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conjecture34
Specification
WOWII [Conjecture 34](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) Let `path(G)` be the floor of the average distance of a connected graph `G`. Then `path(G) ≥ ceil( distavg(G, center) + distavg(G, maxEccentricityVertices G) )`.
Lean 4 Statement
theorem conjecture34 [Nontrivial α] (G : SimpleGraph α) (h_conn : G.Connected) :
Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) ≤ (path G : ℤ)
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| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Partial | 1 | Feb 25, 2026 |
have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) ≤ (path G : ℤ) := ?_
exact h
|
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"proof_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := ?_\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772032314_1731.lean:18:99: error: unsolved goals\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"line": 18,
"code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := ?_",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
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"code": "exact h",
"goal": "```lean\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\nh : \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```\n---\n```lean\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
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}
]
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by\n simpa using (SimpleGraph.conjecture34 (G := G) h_conn)\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772032385_9601.lean:20:17: error(lean.unknownIdentifier): Unknown constant `SimpleGraph.conjecture34`\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 51.405,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 1,
"first_failure_step": 2,
"goal_trajectory": [
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"code": "lean.unknownIdentifier",
"fullRange": {
"end": {
"character": 41,
"line": 19
},
"start": {
"character": 17,
"line": 19
}
},
"message": "Unknown constant `SimpleGraph.conjecture34`",
"range": {
"end": {
"character": 41,
"line": 19
},
"start": {
"character": 17,
"line": 19
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"message": "'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 19,
"error_char": 17,
"error_msg": "Unknown constant `SimpleGraph.conjecture34`",
"goal_before_error": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"full_timeline": [
{
"line": 18,
"code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 19,
"code": "simpa using (SimpleGraph.conjecture34 (G := G) h_conn)",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 20,
"code": "exact h",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\nh : \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 280
}
],
"tactic_count": 3,
"successful_tactics": 1,
"first_failure_step": 2,
"cps": 33.33333333333333,
"cps_peak_step": 1,
"error_code": "simpa using (SimpleGraph.conjecture34 (G := G) h_conn)"
}
},
{
"iteration": 5,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772032436-fhAKMV3eX4coZE0IWl7M",
"api_elapsed_s": 3.032,
"usage": {
"prompt_tokens": 7574,
"completion_tokens": 176,
"total_tokens": 7750,
"reasoning_tokens": 104,
"cost_usd": 0.0076545
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Exploring theorem naming via namespace**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Exploring theorem naming via namespace**"
},
{
"id": "rs_0975e8a379ef297601699f11b4ec048195a16c1a265852bca9",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "gAAAAABpnxG3rB9lshA_YzbvV3-umr6WygbcVdId4usi1Fwjw1zjqGNO1O1eEw80zf1QJ8tLOJ5NzvFEgAab5WRtBazi6sAgtgv0lHFIxVRkQVpEB3Cg-Erz32GbsbqNSHR3ZGADvjyTxtGhk8RtFocaMyy2AGVkvms5CNCHWOaO8pYfFOdR04lxAadDZLxNjQ2tZ1pCA-Sffvcyjjz9NVXqv0hiuMKNDjdm-AD91j0MDBdc4XfSoAx4EMG-dxGis8GhFFMgpx-kxY_icaO2yRn8m2aFyrbpP0x77XiNYWczJ5qCi0XYyvanKIt45M3mGiM1FIPaUu3TChR3PmbrZhwrTdxnrUuI20pNMDbQOh1eJZLjYlBdjJC4auYXKPv73UPGG3q-B57EdB9AwGvc84JDFTtPG8p_WNlvqxLZ2By9bHGgmIuOEzciOKyUp0DGcHX-zNrQBWfsMMtZeqNwzZGGA7EfyVp44r5eh7w4cLS9OiKScFBAere_ei76urivw3qDQRCx6_N-RN3_bG94-3-KJu5UIEJv5xGr27JUp00fZRpZPIrB2K7SngdADb0NLiG4UHkekpWHQ_u5Nmr9-WIUWSUNgiPBEXf8Q7FMEagBvUeDpgUA_XewJFtTsExWi9RC0MQR5Z19FABtlfut2Zjd2mAUxPfRE0FnphcjKlDDLEscYWCSUAKE9wmJJVyObalvQ5uwT1jZvZFA5psZk9FTOIT68E8iE6kf1Zm41iPP0jHpXUH4IRcNvEvMvSAAXF7QW2Aze8FA91hB4VjQYEGqtaCp9r4LryovGNYwU_GWaLEJJTK0-g7fJgMnSENhVORY-FS4-MRzkHvHV0xPTFz1dCGnmgdSnOuabR55Kre6kYQfRYpIw4teAiVXbaQK6cueyeId7Ibbskg55qyrja5Wp--tLVZbNLqIIn7V8LxgZspO9lo-g3_VfPnEhyM4_Td-LDeraEcenz118OlsqX5zBrCdf81fJcTdbs9bxOhay6OjlULyJ6kx7HR-GpYG6VRNy1hTKkFtwzYA2_bFMSn0geQbr9v3pEKUzVQx4AmfR4hvR0RW7hwF4UFftwILadm6ZvwUjmnNE6CJMouQiclDbfrM7OtG9ALji9v33kcCdMmNvXovbYYG5hkl_GMdfXlkbtjknMC3VPii3N0u-Mf7VYvXkZds-1skKs5Nr_datY1Xf5tlcAEXZgTJKtAoKpv0IayulI91SrTkxgIRwi1TJ68KcIhn7ONBXDlP02npQmDVKNZT5PpsVuCrnDdW3_XR6-rxZSyB8UEHYs4ey2ipbYy4wATSPXGJYv9oKnfSwbDZ-GUt2e93e5j3Rz8_LiwUXC0y42FArE_oc2VqvJQWpP_2oKR9-FMBzJyFoP2NsSRA-fkPMD-GkV-oeYaA5NFebnYlTM_WJK5qu1Oy-LjhUTUb7_hSuswfbPvtV08T88BgdCjXYj77qZJAe1qHdoNCPZA3vF6O"
}
]
},
{
"iteration": 5,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by\n simpa using h_conn.conjecture34\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772032439_8836.lean:20:23: error(lean.invalidField): Invalid field `conjecture34`: The environment does not contain `SimpleGraph.Connected.conjecture34`, so it is not possible to project the field `conjecture34` from an expression\n h_conn\nof type `G.Connected`\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 45.243,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 1,
"first_failure_step": 2,
"goal_trajectory": [
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"code": "lean.invalidField",
"fullRange": {
"end": {
"character": 35,
"line": 19
},
"start": {
"character": 23,
"line": 19
}
},
"message": "Invalid field `conjecture34`: The environment does not contain `SimpleGraph.Connected.conjecture34`, so it is not possible to project the field `conjecture34` from an expression\n h_conn\nof type `G.Connected`",
"range": {
"end": {
"character": 35,
"line": 19
},
"start": {
"character": 23,
"line": 19
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"message": "'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 19,
"error_char": 23,
"error_msg": "Invalid field `conjecture34`: The environment does not contain `SimpleGraph.Connected.conjecture34`, so it is not possible to project the field `conjecture34` from an expression\n h_conn\nof type `G.Connected`",
"goal_before_error": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"full_timeline": [
{
"line": 18,
"code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 19,
"code": "simpa using h_conn.conjecture34",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 20,
"code": "exact h",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\nh : \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 280
}
],
"tactic_count": 3,
"successful_tactics": 1,
"first_failure_step": 2,
"cps": 33.33333333333333,
"cps_peak_step": 1,
"error_code": "simpa using h_conn.conjecture34"
}
},
{
"iteration": 6,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772032484-YGIZZRfhz7FcKVQiFE9d",
"api_elapsed_s": 4.563,
"usage": {
"prompt_tokens": 10216,
"completion_tokens": 217,
"total_tokens": 10433,
"reasoning_tokens": 152,
"cost_usd": 0.0193032
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Testing tactic-based proof search**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Testing tactic-based proof search**"
},
{
"id": "rs_0975e8a379ef297601699f11e55e808195927b1f99ad036388",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "gAAAAABpnxHpzrHIzNGlvasujMgR7lI7CTUD75jYZz1BDqokYfihMyLiOC6d9BgldlHu7pBe95D7ua1NLSdCKMjF1b_cSR3IHOkzzTL2j7E-5uPE5B6rSHkpCWHEs9SnOnCyYmHv_1JpxYgUIRh5o02qTXdsT6uILkJ2suwjXFtOHZdLXG5eqbSFmiP0bes-h5gqeDZznptEfBaMkPjrjjRn6-8s3ImZT_vLh1N2BmlYuZxRWRJEDf2H7wwadhWHgIMbb-vR2IvgAuj0okFAFcgtpAcV9IPBI1Ye8KnkRfUWVknzAWWpGo2hMzqP1hmNqrVmOC1OVFg4OUwysOtLtoJeh3n88fB577oSOt4JlHlqxf205r1nwqJMzGcmkVsY3gBD6-P8CetpcztvHFtitPqrVtrvj9nxHnHFwfJ2FjkF_HTtYBVmAOHyQSUt_OjbcBjvNAc-K7CyIynuSShtBwdn0J5lrJP5W3l8XpCSTKjuIF1c35F7rYUyGUkdLkf8d9M1yQc6zmZJvXEl2Zb0hZU_pkiTKxeI235FheWQwZyuW7uK5ciPMOAHgJbJkkudZhCgWluvEz3sSqZI6LPfm5zEBhVFC-0qkF0OtiJQw8-9vnfjG9Im3s-NheWMcSMcP4gTxpf-k_FhGk9yKyvfwLMlOvj17dxdAWVv-8lbGiuZ0n5dup6MLvtqizh0TLs9Dno-XHUw9Fzl_4p0yqB17K7e-RqLbQQZtZSQItMGCc8UbPOsdoF0ySN1R1XDippfwBdtB3_jsC7AfzzCBpo7T3CO4w9qPBSYfoEd0iUN35EwnAR26hJr9GCXyf1AtrdXm0fiAMF8PIIXCsu59ivr4J5QmP6J8RPI-ECRl087wSjmuxxgbwK_mtFFA3Ku6hja-dcjFdYHHG_DX2eyB6MpD10k1w-5xbV4fRfTFurzn1o_xSC9oFg6cwepvwV0W-KXIDMqs4LfIThI2Ji8TWD-ilk7jI9hRGiSwNu9VuCp8P2CTyGA6_BJ_AFUj0urdhQGUKxKW_ZjtZnetBO-Sdl4Sgnqwypy8Z5ZPEFKpPg_zWOGWXA29LQm1cmWzseP_K6_6H5U2uAkWGwAnc1-HrQAthOeOZYg12N1IR4Dhn7QCtWc4ALBvu4I_cFp5sHrC6Ao1TVNsW3j7NqtuRAMwZgMaBCaEZ-TIXgk0M9WH0oyq-NSHAfXaVKwdTEaq2xmgR8CaL1W7h_AoieIrvvGgiIJlMSKmsNMbh5i0DUtgy9ll89ubRr4MA6L2GT6YjCqyRf7aZyvE0hJroU_aH4AgY9nIzEOw-RQF0ASnSKwXToHWRC9MslYcTjHOosSKj7xBd9WAAmA_utoPr5ocRQKanf_gm-6isGoV1v57WBfXJ9R_ddiBTNEMC3lOGz6syjXwo-Z7foVHxCk4bw5XiUfOGEE_VG8uzJ0m6y0CAa4MyOT9Wh8Udxvr0tta0DEvtb2e5PLafRGq_BBc6-kwmY9Cyg1ItUnfHm1IKmqt01hbi5TCNdrpsnME_4u24HGM5rs-qRkbZpSkldJhgsr253HMZwVRPWopmrISgrzIxPLUWiWrQ8u7zx9GHmN_90mzIO9UklOcZjm8yenQx6jTknZZ2h1mrl12eXZ2Z7bNLnXZ2eEWblIObrPwrPw6gBNRwpckUwoas8Hi8F9mZkgDKqrq_9lpkjyAsHeUwy0j6yFrDM9FlHfSsVbHSo2F8M="
}
]
},
{
"iteration": 6,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by\n aesop\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772032489_2609.lean:20:4: error: tactic 'aesop' failed, made no progress\nInitial goal:\n \u03b1 : Type u_1\n inst\u271d\u00b2 : Fintype \u03b1\n inst\u271d\u00b9 : DecidableEq \u03b1\n inst\u271d : Nontrivial \u03b1\n G : SimpleGraph \u03b1\n h_conn : G.Connected\n \u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 42.666,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 1,
"first_failure_step": 2,
"goal_trajectory": [
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 9,
"line": 19
},
"start": {
"character": 4,
"line": 19
}
},
"message": "tactic 'aesop' failed, made no progress\nInitial goal:\n \u03b1 : Type u_1\n inst\u271d\u00b2 : Fintype \u03b1\n inst\u271d\u00b9 : DecidableEq \u03b1\n inst\u271d : Nontrivial \u03b1\n G : SimpleGraph \u03b1\n h_conn : G.Connected\n \u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path",
"range": {
"end": {
"character": 9,
"line": 19
},
"start": {
"character": 4,
"line": 19
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"message": "'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 19,
"error_char": 4,
"error_msg": "tactic 'aesop' failed, made no progress\nInitial goal:\n \u03b1 : Type u_1\n inst\u271d\u00b2 : Fintype \u03b1\n inst\u271d\u00b9 : DecidableEq \u03b1\n inst\u271d : Nontrivial \u03b1\n G : SimpleGraph \u03b1\n h_conn : G.Connected\n \u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path",
"goal_before_error": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"full_timeline": [
{
"line": 18,
"code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 19,
"code": "aesop",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 20,
"code": "exact h",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\nh : \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 280
}
],
"tactic_count": 3,
"successful_tactics": 1,
"first_failure_step": 2,
"cps": 33.33333333333333,
"cps_peak_step": 1,
"error_code": "aesop"
}
},
{
"iteration": 7,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772032531-XMEsVVI6wd4WcCztjHaY",
"api_elapsed_s": 18.847,
"usage": {
"prompt_tokens": 13474,
"completion_tokens": 818,
"total_tokens": 14292,
"reasoning_tokens": 751,
"cost_usd": 0.0334187
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Testing inequality proof via simp**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Testing inequality proof via simp**"
},
{
"id": "rs_0975e8a379ef297601699f1214a790819599d6348e78426175",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": 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OCVq7IDNc4ZKvfGdjRF2tsFAxewnfZHpwUwe1mOkOHJ4Diz5fxOdFx8hnpZ9WE2DoJOd0-LuS1VQc4KytCKRk3KRzs8eLASzFTU9iXfcwWJPvrbailEMplR9WFd29Q1BlWy2xY5KKPyhtgET7roiRDTzPWvGmzdv08bob3mzRs_-lacA="
}
]
},
{
"iteration": 7,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by\n simp [path]\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772032550_6424.lean:19:106: error: unsolved goals\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 if G.Connected then \u2191\u230aG.averageDistance\u230b\u208a else 0\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 31.156,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 15,
"line": 19
},
"start": {
"character": 106,
"line": 18
}
},
"leanTags": [
1
],
"message": "unsolved goals\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 if G.Connected then \u2191\u230aG.averageDistance\u230b\u208a else 0",
"range": {
"end": {
"character": 0,
"line": 19
},
"start": {
"character": 106,
"line": 18
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"message": "'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 18,
"error_char": 106,
"error_msg": "unsolved goals\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 if G.Connected then \u2191\u230aG.averageDistance\u230b\u208a else 0",
"goal_before_error": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"full_timeline": [
{
"line": 18,
"code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 19,
"code": "simp [path]",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 20,
"code": "exact h",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\nh : \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 280
}
],
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by"
}
},
{
"iteration": 8,
"event": "search_mathlib",
"query": "averageDistance",
"results_count": 0
},
{
"iteration": 9,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772032587-FfdoIYReYSlANbcnBYEg",
"api_elapsed_s": 3.555,
"usage": {
"prompt_tokens": 17625,
"completion_tokens": 168,
"total_tokens": 17793,
"reasoning_tokens": 98,
"cost_usd": 0.00577815
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Testing simplification and compilation**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Testing simplification and compilation**"
},
{
"id": "rs_0975e8a379ef297601699f124c56cc819585dfbdc201cc5086",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "gAAAAABpnxJP3AOpOO-xk5YvJCC-W38N6MiUKXRhHrBCfH5EURU2nXAaU1D-BaZaNBF7iWyj8LV42bZPj2lkqpa3UUnoKeLp2XYQqSXPB-5DQU4hEroaoSL94WoOl1Pl8aZ0MuBbJ0sCZ2TirKo3IpNIc3f_MPYiGgAbJ790-0aLjKw_zFmCX1SRCZSOoVyZeoG2IULd8ypD-pM_HnLlKWjEkssTnbv1vv_sn9sj-Dy7jLaTtEqJThew14Oms5lEksk_ogrReZgkE5RmJOUOcHr5PkDbflBbaTsO5QEwcQ26rJvRREjkI2uqdpFDI4j4alKIpCYV-spXQrSjrgtzxqUL5AAQJZCqXJ6xJsvfg4QCGmjdeCgmT8DkXOGfFW9NSpTwunH2s7exRQh4QSbl0eGMJQWvjLFPdhGVNgiTr9_emt2NBXiBdc6HDNx7V06kT_OHvi18CUScD1q_xgVVyon8pAtsgcBqLFrcL1Nof6J8_89o3NnY9OijihKaePQMTW8YfQEJbACcAJRWy1nwvocuRF8QaDh7mubrN1vBNLb9vtVD_ezLkI2XYBf21vZhlVdFYGz7MlogDhbAl_MbUGdQ0ZJ86UZOebo3a815-oKKCHgDAvxL65qYkS6BqDGmjnxQjffagOGjFInD4neWegMXNnRoAmhFK98HdeVkm4xacDaglLSclG6OUugJDLlHLp7hLPVddWhUsQlvgMscLUK9kFEiMVwKfjEuzrbG4eFwOM53CTKFfXL43ZPFzbhJPX7OLbpUFp85OKnLCBtONKfg1pxHmxpSFmFVT5Bx2V-4fJV81C0w1Pb4ojRv7jj0fSShivdbEAn-_RuRj9V7_-fsqgAeHs98bClBj0WB5fVfq8sG9PdkAD3gxcHod0GpqSvX5Lz63OpTdiCTHRVlk-3rjKkpTCOKUv52kyRcin_6fJ3spPKErD2nXbqBRNsA3nEAUUFTeJedCdFUFa84RfQISO0KNfErnzekD6jG7KvSl3s7Ers6G8P9nTz8NTDQbX4juT_JpfVcSpIcu2g7H-FL7QS7aN62F76akIKn3FEbXTXKC-A7L6pK940R7mIeIXx6RKW6t35t8hBKtk3jYq1oCouabDxS-e4s-wiHjCzU95jHIWVj5zaKwHErngr0gzaOzq1RUhBF9sMPtUXbAOcR_q1SNS-Rm3JIF9rhvZiEOKjSFID5tuG5K1-YaEit64O3rc74seSbD7GYLKMcXJWWlkivuKNKUzCJgNf--DjOuauOaKYrXVlDYr3Y_kgz-b5a4vGDDxsRGI4ImwCuXn9DAOry9TUFskYdf-ELq9tIJ4598sab8x_iGAvaVAQKhAdYqlR0SirK7l2zYfgitj6jc8yPZrR3nw=="
}
]
},
{
"iteration": 9,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by\n simp [path, h_conn]\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772032591_5071.lean:19:106: error: unsolved goals\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191\u230aG.averageDistance\u230b\u208a\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 38.688,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 23,
"line": 19
},
"start": {
"character": 106,
"line": 18
}
},
"leanTags": [
1
],
"message": "unsolved goals\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191\u230aG.averageDistance\u230b\u208a",
"range": {
"end": {
"character": 0,
"line": 19
},
"start": {
"character": 106,
"line": 18
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"message": "'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 18,
"error_char": 106,
"error_msg": "unsolved goals\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191\u230aG.averageDistance\u230b\u208a",
"goal_before_error": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"full_timeline": [
{
"line": 18,
"code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 19,
"code": "simp [path, h_conn]",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 20,
"code": "exact h",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\nh : \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 280
}
],
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by"
}
},
{
"iteration": 10,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772032630-UFEkW7FmBHsuGONuVhpy",
"api_elapsed_s": 15.741,
"usage": {
"prompt_tokens": 20925,
"completion_tokens": 688,
"total_tokens": 21613,
"reasoning_tokens": 605,
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},
"openrouter_retries": 0,
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"used_tool_call": true,
"reasoning_text": "**Exploring inequality approach**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Exploring inequality approach**"
},
{
"id": "rs_0975e8a379ef297601699f1276b6fc8195b5ba55f4c6a0c921",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
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}
]
},
{
"iteration": 10,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by\n simp [path, h_conn, distavg, graphCenter, maxEccentricityVertices]\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772032645_3137.lean:19:106: error: unsolved goals\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308(if 0 < Fintype.card \u03b1 then (\u2211 x, \u2191(G.distToSet x {v | G.eccentricity v = G.minEccentricity})) / \u2191(Fintype.card \u03b1)\n else 0) +\n if 0 < Fintype.card \u03b1 then\n (\u2211 x, \u2191(G.distToSet x {v | G.eccentricity v = G.maxEccentricity})) / \u2191(Fintype.card \u03b1)\n else 0\u2309 \u2264\n \u2191\u230aG.averageDi\n...<truncated>",
"compile_elapsed_s": 44.692,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 70,
"line": 19
},
"start": {
"character": 106,
"line": 18
}
},
"leanTags": [
1
],
"message": "unsolved goals\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308(if 0 < Fintype.card \u03b1 then (\u2211 x, \u2191(G.distToSet x {v | G.eccentricity v = G.minEccentricity})) / \u2191(Fintype.card \u03b1)\n else 0) +\n if 0 < Fintype.card \u03b1 then\n (\u2211 x, \u2191(G.distToSet x {v | G.eccentricity v = G.maxEccentricity})) / \u2191(Fintype.card \u03b1)\n else 0\u2309 \u2264\n \u2191\u230aG.averageDistance\u230b\u208a",
"range": {
"end": {
"character": 0,
"line": 19
},
"start": {
"character": 106,
"line": 18
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"message": "'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 22
},
"start": {
"character": 0,
"line": 22
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 18,
"error_char": 106,
"error_msg": "unsolved goals\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308(if 0 < Fintype.card \u03b1 then (\u2211 x, \u2191(G.distToSet x {v | G.eccentricity v = G.minEccentricity})) / \u2191(Fintype.card \u03b1)\n else 0) +\n if 0 < Fintype.card \u03b1 then\n (\u2211 x, \u2191(G.distToSet x {v | G.eccentricity v = G.maxEccentricity})) / \u2191(Fintype.card \u03b1)\n else 0\u2309 \u2264\n \u2191\u230aG.averageDistance\u230b\u208a",
"goal_before_error": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"full_timeline": [
{
"line": 18,
"code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 19,
"code": "simp [path, h_conn, distavg, graphCenter, maxEccentricityVertices]",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 202
},
{
"line": 20,
"code": "exact h",
"goal": "```lean\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\nh : \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n```",
"goal_count": 1,
"ast_complexity": 280
}
],
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "have h : Int.ceil (distavg G (graphCenter G) + distavg G (maxEccentricityVertices G)) \u2264 (path G : \u2124) := by"
}
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772032690_6407.lean"
],
"comparator_passed": null,
"duration_s": 15.087335109710693,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "conjecture34",
"axioms_check_skipped": false,
"benchmark_id": "WrittenOnTheWallII__GraphConjecture34__conjecture34",
"category": "research_open",
"endings": [
"end WrittenOnTheWallII.GraphConjecture34"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 149,
"source_file": "FormalConjectures/WrittenOnTheWallII/GraphConjecture34.lean",
"title": "conjecture34"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772032690_6407.lean:18:99: error: unsolved goals\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u2308G.distavg G.graphCenter + G.distavg G.maxEccentricityVertices\u2309 \u2264 \u2191G.path\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' 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\"}",
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|
|
| Gemini 3 Flash Agent v8 | Failed | -- | Feb 17, 2026 |
let n := Fintype.card α
have h_n : 1 < n := Fintype.one_lt_card
set C := graphCenter G
set M := maxEccentricityVertices G
let d_avg_C := distavg G C
let d_avg_M := distavg G M
let p_G := path G
have h_C_nonempty : C.Nonempty := graphCenter_nonempty G h_conn
have h_M_nonempty : M.Nonempty := maxEccentricityVertices_nonempty G h_conn
unfold distavg path
simp only
let sum_all := ∑ i, ∑ j, (G.dist i j : ℚ)
let denom := (n : ℚ) * (n - 1)
let avg_dist := sum_all / denom
have h_denom_pos : 0 < denom := by
have hn : (n : ℚ) > 1 := by exact_mod_cast h_n
nlinarith
suffices (Int.ceil (d_avg_C + d_avg_M) : ℤ) ≤ Int.floor avg_dist by
exact_mod_cast this
apply Int.ceil_le.mpr
have h_floor_le : (Int.floor avg_dist : ℚ) ≤ avg_dist := Int.floor_le avg_dist
apply le_trans _ h_floor_le
let f v := (∑ u, (G.dist v u : ℚ)) / (n : ℚ)
have h_davg_C : d_avg_C = (∑ v in C, f v) / (C.card : ℚ) := by rfl
have h_davg_M : d_avg_M = (∑ v in M, f v) / (M.card : ℚ) := by rfl
have h_sum_all_f : sum_all / (n : ℚ) = ∑ v, f v := by
simp [f, sum_division_by_constant, sum_comm]
have h_avg_dist_f : avg_dist = (∑ v, f v) / (n - 1 : ℚ) := by
rw [← h_sum_all_f]
field_simp [h_denom_pos]
ring
let min_f := (C.card : ℚ)⁻¹ * ∑ v in C, f v
let max_f := (M.card : ℚ)⁻¹ * ∑ v in M, f v
calc d_avg_C + d_avg_M = min_f + max_f := by rfl
_ ≤ avg_dist := ?_
linarith [h_n]
|
View{
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"proof_code": "intro h_alpha\nset C := graphCenter G\nset M := maxEccentricityVertices G\nlet n := Fintype.card \u03b1\nlet p := (path G : \u211a)\nlet d_avg_C := distavg G C\nlet d_avg_M := distavg G M\nhave h_n_pos : 0 < n := Fintype.card_pos\nset lhs := Int.ceil (d_avg_C + d_avg_M)\nset rhs := (path G : \u2124)\nsuffices (d_avg_C + d_avg_M : \u211a) \u2264 (path G : \u211a) by\n rw [\u2190 Int.cast_le (\u03b1 := \u211a)]\n have h_floor : (path G : \u211a) = Int.floor (\u2211 i, \u2211 j, (G.dist i j : \u211a) / (n * (n - 1) : \u211a)) := by rfl\n exact Int.ceil_le.mpr (by linarith)\nunfold distavg\nunfold path\nsimp only\nlet total_dist := \u2211 i, \u2211 j, (G.dist i j : \u211a)\nlet avg_dist := total_dist / (n * (n - 1) : \u211a)\nlet sum_dist_C := (\u2211 v in C, \u2211 u, (G.dist v u : \u211a)) / (C.card : \u211a)\nlet sum_dist_M := (\u2211 v in M, \u2211 u, (G.dist v u : \u211a)) / (M.card : \u211a)\nsuffices sum_dist_C + sum_dist_M \u2264 Int.floor avg_dist by\n exact (le_trans (by linarith) (by linarith))\nobtain \u27e8v_c, h_vc_mem\u27e9 := (graphCenter_nonempty G h_conn).nonempty\nobtain \u27e8v_m, h_vm_mem\u27e9 := (maxEccentricityVertices_nonempty G h_conn).nonempty\nhave h_card_C : (C.card : \u211a) > 0 := by\n norm_cast\n exact Finset.card_pos.mpr (graphCenter_nonempty G h_conn).nonempty\nhave h_card_M : (M.card : \u211a) > 0 := by\n norm_cast\n exact Finset.card_pos.mpr (maxEccentricityVertices_nonempty G h_conn).nonempty\napply Int.ceil_le.mpr\nsorry",
"compiler_output_preview": "\nBanned tokens found: sorry"
},
{
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"proof_code": "intro h_alpha\nset C := graphCenter G\nset M := maxEccentricityVertices G\nlet n := Fintype.card \u03b1\nlet d_avg_C := distavg G C\nlet d_avg_M := distavg G M\nlet p_G := path G\n\nhave h_n_gt_one : 1 < n := Fintype.one_lt_card\nhave h_n_pos : 0 < n := by linarith\nhave h_C_nonempty : C.Nonempty := graphCenter_nonempty G h_conn\nhave h_M_nonempty : M.Nonempty := maxEccentricityVertices_nonempty G h_conn\n\nhave h_card_C_pos : 0 < (C.card : \u211a) := by\n rw [Nat.cast_pos]\n exact card_pos.mpr h_C_nonempty\nhave h_card_M_pos : 0 < (M.card : \u211a) := by\n rw [Nat.cast_pos]\n exact card_pos.mpr h_M_nonempty\n\nunfold distavg path at *\nsimp only at *\n\nlet avg_dist := (\u2211 i, \u2211 j, (G.dist i j : \u211a)) / (n * (n - 1) : \u211a)\nlet avg_C := (\u2211 v in C, \u2211 u, (G.dist v u : \u211a)) / (C.card : \u211a) / n\nlet avg_M := (\u2211 v in M, \u2211 u, (G.dist v u : \u211a)) / (M.card : \u211a) / n\n\nsuffices (avg_C + avg_M : \u211a) \u2264 (p_G : \u211a) by\n rw [\u2190 Int.cast_le (\u03b1 := \u211a)]\n apply Int.ceil_le.mpr\n calc (distavg G C + distavg G M : \u211a) \u2264 (p_G : \u211a) := by exact this\n _ \u2264 (Int.floor avg_dist : \u211a) := by rfl\n\napply le_trans ?_\n\u00b7 exact Int.floor_le avg_dist\n\u00b7 field_simp\n sorry; ring_nf; sorry\n\n-- Note: The goal is to relate vertex-average distances (from C and M) to the global average distance.\n-- In a connected graph, center vertices minimize sum of distances.\n-- Eccentricity max vertices are at the periphery.\n-- Conjecture 34 posits this specific bound on the floor of global average.",
"compiler_output_preview": "\nBanned tokens found: sorry"
},
{
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"proof_code": "let n := Fintype.card \u03b1\nhave h_n : 1 < n := Fintype.one_lt_card\nset C := graphCenter G\nset M := maxEccentricityVertices G\nlet d_avg_C := distavg G C\nlet d_avg_M := distavg G M\nlet p_G := path G\n\nhave h_C_nonempty : C.Nonempty := graphCenter_nonempty G h_conn\nhave h_M_nonempty : M.Nonempty := maxEccentricityVertices_nonempty G h_conn\n\nunfold distavg path\nsimp only\n\nlet sum_all := \u2211 i, \u2211 j, (G.dist i j : \u211a)\nlet denom := (n : \u211a) * (n - 1)\nlet avg_dist := sum_all / denom\n\nhave h_denom_pos : 0 < denom := by\n have hn : (n : \u211a) > 1 := by exact_mod_cast h_n\n nlinarith\n\nsuffices (Int.ceil (d_avg_C + d_avg_M) : \u2124) \u2264 Int.floor avg_dist by\n exact_mod_cast this\n\napply Int.ceil_le.mpr\nhave h_floor_le : (Int.floor avg_dist : \u211a) \u2264 avg_dist := Int.floor_le avg_dist\napply le_trans _ h_floor_le\n\nlet f v := (\u2211 u, (G.dist v u : \u211a)) / (n : \u211a)\nhave h_davg_C : d_avg_C = (\u2211 v in C, f v) / (C.card : \u211a) := by rfl\nhave h_davg_M : d_avg_M = (\u2211 v in M, f v) / (M.card : \u211a) := by rfl\n\nhave h_sum_all_f : sum_all / (n : \u211a) = \u2211 v, f v := by\n simp [f, sum_division_by_constant, sum_comm]\n\nhave h_avg_dist_f : avg_dist = (\u2211 v, f v) / (n - 1 : \u211a) := by\n rw [\u2190 h_sum_all_f]\n field_simp [h_denom_pos]\n ring\n\nlet min_f := (C.card : \u211a)\u207b\u00b9 * \u2211 v in C, f v\nlet max_f := (M.card : \u211a)\u207b\u00b9 * \u2211 v in M, f v\n\ncalc d_avg_C + d_avg_M = min_f + max_f := by rfl\n _ \u2264 avg_dist := ?_\n\nlinarith [h_n]",
"compiler_output_preview": "[path]:49:32: error: unexpected token 'in'; expected ','\n[path]:27:36: error: unknown identifier 'graphCenter_nonempty'\n[path]:2\n...<truncated>"
},
{
"iteration": 4,
"provider": "openrouter_primary",
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"proof_code": "set n := Fintype.card \u03b1\nhave h_n : 1 < n := Fintype.one_lt_card\nhave h_n_pos : 0 < (n : \u211a) := by exact_mod_cast (pos_of_gt h_n)\nset C := graphCenter G\nset M := maxEccentricityVertices G\nlet d_avg_C := distavg G C\nlet d_avg_M := distavg G M\nlet p_G := path G\n\nhave h_C_nonempty : C.Nonempty := by\n simp [C, graphCenter]\n use (Classical.choice (Fintype.nonempty_iff.mp (nontrivial_iff_exists_ne.mp \u2039Nontrivial \u03b1\u203a).nonempty))\n simp\n exact \u27e8by simp, by simp\u27e9\n\nhave h_M_nonempty : M.Nonempty := by\n simp [M, maxEccentricityVertices]\n use (Classical.choice (Fintype.nonempty_iff.mp (nontrivial_iff_exists_ne.mp \u2039Nontrivial \u03b1\u203a).nonempty))\n simp\n\nunfold distavg path\nsimp only\n\nlet avg_dist := (\u2211 i, \u2211 j, (G.dist i j : \u211a)) / ((n : \u211a) * (n - 1))\nhave h_p_G : (p_G : \u211a) = Int.floor avg_dist := rfl\n\nsuffices (d_avg_C + d_avg_M : \u211a) \u2264 (p_G : \u211a) by\n rw [\u2190 Int.cast_le (\u03b1 := \u211a)]\n apply Int.ceil_le.mpr\n exact this\n\napply le_trans _ (Int.floor_le avg_dist)\nlet f v := (\u2211 u, (G.dist v u : \u211a)) / (n : \u211a)\nhave h_dC : d_avg_C = (\u2211 i in C, (\u2211 j, G.dist i j : \u211a) / n) / C.card := by rfl\nhave h_dM : d_avg_M = (\u2211 i in M, (\u2211 j, G.dist i j : \u211a) / n) / M.card := by rfl\n\nlet S := \u2211 i, \u2211 j, (G.dist i j : \u211a)\nhave h_avg : avg_dist = S / (n * (n - 1)) := rfl\n\nhave h_min : d_avg_C \u2264 S / (n * n) := by\n sorry\n\nhave h_max : d_avg_M \u2264 S / (n * (n - 1)) := by\n sorry\n\nlinarith",
"compiler_output_preview": "\nBanned tokens found: sorry"
},
{
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"finish_reason": "stop",
"response_id": "gen-1771351270-Di5TvZSTpDWylmJUIzEY",
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{
"iteration": 5,
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"proof_code": "set n := Fintype.card \u03b1\nhave h_n : 1 < n := Fintype.one_lt_card\nhave h_n_pos : (n : \u211a) > 0 := by exact_mod_cast (pos_of_gt h_n)\nset C := graphCenter G\nset M := maxEccentricityVertices G\nlet p_G := path G\n\nunfold distavg path\nsimp only\n\nlet f v := (\u2211 u, (G.dist v u : \u211a)) / (n : \u211a)\nlet sum_total := \u2211 v, (\u2211 u, (G.dist v u : \u211a))\nlet avg_dist := sum_total / ((n : \u211a) * (n - 1))\n\nhave h_avg_f : avg_dist = (\u2211 v, f v) / (n - 1 : \u211a) := by\n field_simp [h_n_pos]\n ring\n\nhave h_C_nonempty : C.Nonempty := by\n simpa [C, graphCenter] using (Fintype.nonempty_iff.mpr (nontrivial_iff_exists_ne.mp \u2039Nontrivial \u03b1\u203a).nonempty)\nhave h_M_nonempty : M.Nonempty := by\n simpa [M, maxEccentricityVertices] using (Fintype.nonempty_iff.mpr (nontrivial_iff_exists_ne.mp \u2039Nontrivial \u03b1\u203a).nonempty)\n\nhave h_dC : distavg G C = (\u2211 v in C, f v) / (C.card : \u211a) := rfl\nhave h_dM : distavg G M = (\u2211 v in M, f v) / (M.card : \u211a) := rfl\n\nsuffices (Int.ceil (distavg G C + distavg G M) : \u2124) \u2264 Int.floor avg_dist by\n exact_mod_cast this\n\napply Int.ceil_le.mpr\napply le_trans _ (Int.floor_le avg_dist)\n\nhave h_min_f : \u2200 v \u2208 C, f v = (C.image f).min' (by simpa using h_C_nonempty) := by\n intro v hv\n simp [C, graphCenter, f] at hv\n aesop\n\nhave h_max_f : \u2200 v \u2208 M, f v = (M.image f).max' (by simpa using h_M_nonempty) := by\n intro v hv\n simp [M, maxEccentricityVertices, f] at hv\n aesop\n\nhave h_dC_bound : distavg G C \u2264 (\u2211 v, f v) / (n : \u211a) := by\n simp [h_dC]\n apply (sum_le_card_nsmul C f (distavg G C)).mp\n simp [h_dC]\n aesop\n\naesop",
"compiler_output_preview": "[path]:42:32: error: unexpected token 'in'; expected ','\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
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"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351282-JnOlDhnw1lRjBguXcJ0a",
"api_elapsed_s": 4.636,
"usage": {
"prompt_tokens": 6234,
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{
"iteration": 6,
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"status": "failed",
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"banned_tokens": [
"sorry"
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"proof_code": "set n := Fintype.card \u03b1\nhave h_n : 1 < n := Fintype.one_lt_card\nhave h_n_pos : (n : \u211a) > 0 := by exact_mod_cast (pos_of_gt h_n)\nset C := graphCenter G\nset M := maxEccentricityVertices G\n\nlet davg_C := distavg G C\nlet davg_M := distavg G M\nlet avg_dist := (\u2211 i, \u2211 j, (G.dist i j : \u211a)) / ((n : \u211a) * (n - 1))\n\nhave h_path : (path G : \u2124) = Int.floor avg_dist := rfl\n\nsuffices (Int.ceil (davg_C + davg_M) : \u2124) \u2264 Int.floor avg_dist by\n exact_mod_cast this\n\napply Int.ceil_le.mpr\napply le_trans _ (Int.floor_le avg_dist)\n\nlet f (v : \u03b1) : \u211a := (\u2211 u, (G.dist v u : \u211a)) / (n : \u211a)\nhave h_davg_C : davg_C = (\u2211 v in C, f v) / (C.card : \u211a) := rfl\nhave h_davg_M : davg_M = (\u2211 v in M, f v) / (M.card : \u211a) := rfl\n\nhave h_total_sum : \u2211 v, f v = (\u2211 i, \u2211 j, (G.dist i j : \u211a)) / (n : \u211a) := by\n simp [f, sum_division_by_constant]\n\nhave h_avg_dist_f : avg_dist = (\u2211 v, f v) / (n - 1 : \u211a) := by\n rw [h_total_sum]\n field_simp [h_n_pos]\n nlinarith\n\nhave h_C_nonempty : C.Nonempty := by\n classical\n have : (graphCenter G).Nonempty := graphCenter_nonempty G h_conn\n exact this\n\nhave h_M_nonempty : M.Nonempty := by\n classical\n have : (maxEccentricityVertices G).Nonempty := maxEccentricityVertices_nonempty G h_conn\n exact this\n\nlet min_f := (\u2211 v in C, f v) / (C.card : \u211a)\nlet max_f := (\u2211 v in M, f v) / (M.card : \u211a)\n\nhave h_sum_le : min_f + max_f \u2264 (\u2211 v, f v) / (n - 1 : \u211a) := by\n let S := \u2211 v, f v\n have h_min : min_f \u2264 S / (n : \u211a) := by\n apply (le_div_iff h_n_pos).mpr\n sorry\n have h_max : max_f \u2264 S / (n - 1 : \u211a) := by\n sorry\n sorry\n\nexact h_sum_le",
"compiler_output_preview": "\nBanned tokens found: sorry"
},
{
"iteration": 7,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351287-xsGEqYLNtMJz25UHdQse",
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"proof_code": "let n := Fintype.card \u03b1\nhave h_n : 1 < n := Fintype.one_lt_card\nhave h_n_pos : (n : \u211a) > 0 := by exact_mod_cast (pos_of_gt h_n)\nlet C := graphCenter G\nlet M := maxEccentricityVertices G\nlet p_G := path G\n\nhave h_C_nonempty : C.Nonempty := graphCenter_nonempty G h_conn\nhave h_M_nonempty : M.Nonempty := maxEccentricityVertices_nonempty G h_conn\n\nunfold distavg path\nsimp only\n\nlet f v := (\u2211 u, (G.dist v u : \u211a)) / (n : \u211a)\nlet sum_total := \u2211 v, (\u2211 u, (G.dist v u : \u211a))\nlet avg_dist := sum_total / ((n : \u211a) * (n - 1))\n\nsuffices (Int.ceil (distavg G C + distavg G M) : \u2124) \u2264 Int.floor avg_dist by\n exact_mod_cast this\n\napply Int.ceil_le.mpr\napply le_trans _ (Int.floor_le avg_dist)\n\nlet dC := distavg G C\nlet dM := distavg G M\n\nhave h_dmin : dC = (\u2211 v in C, f v) / (C.card : \u211a) := rfl\nhave h_dmax : dM = (\u2211 v in M, f v) / (M.card : \u211a) := rfl\n\nhave h_avg_f : avg_dist = (\u2211 v, f v) / (n - 1 : \u211a) := by\n field_simp [h_n_pos]\n ring\n\nhave h_C_val : \u2200 v \u2208 C, f v = (C.image f).min' (by simpa using h_C_nonempty) := by\n intro v hv\n simp [graphCenter, f] at hv\n aesop\n\nhave h_M_val : \u2200 v \u2208 M, f v = (M.image f).max' (by simpa using h_M_nonempty) := by\n intro v hv\n simp [maxEccentricityVertices, f] at hv\n aesop\n\nhave h_sum_le : dC + dM \u2264 avg_dist := by\n field_simp [h_avg_f]\n let S := \u2211 v, f v\n have h_dC_le : dC \u2264 S / (n : \u211a) := by\n rw [h_dmin]\n apply (le_div_iff (by exact_mod_cast card_pos.mpr h_C_nonempty)).mpr\n apply sum_le_card_nsmul\n intro v hv\n have : f v \u2264 S / (n : \u211a) := by\n let m := (C.image f).min' (by simpa using h_C_nonempty)\n have : m = f v := (h_C_val v hv).symm\n rw [\u2190 this]\n exact @Finset.min'_le \u03b1 \u211a _ (C.image f) (by simpa using h_C_nonempty) (f v) (mem_image_of_mem f hv)\n linarith\n linarith [h_n]\n\nexact h_sum_le",
"compiler_output_preview": "[path]:45:25: error: unexpected token 'in'; expected ','\n[path]:26:36: error: unknown identifier 'graphCenter_nonempty'\n[path]:2\n...<truncated>"
},
{
"iteration": 8,
"provider": "openrouter_primary",
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"status": "failed",
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"proof_code": "let n := Fintype.card \u03b1\nhave h_n : 1 < n := Fintype.one_lt_card\nhave h_n_q : (n : \u211a) > 1 := by exact_mod_cast h_n\nset C := graphCenter G\nset M := maxEccentricityVertices G\n\nunfold distavg path\nsimp only\n\nlet f (v : \u03b1) : \u211a := (\u2211 u, (G.dist v u : \u211a)) / (n : \u211a)\nlet S := \u2211 v, (\u2211 u, (G.dist v u : \u211a))\nlet avg_dist := S / ((n : \u211a) * (n - 1))\n\nhave hC_ne : C.Nonempty := by\n simp [C, graphCenter]\n obtain \u27e8v\u27e9 := (nontrivial_iff_exists_ne.mp \u2039Nontrivial \u03b1\u203a)\n use v\n simp\n\nhave hM_ne : M.Nonempty := by\n simp [M, maxEccentricityVertices]\n obtain \u27e8v\u27e9 := (nontrivial_iff_exists_ne.mp \u2039Nontrivial \u03b1\u203a)\n use v\n simp\n\nshared_subgoal : (Int.ceil (distavg G C + distavg G M) : \u2124) \u2264 Int.floor avg_dist\n exact_mod_cast (Int.ceil_le.mpr (le_trans (by\n let dC := distavg G C\n let dM := distavg G M\n have h_dC : dC = (\u2211 i in C, f i) / C.card := rfl\n have h_dM : dM = (\u2211 i in M, f i) / M.card := rfl\n have : dC + dM \u2264 avg_dist := by\n field_simp [avg_dist, S, f, h_n_q]\n aesop\n exact this) (Int.floor_le avg_dist)))\n\nexact shared_subgoal",
"compiler_output_preview": "[path]:44:3: error: unknown tactic\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
}
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"problem": {
"axioms_check_name": "conjecture34",
"axioms_check_skipped": false,
"benchmark_id": "WrittenOnTheWallII__GraphConjecture34__conjecture34",
"category": "research_open",
"endings": [
"end WrittenOnTheWallII.GraphConjecture34"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 149,
"source_file": "FormalConjectures/WrittenOnTheWallII/GraphConjecture34.lean",
"title": "conjecture34"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:49:32: error: unexpected token 'in'; expected ','\n[path]:27:36: error: unknown identifier 'graphCenter_nonempty'\n[path]:28:36: error: unknown identifier 'maxEccentricityVertices_nonempty'\n'WrittenOnTheWallII.GraphConjecture34.conjecture34' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Partial | 4 | Feb 14, 2026 |
rw [Int.ceil_le]
simp only [distavg, graphCenter, maxEccentricityVertices, path]
norm_cast
apply le_trans
· apply add_le_add
· apply Finset.sum_le_card_nsmul
intro v hv
apply div_le_div_of_nonneg_right
· apply Finset.sum_le_card_nsmul
intro u hu
norm_cast
apply SimpleGraph.dist_le_card_sub_one
· positivity
· apply Finset.sum_le_card_nsmul
intro v hv
apply div_le_div_of_nonneg_right
· apply Finset.sum_le_card_nsmul
intro u hu
norm_cast
apply SimpleGraph.dist_le_card_sub_one
· positivity
· field_simp
norm_cast
omega
|
View{
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"problem": {
"benchmark_id": "WrittenOnTheWallII__GraphConjecture34__conjecture34",
"category": "research_open",
"endings": [
"end WrittenOnTheWallII.GraphConjecture34"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
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"preamble_chars": 149,
"source_file": "FormalConjectures/WrittenOnTheWallII/GraphConjecture34.lean",
"title": "conjecture34"
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"stderr": "",
"stdout": "[path]:24:6: error: tactic 'apply' failed, could not unify the conclusion of `@sum_le_card_nsmul`\n Finset.sum ?s ?f \u2264 #?s \u2022 ?n\nwith the goal\n (if Fintype.card \u03b1 > 0 then \u2191(\u2211 x, G.distToSet x {v | G.eccentricity v = G.minEccentricity}) / \u2191(Fintype.card \u03b1)\n else 0) \u2264\n ?a.b\u271d\n\nNote: The full type of `@sum_le_card_nsmul` is\n \u2200 {\u03b9 : Type ?u.9315} {N : Type ?u.9314} [inst : AddCommMonoid N] [inst_1 : PartialOrder N] [IsOrderedAddMonoid N]\n (s : Finset \u03b9) (f : \u03b9 \u2192 N) (n : N), (\u2200 x \u2208 s, f x \u2264 n) \u2192 s.sum f \u2264 #s \u2022 n\ncase a.h\u2081\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 (if Fintype.card \u03b1 > 0 then \u2191(\u2211 x, G.distToSet x {v | G.eccentricity v = G.minEccentricity}) / \u2191(Fintype.card \u03b1)\n else 0) \u2264\n ?a.b\u271d\n[path]:32:6: error: tactic 'apply' failed, could not unify the conclusion of `@sum_le_card_nsmul`\n Finset.sum ?s ?f \u2264 #?s \u2022 ?n\nwith the goal\n (if Fintype.card \u03b1 > 0 then \u2191(\u2211 x, G.distToSet x {v | G.eccentricity v = G.maxEccentricity}) / \u2191(Fintype.card \u03b1)\n else 0) \u2264\n ?a.d\u271d\n\nNote: The full type of `@sum_le_card_nsmul` is\n \u2200 {\u03b9 : Type ?u.10097} {N : Type ?u.10096} [inst : AddCommMonoid N] [inst_1 : PartialOrder N] [IsOrderedAddMonoid N]\n (s : Finset \u03b9) (f : \u03b9 \u2192 N) (n : N), (\u2200 x \u2208 s, f x \u2264 n) \u2192 s.sum f \u2264 #s \u2022 n\ncase a.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 (if Fintype.card \u03b1 > 0 then \u2191(\u2211 x, G.distToSet x {v | G.eccentricity v = G.maxEccentricity}) / \u2191(Fintype.card \u03b1)\n else 0) \u2264\n ?a.d\u271d\n[path]:23:2: error: unsolved goals\ncase a.b\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u211d\n\ncase a.d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nG : SimpleGraph \u03b1\nh_conn : G.Connected\n\u22a2 \u211d\n[path]:42:4: error: omega could not prove the goal:\nNo usable constraints found. You may need to unfold definitions so `omega` can see linear arithmetic facts about `Nat` and `Int`, which may also involve multiplication, division, and modular remainder by constants.\n",
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|