rw [pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nrw [\u2190 hab]\nshow IsNilpotent (a + b)",
"compiler_output_preview": "[path]:51:43: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nh_ra_nil : \u2200 (r : R), IsNilpotent (r * a)\nh_rb_nil : \u2200 (r : R), IsNilpotent (r * b)\nr : R\nn : \u2115\nhn_pow : (r * a) ^ n = 0\nk : \u2115\nih : (a * r) ^ k * a = a * (r * a) ^ k\n\u22a2 (a * r) ^ k * (a * r) * a = (a * r) ^ k * a * (r *\n...<truncated>"
},
{
"iteration": 6,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 7.554,
"usage": {
"prompt_tokens": 39582,
"completion_tokens": 734,
"total_tokens": 40685,
"reasoning_tokens": 369,
"cost_usd": 0.0
},
"thoughts_tokens": 369,
"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": 3,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro i hi\nobtain \u27e8a, ha, b, hb, hab\u27e9 := Submodule.mem_sup.1 hi\nhave h_ra_nil (r : R) : IsNilpotent (r * a) := hI (r \u2022 a) (Submodule.smul_mem I r ha)\nhave h_rb_nil (r : R) : IsNilpotent (r * b) := hJ (r \u2022 b) (Submodule.smul_mem J r hb)\nhave h_ar_nil (r : R) : IsNilpotent (a * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_ra_nil r\n use n + 1\n have h_pow (k : \u2115) : (a * r) ^ k * a = a * (r * a) ^ k := by\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih =>\n calc (a * r) ^ (k + 1) * a\n _ = (a * r) ^ k * (a * r) * a := by rw [pow_succ]\n _ = (a * r) ^ k * a * (r * a) := by rw [mul_assoc]\n _ = a * (r * a) ^ k * (r * a) := by rw [ih]\n _ = a * ((r * a) ^ k * (r * a)) := by rw [mul_assoc]\n _ = a * (r * a) ^ (k + 1) := by rw [pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nhave h_br_nil (r : R) : IsNilpotent (b * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_rb_nil r\n use n + 1\n have h_pow (k : \u2115) : (b * r) ^ k * b = b * (r * b) ^ k := by\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih =>\n calc (b * r) ^ (k + 1) * b\n _ = (b * r) ^ k * (b * r) * b := by rw [pow_succ]\n _ = (b * r) ^ k * b * (r * b) := by rw [mul_assoc]\n _ = b * (r * b) ^ k * (r * b) := by rw [ih]\n _ = b * ((r * b) ^ k * (r * b)) := by rw [mul_assoc]\n _ = b * (r * b) ^ (k + 1) := by rw [pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nrw [\u2190 hab]\nshow IsNilpotent (a + b)",
"compiler_output_preview": "[path]:50:43: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nh_ra_nil : \u2200 (r : R), IsNilpotent (r * a)\nh_rb_nil : \u2200 (r : R), IsNilpotent (r * b)\nr : R\nn : \u2115\nhn_pow : (r * a) ^ n = 0\nk : \u2115\nih : (a * r) ^ k * a = a * (r * a) ^ k\n\u22a2 (a * r) ^ k * (a * r * a) = (a * r) ^ k * a * (r *\n...<truncated>"
},
{
"iteration": 7,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 6.714,
"usage": {
"prompt_tokens": 41549,
"completion_tokens": 734,
"total_tokens": 42402,
"reasoning_tokens": 119,
"cost_usd": 0.0
},
"thoughts_tokens": 119,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 7,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "partial",
"goals_remaining": 3,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro i hi\nobtain \u27e8a, ha, b, hb, hab\u27e9 := Submodule.mem_sup.1 hi\nhave h_ra_nil (r : R) : IsNilpotent (r * a) := hI (r \u2022 a) (Submodule.smul_mem I r ha)\nhave h_rb_nil (r : R) : IsNilpotent (r * b) := hJ (r \u2022 b) (Submodule.smul_mem J r hb)\nhave h_ar_nil (r : R) : IsNilpotent (a * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_ra_nil r\n use n + 1\n have h_pow (k : \u2115) : (a * r) ^ k * a = a * (r * a) ^ k := by\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih =>\n calc (a * r) ^ (k + 1) * a\n _ = (a * r) ^ k * (a * r) * a := by rw [pow_succ]\n _ = (a * r) ^ k * a * (r * a) := by rw [mul_assoc]\n _ = a * (r * a) ^ k * (r * a) := by rw [ih]\n _ = a * ((r * a) ^ k * (r * a)) := by rw [mul_assoc]\n _ = a * (r * a) ^ (k + 1) := by rw [pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nhave h_br_nil (r : R) : IsNilpotent (b * r) := by\n obtain \u27e8n, hn_pow\u27e9 := h_rb_nil r\n use n + 1\n have h_pow (k : \u2115) : (b * r) ^ k * b = b * (r * b) ^ k := by\n induction k with\n | zero => rw [pow_zero, pow_zero, one_mul, mul_one]\n | succ k ih =>\n calc (b * r) ^ (k + 1) * b\n _ = (b * r) ^ k * (b * r) * b := by rw [pow_succ]\n _ = (b * r) ^ k * b * (r * b) := by rw [mul_assoc]\n _ = b * (r * b) ^ k * (r * b) := by rw [ih]\n _ = b * ((r * b) ^ k * (r * b)) := by rw [mul_assoc]\n _ = b * (r * b) ^ (k + 1) := by rw [pow_succ]\n rw [pow_succ, \u2190 mul_assoc, h_pow n, hn_pow, mul_zero, zero_mul]\nrw [\u2190 hab]\nshow IsNilpotent (a + b)",
"compiler_output_preview": "[path]:50:43: error: unsolved goals\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nh_ra_nil : \u2200 (r : R), IsNilpotent (r * a)\nh_rb_nil : \u2200 (r : R), IsNilpotent (r * b)\nr : R\nn : \u2115\nhn_pow : (r * a) ^ n = 0\nk : \u2115\nih : (a * r) ^ k * a = a * (r * a) ^ k\n\u22a2 (a * r) ^ k * (a * r * a) = (a * r) ^ k * a * (r *\n...<truncated>"
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"[path]
],
"duration_s": 7.02988076210022,
"goals_remaining": 3,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"axioms_check_name": "KotheConjecture",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Koethe__KotheConjecture",
"category": "research_open",
"endings": [
"end Koethe"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 755,
"source_file": "FormalConjectures/Wikipedia/Koethe.lean",
"title": "KotheConjecture"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:47:18: error: unsolved goals\ncase succ\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nh_ra_nil : \u2200 (r : R), IsNilpotent (r * a)\nh_rb_nil : \u2200 (r : R), IsNilpotent (r * b)\nr : R\nn : \u2115\nhn_pow : (r * a) ^ n = 0\nk : \u2115\nih : (a * r) ^ k * a = a * (r * a) ^ k\n\u22a2 (a * r) ^ k * (a * (r * a)) = a * ((r * a) ^ k * (r * a))\n[path]:55:18: error: unsolved goals\ncase succ\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nh_ra_nil : \u2200 (r : R), IsNilpotent (r * a)\nh_rb_nil : \u2200 (r : R), IsNilpotent (r * b)\nh_ar_nil : \u2200 (r : R), IsNilpotent (a * r)\nr : R\nn : \u2115\nhn_pow : (r * b) ^ n = 0\nk : \u2115\nih : (b * r) ^ k * b = b * (r * b) ^ k\n\u22a2 (b * r) ^ k * (b * (r * b)) = b * ((r * b) ^ k * (r * b))\n[path]:36:89: error: unsolved goals\ncase intro.intro.intro.intro\nR : Type u_1\ninst\u271d : Ring R\nI J : Ideal R\nhI : IsNil I\nhJ : IsNil J\ni : R\nhi : i \u2208 I + J\na : R\nha : a \u2208 I\nb : R\nhb : b \u2208 J\nhab : a + b = i\nh_ra_nil : \u2200 (r : R), IsNilpotent (r * a)\nh_rb_nil : \u2200 (r : R), IsNilpotent (r * b)\nh_ar_nil : \u2200 (r : R), IsNilpotent (a * r)\nh_br_nil : \u2200 (r : R), IsNilpotent (b * r)\n\u22a2 IsNilpotent (a + b)\n[path]:47:37: warning: This simp argument is unused:\n \u2190 mul_assoc\n\nHint: Omit it from the simp argument list.\n simp [pow_succ, \u2190\u0335 \u0335m\u0335u\u0335l\u0335_\u0335a\u0335s\u0335s\u0335o\u0335c\u0335,\u0335 \u0335ih, mul_assoc]\n\nNote: This linter can be disabled with `set_option linter.unusedSimpArgs false`\n[path]:47:50: warning: This simp argument is unused:\n ih\n\nHint: Omit it from the simp argument list.\n simp [pow_succ, \u2190 mul_assoc, i\u0335h\u0335,\u0335 \u0335mul_assoc]\n\nNote: This linter can be disabled with `set_option linter.unusedSimpArgs false`\n[path]:55:37: warning: This simp argument is unused:\n \u2190 mul_assoc\n\nHint: Omit it from the simp argument list.\n simp [pow_succ, \u2190\u0335 \u0335m\u0335u\u0335l\u0335_\u0335a\u0335s\u0335s\u0335o\u0335c\u0335,\u0335 \u0335ih, mul_assoc]\n\nNote: This linter can be disabled with `set_option linter.unusedSimpArgs false`\n[path]:55:50: warning: This simp argument is unused:\n ih\n\nHint: Omit it from the simp argument list.\n simp [pow_succ, \u2190 mul_assoc, i\u0335h\u0335,\u0335 \u0335mul_assoc]\n\nNote: This linter can be disabled with `set_option linter.unusedSimpArgs false`\n'Koethe.KotheConjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
Gemini 3 Flash Preview
Partial
1
Feb 14, 2026
rcases hJ y hy with ⟨n, hn⟩
use n
simp [f, hn]
have hfz : IsNilpotent (f (x + y)) := by
have hfx : f x = 0 := by
simp [f, Quotient.eq_zero_iff_mem.mpr hx]
rw [map_add, hfx, zero_add]
exact hfy
rcases hfz with ⟨n, hn⟩
replace hn : (x + y)^n ∈ I := by
rw [← Quotient.eq_zero_iff_mem]
exact hn
rcases hI ((x + y)^n) hn with ⟨m, hm⟩
use n * m
rw [pow_mul, hm]
