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P_ne_NP
Specification
**P ≠ NP**: The conjecture that the complexity classes P and NP are not equal.
Lean 4 Statement
theorem P_ne_NP : P ≠ NP
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Evaluation Results
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| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Failed | -- | Feb 25, 2026 |
have h : P ≠ NP := ?_
exact h
|
View{
"agent_telemetry": {
"compilations": 4,
"iterations": 10,
"duration_s": 303.9,
"best_status": "failed",
"best_goals": null,
"max_iterations": 10,
"model": "openai/gpt-5.3-codex",
"error": null,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"usage": {
"prompt_tokens": 104991,
"completion_tokens": 6419,
"total_tokens": 111410,
"reasoning_tokens": 6089,
"cost_usd": 0.16473625
},
"reasoning_effort": "high",
"response_trace_count": 14
},
"response_trace": [
{
"iteration": 1,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772003920-rcvi7zVGYqr2174V3NBq",
"api_elapsed_s": 63.857,
"usage": {
"prompt_tokens": 1992,
"completion_tokens": 2936,
"total_tokens": 4928,
"reasoning_tokens": 2902,
"cost_usd": 0.0429772
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Analyzing theorem proof feasibility**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Analyzing theorem proof feasibility**"
},
{
"id": "rs_0585c1b11397ca8601699ea25122088195ad21bfd2b12d6216",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
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"
}
]
},
{
"iteration": 1,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h : P \u2260 NP := ?_\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772003984_9979.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772003984_9979.lean:76:8: warning: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772003984_9979.lean:88:2: error: Type mismatch\n h\nhas type\n P \u2260 NP\nbut is expected to have type\n {L | L\u1d9c \u2208 P} = P\n'ComplexityTheory.P_ne_NP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 43.089,
"tactic_replay": {
"tactic_count": 2,
"successful_tactics": 1,
"first_failure_step": 2,
"goal_trajectory": [
1,
2
],
"max_goals": 2,
"final_goal_count": 2,
"goals_opened": 1,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 9,
"line": 87
},
"start": {
"character": 2,
"line": 87
}
},
"message": "Type mismatch\n h\nhas type\n P \u2260 NP\nbut is expected to have type\n {L | L\u1d9c \u2208 P} = P",
"range": {
"end": {
"character": 9,
"line": 87
},
"start": {
"character": 2,
"line": 87
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 89
},
"start": {
"character": 0,
"line": 89
}
},
"message": "'ComplexityTheory.P_ne_NP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 89
},
"start": {
"character": 0,
"line": 89
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 87,
"error_char": 2,
"error_msg": "Type mismatch\n h\nhas type\n P \u2260 NP\nbut is expected to have type\n {L | L\u1d9c \u2208 P} = P",
"goal_before_error": "```lean\ncase refine_2\nh : P \u2260 NP\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```\n---\n```lean\ncase refine_1\n\u22a2 P \u2260 NP\n```",
"full_timeline": [
{
"line": 86,
"code": "have h : P \u2260 NP := ?_",
"goal": "```lean\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```",
"goal_count": 1,
"ast_complexity": 30
},
{
"line": 87,
"code": "exact h",
"goal": "```lean\ncase refine_2\nh : P \u2260 NP\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```\n---\n```lean\ncase refine_1\n\u22a2 P \u2260 NP\n```",
"goal_count": 2,
"ast_complexity": 94
}
],
"tactic_count": 2,
"successful_tactics": 1,
"first_failure_step": 2,
"cps": 50.0,
"cps_peak_step": 1,
"error_code": "exact h"
}
},
{
"iteration": 2,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772004027-RrCT9MsIIMj1a1cYVKJK",
"api_elapsed_s": 32.13,
"usage": {
"prompt_tokens": 6331,
"completion_tokens": 1288,
"total_tokens": 7619,
"reasoning_tokens": 1265,
"cost_usd": 0.02911125
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Analyzing compile errors from earlier theorems**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Analyzing compile errors from earlier theorems**"
},
{
"id": "rs_0585c1b11397ca8601699ea2bc34908195a7f1f80cb526bd0b",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": 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}
]
},
{
"iteration": 2,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "rfl",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772004059_6274.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772004059_6274.lean:76:8: warning: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772004059_6274.lean:87:2: error: Tactic `rfl` failed: The left-hand side\n {L | L\u1d9c \u2208 P}\nis not definitionally equal to the right-hand side\n P\n\n\u22a2 {L | L\u1d9c \u2208 P} = P\n'ComplexityTheory.P_ne_NP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 45.651,
"tactic_replay": {
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 5,
"line": 86
},
"start": {
"character": 2,
"line": 86
}
},
"message": "Tactic `rfl` failed: The left-hand side\n {L | L\u1d9c \u2208 P}\nis not definitionally equal to the right-hand side\n P\n\n\u22a2 {L | L\u1d9c \u2208 P} = P",
"range": {
"end": {
"character": 5,
"line": 86
},
"start": {
"character": 2,
"line": 86
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 88
},
"start": {
"character": 0,
"line": 88
}
},
"message": "'ComplexityTheory.P_ne_NP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 88
},
"start": {
"character": 0,
"line": 88
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 86,
"error_char": 2,
"error_msg": "Tactic `rfl` failed: The left-hand side\n {L | L\u1d9c \u2208 P}\nis not definitionally equal to the right-hand side\n P\n\n\u22a2 {L | L\u1d9c \u2208 P} = P",
"goal_before_error": "```lean\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```",
"full_timeline": [
{
"line": 86,
"code": "rfl",
"goal": "```lean\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```",
"goal_count": 1,
"ast_complexity": 30
}
],
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "rfl"
}
},
{
"iteration": 3,
"event": "search_mathlib",
"query": "TM2ComputableInPolyTime",
"results_count": 1
},
{
"iteration": 4,
"event": "search_mathlib",
"query": "ComputableInPolyTime",
"results_count": 0
},
{
"iteration": 5,
"event": "search_mathlib",
"query": "TM2Computable",
"results_count": 0
},
{
"iteration": 6,
"event": "search_mathlib",
"query": "coP_eq_P",
"results_count": 0
},
{
"iteration": 7,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772004127-TOijyD5CDAgEQmUDnh1K",
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"usage": {
"prompt_tokens": 10349,
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},
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}
]
},
{
"iteration": 7,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "ext L\nconstructor\n\u00b7 intro h\n have h1 : L \u2208 P := ?_\n exact h1\n\u00b7 intro h\n have h2 : L\u1d9c \u2208 P := ?_\n exact h2",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772004132_3257.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772004132_3257.lean:76:8: warning: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772004132_3257.lean:89:2: error: unsolved goals\ncase h.mp.refine_1\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n/app/asi_portal/verifier/quick_1772004132_3257.lean:92:2: error: unsolved goals\ncase h.mpr.refine_1\nL : DecisionProblem\nh : L \u2208 P\n\u22a2 L\u1d9c \u2208 P\n'ComplexityTheory.P_ne_NP' depe\n...<truncated>",
"compile_elapsed_s": 31.883,
"tactic_replay": {
"tactic_count": 8,
"successful_tactics": 2,
"first_failure_step": 3,
"goal_trajectory": [
1,
1,
2,
1,
2,
1,
1,
2
],
"max_goals": 2,
"final_goal_count": 2,
"goals_opened": 3,
"goals_closed": 2
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 12,
"line": 90
},
"start": {
"character": 2,
"line": 88
}
},
"leanTags": [
1
],
"message": "unsolved goals\ncase h.mp.refine_1\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P",
"range": {
"end": {
"character": 0,
"line": 89
},
"start": {
"character": 2,
"line": 88
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 12,
"line": 93
},
"start": {
"character": 2,
"line": 91
}
},
"leanTags": [
1
],
"message": "unsolved goals\ncase h.mpr.refine_1\nL : DecisionProblem\nh : L \u2208 P\n\u22a2 L\u1d9c \u2208 P",
"range": {
"end": {
"character": 0,
"line": 92
},
"start": {
"character": 2,
"line": 91
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 95
},
"start": {
"character": 0,
"line": 95
}
},
"message": "'ComplexityTheory.P_ne_NP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 95
},
"start": {
"character": 0,
"line": 95
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 88,
"error_char": 2,
"error_msg": "unsolved goals\ncase h.mp.refine_1\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P",
"goal_before_error": "```lean\ncase h.mp\nL : DecisionProblem\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P} \u2192 L \u2208 P\n```\n---\n```lean\ncase h.mpr\nL : DecisionProblem\n\u22a2 L \u2208 P \u2192 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"full_timeline": [
{
"line": 86,
"code": "ext L",
"goal": "```lean\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```",
"goal_count": 1,
"ast_complexity": 30
},
{
"line": 87,
"code": "constructor",
"goal": "```lean\ncase h\nL : DecisionProblem\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P} \u2194 L \u2208 P\n```",
"goal_count": 1,
"ast_complexity": 65
},
{
"line": 88,
"code": "\u00b7 intro h",
"goal": "```lean\ncase h.mp\nL : DecisionProblem\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P} \u2192 L \u2208 P\n```\n---\n```lean\ncase h.mpr\nL : DecisionProblem\n\u22a2 L \u2208 P \u2192 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"goal_count": 2,
"ast_complexity": 142
},
{
"line": 89,
"code": "have h1 : L \u2208 P := ?_",
"goal": "```lean\ncase h.mp\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n```",
"goal_count": 1,
"ast_complexity": 70
},
{
"line": 90,
"code": "exact h1",
"goal": "```lean\ncase h.mp.refine_2\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\nh1 : L \u2208 P\n\u22a2 L \u2208 P\n```\n---\n```lean\ncase h.mp.refine_1\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n```",
"goal_count": 2,
"ast_complexity": 174
},
{
"line": 91,
"code": "\u00b7 intro h",
"goal": "```lean\ncase h.mpr\nL : DecisionProblem\n\u22a2 L \u2208 P \u2192 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"goal_count": 1,
"ast_complexity": 69
},
{
"line": 92,
"code": "have h2 : L\u1d9c \u2208 P := ?_",
"goal": "```lean\ncase h.mpr\nL : DecisionProblem\nh : L \u2208 P\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"goal_count": 1,
"ast_complexity": 71
},
{
"line": 93,
"code": "exact h2",
"goal": "```lean\ncase h.mpr.refine_2\nL : DecisionProblem\nh : L \u2208 P\nh2 : L\u1d9c \u2208 P\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P}\n```\n---\n```lean\ncase h.mpr.refine_1\nL : DecisionProblem\nh : L \u2208 P\n\u22a2 L\u1d9c \u2208 P\n```",
"goal_count": 2,
"ast_complexity": 167
}
],
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"successful_tactics": 2,
"first_failure_step": 3,
"cps": 25.0,
"cps_peak_step": 2,
"error_code": "\u00b7 intro h"
}
},
{
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"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772004164-lTvuP9oGqlDNx0FN7fbt",
"api_elapsed_s": 18.382,
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"summary": "**Assessing compile budget and proof attempts**"
},
{
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}
]
},
{
"iteration": 8,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
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"proof_code": "ext L\nconstructor\n\u00b7 intro h\n simpa [P] using h\n\u00b7 intro h\n simpa [P] using h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772004182_1815.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772004182_1815.lean:76:8: warning: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772004182_1815.lean:90:4: error: Type mismatch: After simplification, term\n h\n has type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\u1d9c\nbut is expected to have type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\n/app/asi_portal/verifier/quick_17\n...<truncated>",
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2,
1,
1,
1
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{
"fullRange": {
"end": {
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},
"message": "Type mismatch: After simplification, term\n h\n has type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\u1d9c\nbut is expected to have type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L",
"range": {
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{
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"message": "Type mismatch: After simplification, term\n h\n has type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\nbut is expected to have type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\u1d9c",
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{
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"message": "'ComplexityTheory.P_ne_NP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
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},
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],
"error_line": 89,
"error_char": 4,
"error_msg": "Type mismatch: After simplification, term\n h\n has type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\u1d9c\nbut is expected to have type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L",
"goal_before_error": "```lean\ncase h.mp\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n```",
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{
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"goal_count": 1,
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},
{
"line": 87,
"code": "constructor",
"goal": "```lean\ncase h\nL : DecisionProblem\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P} \u2194 L \u2208 P\n```",
"goal_count": 1,
"ast_complexity": 65
},
{
"line": 88,
"code": "\u00b7 intro h",
"goal": "```lean\ncase h.mp\nL : DecisionProblem\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P} \u2192 L \u2208 P\n```\n---\n```lean\ncase h.mpr\nL : DecisionProblem\n\u22a2 L \u2208 P \u2192 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"goal_count": 2,
"ast_complexity": 142
},
{
"line": 89,
"code": "simpa [P] using h",
"goal": "```lean\ncase h.mp\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n```",
"goal_count": 1,
"ast_complexity": 70
},
{
"line": 90,
"code": "\u00b7 intro h",
"goal": "```lean\ncase h.mpr\nL : DecisionProblem\n\u22a2 L \u2208 P \u2192 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"goal_count": 1,
"ast_complexity": 69
},
{
"line": 91,
"code": "simpa [P] using h",
"goal": "```lean\ncase h.mpr\nL : DecisionProblem\nh : L \u2208 P\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"goal_count": 1,
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}
],
"tactic_count": 6,
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"first_failure_step": 4,
"cps": 50.0,
"cps_peak_step": 3,
"error_code": "simpa [P] using h"
}
},
{
"iteration": 9,
"event": "search_mathlib",
"query": "compl TM2",
"results_count": 0
},
{
"iteration": 10,
"event": "search_mathlib",
"query": "toTM2ComputableInTime",
"results_count": 1
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772004224_2252.lean"
],
"comparator_passed": null,
"duration_s": 21.318677186965942,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "P_ne_NP",
"axioms_check_skipped": false,
"benchmark_id": "Millenium__PvsNP__P_ne_NP",
"category": "research_open",
"endings": [
"end ComplexityTheory"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 1987,
"source_file": "FormalConjectures/Millenium/PvsNP.lean",
"title": "P_ne_NP"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772004224_2252.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772004224_2252.lean:76:8: warning: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772004224_2252.lean:88:2: error: Type mismatch\n h\nhas type\n P \u2260 NP\nbut is expected to have type\n {L | L\u1d9c \u2208 P} = P\n'ComplexityTheory.P_ne_NP' 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 |
ext L
constructor
· intro h
exact L h
· intro h
exact L h
|
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"reasoning_text": "**Defining the Complement Set**\n\nI'm trying to figure out what `coP` is, given the theorem statement involving it and set `P`. The theorem seems to suggest `coP` represents a set of languages whose complements are in `P`. I'm starting to formalize this connection between `coP`, complements, and `P` to get a better handle on the problem, before diving into the actual proof.\n\n\n**Refining the Approach**\n\nI'm now contemplating how to actually demonstrate this equality. Since `P` is defined in terms of polynomial-time computability, I'm considering how boolean negation and function composition come into play, as the context hints. I also need to figure out how to leverage the properties of `IsComputableInPolyTime` to prove the theorem.\n\n\n**Unraveling Theorem's Implications**\n\nNow, I'm examining the theorem statement and its nuances. I've realized that `coP` isn't explicitly defined, which is important. Also, the context mentions boolean negation and composition, so it might be related to the characteristics of `L` and `L\u1d9c`. I will examine the types in the definitions for `L\u1d9c`, before trying to see if I can get some lemmas to build my proof from. It's becoming clearer where to focus.\n\n\n",
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"text": "**Defining the Complement Set**\n\nI'm trying to figure out what `coP` is, given the theorem statement involving it and set `P`. The theorem seems to suggest `coP` represents a set of languages whose complements are in `P`. I'm starting to formalize this connection between `coP`, complements, and `P` to get a better handle on the problem, before diving into the actual proof.\n\n\n**Refining the Approach**\n\nI'm now contemplating how to actually demonstrate this equality. Since `P` is defined in terms of polynomial-time computability, I'm considering how boolean negation and function composition come into play, as the context hints. I also need to figure out how to leverage the properties of `IsComputableInPolyTime` to prove the theorem.\n\n\n**Unraveling Theorem's Implications**\n\nNow, I'm examining the theorem statement and its nuances. I've realized that `coP` isn't explicitly defined, which is important. Also, the context mentions boolean negation and composition, so it might be related to the characteristics of `L` and `L\u1d9c`. I will examine the types in the definitions for `L\u1d9c`, before trying to see if I can get some lemmas to build my proof from. It's becoming clearer where to focus.\n\n\n"
},
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},
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{
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{
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"goal_before_error": "```lean\ncase h.mp\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n```",
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{
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"ast_complexity": 65
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{
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"goal": "```lean\ncase h.mp\nL : DecisionProblem\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P} \u2192 L \u2208 P\n```\n---\n```lean\ncase h.mpr\nL : DecisionProblem\n\u22a2 L \u2208 P \u2192 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"goal_count": 2,
"ast_complexity": 142
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{
"line": 89,
"code": "have h1 := @IsComputableInPolyTime.comp",
"goal": "```lean\ncase h.mp\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n```",
"goal_count": 1,
"ast_complexity": 70
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{
"line": 90,
"code": "exact L h",
"goal": "```lean\ncase h.mp\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n```",
"goal_count": 1,
"ast_complexity": 70
},
{
"line": 91,
"code": "\u00b7 intro h",
"goal": "```lean\ncase h.mpr\nL : DecisionProblem\n\u22a2 L \u2208 P \u2192 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"goal_count": 1,
"ast_complexity": 69
},
{
"line": 92,
"code": "exact L h",
"goal": "```lean\ncase h.mpr\nL : DecisionProblem\nh : L \u2208 P\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P}\n```",
"goal_count": 1,
"ast_complexity": 71
}
],
"tactic_count": 7,
"successful_tactics": 3,
"first_failure_step": 4,
"cps": 42.857142857142854,
"cps_peak_step": 3,
"error_code": "have h1 := @IsComputableInPolyTime.comp"
}
},
{
"iteration": 9,
"event": "search_mathlib",
"query": "Turing.TM2ComputableInPolyTime",
"results_count": 5
},
{
"iteration": 10,
"event": "search_mathlib",
"query": "compComputableInPolyTime",
"results_count": 0
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
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"comparator_passed": null,
"duration_s": 16.722581386566162,
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"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "P_ne_NP",
"axioms_check_skipped": false,
"benchmark_id": "Millenium__PvsNP__P_ne_NP",
"category": "research_open",
"endings": [
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],
"imports": [
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"preamble_chars": 1987,
"source_file": "FormalConjectures/Millenium/PvsNP.lean",
"title": "P_ne_NP"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1771906992_4812.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1771906992_4812.lean:76:8: warning: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1771906992_4812.lean:90:12: error: Application type mismatch: The argument\n h\nhas type\n L \u2208 {L | L\u1d9c \u2208 P}\nof sort `Prop` but is expected to have type\n List Bool\nof sort `Type` in the application\n L h\n/app/asi_portal/verifier/quick_1771906992_4812.lean:92:12: error: Application type mismatch: The argument\n h\nhas type\n L \u2208 P\nof sort `Prop` but is expected to have type\n List Bool\nof sort `Type` in the application\n L h\n'ComplexityTheory.P_ne_NP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Agent v8 | Failed | -- | Feb 17, 2026 |
ext L
simp [P]
constructor
· intro h
have h_not : IsComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not := IsComputableInPolyTime.bool_not
have h_comp : L = Bool.not ∘ Lᶜ := by ext x; simp
rw [h_comp]
exact IsComputableInPolyTime.comp h_not h
· intro h
have h_not : IsComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not := IsComputableInPolyTime.bool_not
have h_comp : Lᶜ = Bool.not ∘ L := by ext x; simp
rw [h_comp]
exact IsComputableInPolyTime.comp h_not h
|
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"proof_code": "ext L\n simp [P]\n constructor\n \u00b7 intro h\n have h_not : IsComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not := IsComputableInPolyTime.bool_not\n have h_comp : L = Bool.not \u2218 L\u1d9c := by ext x; simp\n rw [h_comp]\n exact IsComputableInPolyTime.comp h_not h\n \u00b7 intro h\n have h_not : IsComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not := IsComputableInPolyTime.bool_not\n have h_comp : L\u1d9c = Bool.not \u2218 L := by ext x; simp\n rw [h_comp]\n exact IsComputableInPolyTime.comp h_not h",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:76:8: warning: declaration uses 'sorry'\n[path]:91:92: error: unknown constant 'ComplexityTheory.IsComputableInPolyTime.bool_not'\n[path]
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"proof_code": "ext L\n constructor\n \u00b7 intro h\n simp [P, IsComputableInPolyTime] at h \u22a2\n have h_not : Nonempty (TM2ComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not) := \u27e8TM2ComputableInPolyTime.bool_not\u27e9\n apply Nonempty.map2 (fun tm_not tm_L_not => tm_not.comp tm_L_not) h_not h\n \u00b7 intro h\n simp [P, IsComputableInPolyTime] at h \u22a2\n have h_not : Nonempty (TM2ComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not) := \u27e8TM2ComputableInPolyTime.bool_not\u27e9\n apply Nonempty.map2 (fun tm_not tm_L => tm_not.comp tm_L) h_not h",
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"proof_code": "ext L\n simp [P, IsComputableInPolyTime]\n constructor\n \u00b7 intro h\n have : L = Bool.not \u2218 L\u1d9c := by ext; simp\n rw [this]\n -- try to use some very general tactics\n simp [h]\n aesop\n \u00b7 intro h\n have : L\u1d9c = Bool.not \u2218 L := by ext; simp\n rw [this]\n simp [h]\n aesop",
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"proof_code": "ext L\n constructor\n \u00b7 intro h\n simp [P] at h \u22a2\n have : L = L\u1d9c\u1d9c := by ext; simp\n rw [this]\n exact isComputableInPolyTime_comp isComputableInPolyTime_not h\n \u00b7 intro h\n simp [P] at h \u22a2\n exact isComputableInPolyTime_comp isComputableInPolyTime_not h",
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"proof_code": "ext L\n constructor\n \u00b7 intro h\n simp [P, IsComputableInPolyTime] at h \u22a2\n have h_not : Nonempty (TM2ComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not) := \u27e8TM2ComputableInPolyTime.bool_not\u27e9\n exact Nonempty.map2 (fun tm_not tm_L_not => tm_not.comp tm_L_not) h_not h\n \u00b7 intro h\n simp [P, IsComputableInPolyTime] at h \u22a2\n have h_not : Nonempty (TM2ComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not) := \u27e8TM2ComputableInPolyTime.bool_not\u27e9\n exact Nonempty.map2 (fun tm_not tm_L => tm_not.comp tm_L) h_not h",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:76:8: warning: declaration uses 'sorry'\n[path]:91:105: error: unknown constant 'Turing.TM2ComputableInPolyTime.bool_not'\n[path]
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"proof_code": "ext L\n simp [P]\n constructor\n \u00b7 intro h\n have : L = L\u1d9c\u1d9c := by ext; simp\n rw [this]\n exact isComputableInPolyTime_compl h\n \u00b7 intro h\n exact isComputableInPolyTime_compl h",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:76:8: warning: declaration uses 'sorry'\n[path]:93:10: error: unknown identifier 'isComputableInPolyTime_compl'\n[path]
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"proof_code": "ext L\n constructor\n \u00b7 intro h\n simp [P, IsComputableInPolyTime] at h \u22a2\n obtain \u27e8tm_not\u27e9 : Nonempty (TM2ComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not) := \u27e8TM2ComputableInPolyTime.bool_not\u27e9\n obtain \u27e8tm_L_not\u27e9 := h\n exact \u27e8tm_not.comp tm_L_not\u27e9\n \u00b7 intro h\n simp [P, IsComputableInPolyTime] at h \u22a2\n obtain \u27e8tm_not\u27e9 : Nonempty (TM2ComputableInPolyTime finEncodingBoolBool finEncodingBoolBool Bool.not) := \u27e8TM2ComputableInPolyTime.bool_not\u27e9\n obtain \u27e8tm_L\u27e9 := h\n exact \u27e8tm_not.comp tm_L\u27e9",
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"proof_code": "ext L\n constructor\n \u00b7 intro h\n simp [P] at h \u22a2\n rw [\u2190 compl_compl L] at h\n exact isComputableInPolyTime_not_iff.mp h\n \u00b7 intro h\n simp [P] at h \u22a2\n exact isComputableInPolyTime_not_iff.mpr h",
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"stdout": "[path]:68:8: error: declaration uses 'sorry'\n[path]:76:8: warning: declaration uses 'sorry'\n[path]:91:92: error: unknown constant 'ComplexityTheory.IsComputableInPolyTime.bool_not'\n[path]:94:38: error: Application type mismatch: In the application\n IsComputableInPolyTime \u2218 h_not\nthe argument\n h_not\nhas type\n IsComputableInPolyTime finEncodingBoolBool finEncodingBoolBool not : Prop\nbut is expected to have type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\u1d9c \u2192 FinEncoding ?m.1707 : Type 1\n[path]:96:92: error: unknown constant 'ComplexityTheory.IsComputableInPolyTime.bool_not'\n[path]:99:38: error: Application type mismatch: In the application\n IsComputableInPolyTime \u2218 h_not\nthe argument\n h_not\nhas type\n IsComputableInPolyTime finEncodingBoolBool finEncodingBoolBool not : Prop\nbut is expected to have type\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L \u2192 FinEncoding ?m.4861 : Type 1\n'ComplexityTheory.P_ne_NP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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|
| Gemini 3 Flash Agent v7 | Failed | -- | Feb 16, 2026 |
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|
|
| Gemini 3 Flash Preview | Partial | 1 | Feb 14, 2026 |
ext x
simp only [Set.compl_apply, Bool.not_not]
rw [h_compl]
unfold IsComputableInPolyTime at *
cases' h with tm
constructor
let f_not : Bool → Bool := fun b => !b
have h_not : TM2ComputableInPolyTime finEncodingBoolBool finEncodingBoolBool f_not := by
apply TM2ComputableInPolyTime.const
apply TM2ComputableInPolyTime.comp tm h_not
· intro h
simp only [Set.mem_setOf_eq] at h
simp only [Set.mem_setOf_eq]
unfold IsComputableInPolyTime at *
cases' h with tm
constructor
let f_not : Bool → Bool := fun b => !b
have h_not : TM2ComputableInPolyTime finEncodingBoolBool finEncodingBoolBool f_not := by
apply TM2ComputableInPolyTime.const
apply TM2ComputableInPolyTime.comp tm h_not
|
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