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NP_ne_coNP
Specification
**NP ≠ coNP**: The conjecture that the complexity classes NP and coNP are not equal.
Lean 4 Statement
theorem NP_ne_coNP : NP ≠ coNP
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Evaluation Results
Recent Submissions
| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Failed | -- | Feb 25, 2026 |
rfl
|
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"event": "search_mathlib",
"query": "coP_eq_P",
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},
{
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"event": "search_mathlib",
"query": "NP_ne_coNP",
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},
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},
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"goal_before_error": "```lean\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```",
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}
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}
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},
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"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_1772003978_4013.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772003978_4013.lean:81:2: 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_1772003978_4013.lean:84:2: error: unsolved goals\ncase h.mpr.refine_1\nL : DecisionProblem\nh : L \u2208 P\n\u22a2 L\u1d9c \u2208 P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 31.662,
"tactic_replay": {
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"first_failure_step": 3,
"goal_trajectory": [
1,
1,
2,
1,
2,
1,
1,
2
],
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{
"fullRange": {
"end": {
"character": 12,
"line": 82
},
"start": {
"character": 2,
"line": 80
}
},
"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": {
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"line": 81
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"line": 80
}
},
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"source": "Lean 4"
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{
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"character": 12,
"line": 85
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"message": "unsolved goals\ncase h.mpr.refine_1\nL : DecisionProblem\nh : L \u2208 P\n\u22a2 L\u1d9c \u2208 P",
"range": {
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"character": 0,
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{
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"end": {
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"line": 87
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},
"message": "'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
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"line": 87
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"line": 87
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],
"error_line": 80,
"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": 78,
"code": "ext L",
"goal": "```lean\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```",
"goal_count": 1,
"ast_complexity": 30
},
{
"line": 79,
"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": 80,
"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": 81,
"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": 82,
"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": 83,
"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": 84,
"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": 85,
"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
}
],
"tactic_count": 8,
"successful_tactics": 2,
"first_failure_step": 3,
"cps": 25.0,
"cps_peak_step": 2,
"error_code": "\u00b7 intro h"
}
},
{
"iteration": 8,
"event": "search_mathlib",
"query": "finEncodingBoolBool",
"results_count": 1
},
{
"iteration": 9,
"event": "search_mathlib",
"query": "TM2ComputableInPolyTime.not",
"results_count": 0
},
{
"iteration": 10,
"event": "search_mathlib",
"query": "Bool.not",
"results_count": 1
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772004019_2953.lean"
],
"comparator_passed": null,
"duration_s": 15.084689617156982,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "NP_ne_coNP",
"axioms_check_skipped": false,
"benchmark_id": "Millenium__PvsNP__NP_ne_coNP",
"category": "research_open",
"endings": [
"end ComplexityTheory"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 1987,
"source_file": "FormalConjectures/Millenium/PvsNP.lean",
"title": "NP_ne_coNP"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772004019_2953.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1772004019_2953.lean:79:2: 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.NP_ne_coNP' 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 ?_
· intro h
exact ?_
|
View{
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"response_trace": [
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{
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"reasoning_text": "**Analyzing the Complexity**\n\nI'm currently focused on proving the equality `{ L | L\u1d9c \u2208 P } = P`. Contextually, I'm examining definitions like `DecisionProblem` and `ComplexityClass`, along with the `IsComputableIn` definition. I need to understand how these relate to the initial set theory theorem. I plan to use the definitions provided in the context to determine the proper strategy.\n\n\n**Clarifying the Complementation**\n\nI've clarified the definition of `L\u1d9c` as the pointwise complement of the decision problem L, which is a function from `List Bool` to `Bool`. It leverages `Pi.instHasCompl`, thus allowing `(L\u1d9c) x = \u00ac(L x)`. This clarifies how complementation applies in the context. I'm now set to prove `{ L | L\u1d9c \u2208 P } = P`, which now means proving `(fun x => \u00ac L x) \u2208 P \u2194 L \u2208 P`. I will try to test this skeleton.\n\n\n",
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"text": "**Analyzing the Complexity**\n\nI'm currently focused on proving the equality `{ L | L\u1d9c \u2208 P } = P`. Contextually, I'm examining definitions like `DecisionProblem` and `ComplexityClass`, along with the `IsComputableIn` definition. I need to understand how these relate to the initial set theory theorem. I plan to use the definitions provided in the context to determine the proper strategy.\n\n\n**Clarifying the Complementation**\n\nI've clarified the definition of `L\u1d9c` as the pointwise complement of the decision problem L, which is a function from `List Bool` to `Bool`. It leverages `Pi.instHasCompl`, thus allowing `(L\u1d9c) x = \u00ac(L x)`. This clarifies how complementation applies in the context. I'm now set to prove `{ L | L\u1d9c \u2208 P } = P`, which now means proving `(fun x => \u00ac L x) \u2208 P \u2194 L \u2208 P`. I will try to test this skeleton.\n\n\n"
},
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},
{
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"proof_code": "ext L\nconstructor\n\u00b7 intro h\n sorry\n\u00b7 intro h\n sorry",
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},
{
"iteration": 4,
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"compiler_output_preview": "/app/asi_portal/verifier/quick_1771906669_6030.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1771906669_6030.lean:82:10: error: don't know how to synthesize placeholder\ncontext:\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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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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{
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"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```",
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{
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"code": "exact ?_",
"goal": "```lean\ncase h.mp\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n```",
"goal_count": 1,
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},
{
"line": 82,
"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,
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{
"line": 83,
"code": "exact ?_",
"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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},
{
"iteration": 5,
"event": "search_mathlib",
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{
"iteration": 6,
"event": "search_mathlib",
"query": "TM2ComputableInPolyTime.comp",
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},
{
"iteration": 7,
"event": "search_mathlib",
"query": "ComplexityTheory.IsComputableInPolyTime",
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},
{
"iteration": 8,
"event": "search_mathlib",
"query": "Turing.TM2ComputableInPolyTime",
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"reasoning_text": "**Discovering Conjecture's Source**\n\nI've hit a snag. There's no immediately obvious theorem for `TM2ComputableInPolyTime.comp`. However, I've found a matching theorem in the repository named `coP_eq_P` within `ComplexityTheory`. Could this be the relevant conjecture? I need to investigate further.\n\n\n",
"reasoning_details": [
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"index": 0,
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"text": "**Discovering Conjecture's Source**\n\nI've hit a snag. There's no immediately obvious theorem for `TM2ComputableInPolyTime.comp`. However, I've found a matching theorem in the repository named `coP_eq_P` within `ComplexityTheory`. Could this be the relevant conjecture? I need to investigate further.\n\n\n"
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"compiler_output_preview": "/app/asi_portal/verifier/quick_1771906742_6262.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1771906742_6262.lean:80:8: error: don't know how to synthesize placeholder\ncontext:\nx : \u2115 := 1\n\u22a2 {L | L\u1d9c \u2208 P} = P\n/app/asi_portal/verifier/quick_1771906742_6262.lean:78:26: error: unsolved goals\nx : \u2115 := 1\n\u22a2 {L | L\u1d9c \u2208 P} = P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"error_msg": "unsolved goals\nx : \u2115 := 1\n\u22a2 {L | L\u1d9c \u2208 P} = P",
"goal_before_error": null,
"full_timeline": [
{
"line": 78,
"code": "let x := 1",
"goal": "```lean\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```",
"goal_count": 1,
"ast_complexity": 30
},
{
"line": 79,
"code": "exact ?_",
"goal": "```lean\nx : \u2115 := 1\n\u22a2 {L | L\u1d9c \u2208 P} = P\n```",
"goal_count": 1,
"ast_complexity": 41
}
],
"tactic_count": 2,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "{ L | L\u1d9c \u2208 P } = P := by"
}
},
{
"iteration": 10,
"event": "search_mathlib",
"query": "TM2ComputableInPolyTime",
"results_count": 1
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1771906814_9447.lean"
],
"comparator_passed": null,
"duration_s": 14.073882579803467,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "NP_ne_coNP",
"axioms_check_skipped": false,
"benchmark_id": "Millenium__PvsNP__NP_ne_coNP",
"category": "research_open",
"endings": [
"end ComplexityTheory"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 1987,
"source_file": "FormalConjectures/Millenium/PvsNP.lean",
"title": "NP_ne_coNP"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1771906814_9447.lean:68:8: error: declaration uses 'sorry'\n/app/asi_portal/verifier/quick_1771906814_9447.lean:82:10: error: don't know how to synthesize placeholder\ncontext:\nL : DecisionProblem\nh : L \u2208 {L | L\u1d9c \u2208 P}\n\u22a2 L \u2208 P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Agent v8 | Failed | -- | Feb 17, 2026 |
intro h
have h_closed : ∀ L, L ∈ NP ↔ Lᶜ ∈ NP := by
intro L
constructor
· intro hL
rw [← h]
simp [coNP]
exact hL
· intro hLc
simp [coNP] at hLc
rw [h] at hLc
exact hLc
simp [NP, coNP, P] at *
|
View{
"agent_telemetry": {
"compilations": 8,
"iterations": 8,
"duration_s": 217.43,
"best_status": "failed",
"best_goals": null,
"max_iterations": 8,
"model": "gemini-3-flash-preview",
"error": null,
"provider": "gemini_direct",
"finish_reason": "STOP",
"usage": {
"prompt_tokens": 184127,
"completion_tokens": 624,
"total_tokens": 210748,
"reasoning_tokens": 25997,
"cost_usd": 0.0
},
"reasoning_effort": "",
"response_trace_count": 16
},
"response_trace": [
{
"iteration": 1,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 102.905,
"usage": {
"prompt_tokens": 2307,
"completion_tokens": 113,
"total_tokens": 20567,
"reasoning_tokens": 18147,
"cost_usd": 0.0
},
"thoughts_tokens": 18147,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 1,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro h\nhave h_closed : \u2200 L, L \u2208 NP \u2194 L\u1d9c \u2208 NP := by\n intro L\n constructor\n \u00b7 intro hL\n rw [\u2190 h]\n simp [coNP]\n exact hL\n \u00b7 intro hLc\n simp [coNP] at hLc\n rw [h] at hLc\n exact hLc\nsimp [NP, coNP, P] at *",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:79:8: error: tactic 'introN' failed, insufficient number of binders\n\u22a2 {L | L\u1d9c \u2208 P} = P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 2,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 8.506,
"usage": {
"prompt_tokens": 20783,
"completion_tokens": 59,
"total_tokens": 21946,
"reasoning_tokens": 1104,
"cost_usd": 0.0
},
"thoughts_tokens": 1104,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 2,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "try (intro h)\ntry (apply Set.ext; intro L)\ntry (simp [coNP] at *)\ntry (rw [\u2190 h] at *)\ntry (simp at *)",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:78:26: error: unsolved goals\ncase h\nL : DecisionProblem\n\u22a2 L\u1d9c \u2208 P \u2194 L \u2208 P\n[path]:81:13: warning: This simp argument is unused:\n coNP\n\nHint: Omit \n...<truncated>"
},
{
"iteration": 3,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 13.236,
"usage": {
"prompt_tokens": 22289,
"completion_tokens": 106,
"total_tokens": 24238,
"reasoning_tokens": 1843,
"cost_usd": 0.0
},
"thoughts_tokens": 1843,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 3,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "try (intro h_eq)\ntry (ext L; constructor)\ntry (\n intro hL\n simp [P, IsComputableInPolyTime] at *\n cases hL\n fconstructor\n assumption\n)\ntry (\n intro hL\n simp [P, IsComputableInPolyTime] at *\n cases hL\n fconstructor\n assumption\n)",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:78:26: error: unsolved goals\ncase h.mp\nL : DecisionProblem\n\u22a2 L \u2208 {L | L\u1d9c \u2208 P} \u2192 L \u2208 P\n\ncase h.mpr\nL : DecisionProblem\n\u22a2 L \u2208 P \u2192 L \u2208 {L | L\u1d9c \u2208 P}\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 4,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 14.555,
"usage": {
"prompt_tokens": 24492,
"completion_tokens": 40,
"total_tokens": 26668,
"reasoning_tokens": 2136,
"cost_usd": 0.0
},
"thoughts_tokens": 2136,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "all_goals (try intro)\nall_goals (try ext)\nall_goals (try simp)",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:78:26: error: unsolved goals\ncase h\nx\u271d : DecisionProblem\n\u22a2 x\u271d\u1d9c \u2208 P \u2194 x\u271d \u2208 P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 5,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 6.862,
"usage": {
"prompt_tokens": 26891,
"completion_tokens": 39,
"total_tokens": 27818,
"reasoning_tokens": 888,
"cost_usd": 0.0
},
"thoughts_tokens": 888,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 5,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "try (intro h; aesop)\ntry (apply Set.ext; intro L; aesop)",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:80:31: warning: aesop: failed to prove the goal after exhaustive search.\n[path]:78:26: error: unsolved goals\ncase h.mp\nL : DecisionProblem\na : L\u1d9c\n...<truncated>"
},
{
"iteration": 6,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 6.063,
"usage": {
"prompt_tokens": 28136,
"completion_tokens": 62,
"total_tokens": 28936,
"reasoning_tokens": 738,
"cost_usd": 0.0
},
"thoughts_tokens": 738,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 6,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "try (intro h; cases h)\ntry (ext L; constructor; intro hL; cases hL; fconstructor; assumption)\ntry (intro hL; cases hL; fconstructor; assumption)",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:78:26: error: unsolved goals\n\u22a2 {L | L\u1d9c \u2208 P} = P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 7,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 5.43,
"usage": {
"prompt_tokens": 29149,
"completion_tokens": 94,
"total_tokens": 29867,
"reasoning_tokens": 624,
"cost_usd": 0.0
},
"thoughts_tokens": 624,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 7,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "try (intro h_eq)\ntry (\n ext L\n constructor\n \u00b7 intro hLc; simp [P, IsComputableInPolyTime] at *; cases hLc; fconstructor; assumption\n \u00b7 intro hL; simp [P, IsComputableInPolyTime] at *; cases hL; fconstructor; assumption\n)",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:78:26: error: unsolved goals\n\u22a2 {L | L\u1d9c \u2208 P} = P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 8,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 5.207,
"usage": {
"prompt_tokens": 30080,
"completion_tokens": 111,
"total_tokens": 30708,
"reasoning_tokens": 517,
"cost_usd": 0.0
},
"thoughts_tokens": 517,
"gemini_retries": 0,
"has_function_call": true,
"response_text_len": 0,
"model_text_preview": ""
},
{
"iteration": 8,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "try (intro h_eq; simp [NP, coNP, P] at *; aesop)\ntry (\n apply Set.ext\n intro L\n constructor\n \u00b7 intro h; simp [P, IsComputableInPolyTime] at *; cases h; fconstructor; assumption\n \u00b7 intro h; simp [P, IsComputableInPolyTime] at *; cases h; fconstructor; assumption\n)",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:78:26: error: unsolved goals\n\u22a2 {L | L\u1d9c \u2208 P} = P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"[path]
],
"duration_s": 6.721222400665283,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"axioms_check_name": "NP_ne_coNP",
"axioms_check_skipped": false,
"benchmark_id": "Millenium__PvsNP__NP_ne_coNP",
"category": "research_open",
"endings": [
"end ComplexityTheory"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 1987,
"source_file": "FormalConjectures/Millenium/PvsNP.lean",
"title": "NP_ne_coNP"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:68:8: error: declaration uses 'sorry'\n[path]:79:8: error: tactic 'introN' failed, insufficient number of binders\n\u22a2 {L | L\u1d9c \u2208 P} = P\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Agent v7 | Failed | -- | Feb 16, 2026 |
ext L
simp [P]
constructor
· intro h
have : L = (fun b => !b) ∘ (fun x => !(L x)) := by
ext x
simp
rw [this]
apply IsComputableInPolyTime.comp
· exact isComputableInPolyTime_not
· exact h
· intro h
have : (fun x => !(L x)) = (fun b => !b) ∘ L := rfl
rw [this]
apply IsComputableInPolyTime.comp
· exact isComputableInPolyTime_not
· exact h
|
View{
"agent_telemetry": {
"compilations": 8,
"iterations": 8,
"duration_s": 215.76,
"best_status": "failed",
"best_goals": null,
"max_iterations": 8,
"model": "gemini-3-flash-preview",
"error": null,
"provider": "gemini_direct",
"finish_reason": "STOP",
"usage": {
"prompt_tokens": 164060,
"completion_tokens": 776,
"total_tokens": 190123,
"reasoning_tokens": 25287,
"cost_usd": 0.0
},
"reasoning_effort": "",
"response_trace_count": 16
},
"response_trace": [
{
"iteration": 1,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 49.514,
"usage": {
"prompt_tokens": 2307,
"completion_tokens": 160,
"total_tokens": 10330,
"reasoning_tokens": 7863,
"cost_usd": 0.0
},
"has_function_call": true,
"response_text_len": 0
},
{
"iteration": 1,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "ext L\nsimp [P]\nconstructor\n\u00b7 intro h\n have : L = (fun b => !b) \u2218 (fun x => !(L x)) := by\n ext x\n simp\n rw [this]\n apply IsComputableInPolyTime.comp\n \u00b7 exact isComputableInPolyTime_not\n \u00b7 exact h\n\u00b7 intro h\n have : (fun x => !(L x)) = (fun b => !b) \u2218 L := rfl\n rw [this]\n apply IsComputableInPolyTime.comp\n \u00b7 exact isComputableInPolyTime_not\n \u00b7 exact h",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:87:4: error: tactic 'apply' failed, could not unify the conclusion of `Function.comp IsComputableInPolyTime`\n Prop\nwith the goal\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool ((fun b \u21a6 !b) \u2218 fun x \u21a6 !L x)\n\nNote: The full type \n...<truncated>"
},
{
"iteration": 2,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 41.469,
"usage": {
"prompt_tokens": 10908,
"completion_tokens": 67,
"total_tokens": 17765,
"reasoning_tokens": 6790,
"cost_usd": 0.0
},
"has_function_call": true,
"response_text_len": 0
},
{
"iteration": 2,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "ext L\nsimp [P, IsComputableInPolyTime]\nconstructor\n\u00b7 rintro \u27e8tm\u27e9\n exact \u27e8tm.not\u27e9\n\u00b7 rintro \u27e8tm\u27e9\n exact \u27e8tm.not\u27e9",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:83:11: error: Invalid field `not`: The environment does not contain `Turing.TM2ComputableInPolyTime.not`\n tm\nhas type\n TM2ComputableInPolyTime finEncodingListBool finEncodingBoolBool L\u1d9c\n[path]
},
{
"iteration": 3,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 34.067,
"usage": {
"prompt_tokens": 18119,
"completion_tokens": 71,
"total_tokens": 24005,
"reasoning_tokens": 5815,
"cost_usd": 0.0
},
"has_function_call": true,
"response_text_len": 0
},
{
"iteration": 3,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "ext L\nsimp [P, IsComputableInPolyTime]\nconstructor\n\u00b7 rintro \u27e8tm\u27e9\n refine \u27e8tm.comp _\u27e9\n\u00b7 rintro \u27e8tm\u27e9\n refine \u27e8tm.comp _\u27e9",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:83:12: error: Invalid field `comp`: The environment does not contain `Turing.TM2ComputableInPolyTime.comp`\n tm\nhas type\n TM2ComputableInPolyTime finEncodingListBool finEncodingBoolBool L\u1d9c\n[path]
},
{
"iteration": 4,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 3.386,
"usage": {
"prompt_tokens": 24360,
"completion_tokens": 27,
"total_tokens": 24767,
"reasoning_tokens": 380,
"cost_usd": 0.0
},
"has_function_call": true,
"response_text_len": 0
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "ext L\nsimp [P]\naesop",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:81:2: warning: aesop: failed to prove the goal after exhaustive search.\n[path]:78:26: error: unsolved goals\ncase h.mp\nL : DecisionProblem\na : IsC\n...<truncated>"
},
{
"iteration": 5,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 5.2,
"usage": {
"prompt_tokens": 25131,
"completion_tokens": 57,
"total_tokens": 25812,
"reasoning_tokens": 624,
"cost_usd": 0.0
},
"has_function_call": true,
"response_text_len": 0
},
{
"iteration": 5,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "ext L\napply Iff.intro\n\u00b7 intro h\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]:82:10: error: unknown constant 'ComplexityTheory.IsComputableInPolyTime.compl'\n[path]:84:10: error: unknown constant 'ComplexityTheory.IsComputab\n...<truncated>"
},
{
"iteration": 6,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 8.24,
"usage": {
"prompt_tokens": 26100,
"completion_tokens": 137,
"total_tokens": 27347,
"reasoning_tokens": 1110,
"cost_usd": 0.0
},
"has_function_call": true,
"response_text_len": 0
},
{
"iteration": 6,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "ext L\nsimp [P, IsComputableInPolyTime]\nconstructor\n\u00b7 rintro \u27e8tm\u27e9\n refine \u27e8Turing.TM2ComputableInPolyTime.comp _ _ _ _ _ _ tm _\u27e9\n \u00b7 exact Turing.TM2ComputableInPolyTime.bool_not\n\u00b7 rintro \u27e8tm\u27e9\n refine \u27e8Turing.TM2ComputableInPolyTime.comp _ _ _ _ _ _ tm _\u27e9\n \u00b7 exact Turing.TM2ComputableInPolyTime.bool_not",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:83:12: error: Function expected at\n (TM2ComputableInPolyTime \u2218 ?m.1640) ?m.1641 ?m.1642 ?m.1643\nbut this term has type\n Type 1\n\nNote: Expected a function because this term is being applied to the argument\n _\n[path]
},
{
"iteration": 7,
"provider": "gemini_direct",
"finish_reason": "STOP",
"api_elapsed_s": 8.616,
"usage": {
"prompt_tokens": 27877,
"completion_tokens": 46,
"total_tokens": 29142,
"reasoning_tokens": 1219,
"cost_usd": 0.0
},
"has_function_call": true,
"response_text_len": 0
},
{
"iteration": 7,
"event": "compile_proof",
"provider": "gemini_direct",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "ext L\nsimp [P]\napply Iff.intro\n\u00b7 intro h\n exact?\n\u00b7 intro h\n exact?",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n\nINTERNAL PANIC: out of memory\n"
},
{
"iteration": 8,
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"proof_code": "ext L\nsimp [P]\napply Iff.intro\n\u00b7 intro h\n have h_not : IsComputableInPolyTime finEncodingBoolBool finEncodingBoolBool (fun b => !b) := by\n -- let's try to see if this is an instance or a lemma\n simp [IsComputableInPolyTime]\n exact \u27e8TM2ComputableInPolyTime.bool_not\u27e9\n exact IsComputableInPolyTime.comp _ _ _ _ _ h h_not\n\u00b7 intro h\n have h_not : IsComputableInPolyTime finEncodingBoolBool finEncodingBoolBool (fun b => !b) := by\n simp [IsComputableInPolyTime]\n exact \u27e8TM2ComputableInPolyTime.bool_not\u27e9\n exact IsComputableInPolyTime.comp _ _ _ _ _ h h_not",
"compiler_output_preview": "[path]:68:8: error: declaration uses 'sorry'\n[path]:86:13: error: unknown constant 'Turing.TM2ComputableInPolyTime.bool_not'\n[path]:87:10: error: Function expected at\n (IsComputableInPolyTime \u2218 ?m\n...<truncated>"
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"category": "research_open",
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"stdout": "[path]:68:8: error: declaration uses 'sorry'\n[path]:87:4: error: tactic 'apply' failed, could not unify the conclusion of `Function.comp IsComputableInPolyTime`\n Prop\nwith the goal\n IsComputableInPolyTime finEncodingListBool finEncodingBoolBool ((fun b \u21a6 !b) \u2218 fun x \u21a6 !L x)\n\nNote: The full type of `Function.comp IsComputableInPolyTime` is\n (?m.1563 \u2192 FinEncoding ?m.1552) \u2192 ?m.1563 \u2192 FinEncoding ?m.1553 \u2192 (?m.1552 \u2192 ?m.1553) \u2192 Prop\ncase h.mp\nL : DecisionProblem\nh : IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\u1d9c\nthis : L = (fun b \u21a6 !b) \u2218 fun x \u21a6 !L x\n\u22a2 IsComputableInPolyTime finEncodingListBool finEncodingBoolBool ((fun b \u21a6 !b) \u2218 fun x \u21a6 !L x)\n[path]:92:8: error: tactic 'rewrite' failed, did not find instance of the pattern in the target expression\n fun x \u21a6 !L x\ncase h.mpr\nL : DecisionProblem\nh : IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\nthis : (fun x \u21a6 !L x) = (fun b \u21a6 !b) \u2218 L\n\u22a2 IsComputableInPolyTime finEncodingListBool finEncodingBoolBool L\u1d9c\n'ComplexityTheory.NP_ne_coNP' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
simp [IsComputableInPolyTime]
infer_instance
have h_comp : IsComputableInPolyTime finEncodingListBool finEncodingBoolBool (f ∘ Lᶜ) := by
apply Nonempty.intro
sorry_placeholder_for_composition
convert h_comp
ext x
simp [f]
· intro h
simp [P] at *
rcases h with ⟨tm⟩
let f : Bool → Bool := bnot
have hf : IsComputableInPolyTime finEncodingBoolBool finEncodingBoolBool f := by
simp [IsComputableInPolyTime]
infer_instance
have h_comp : IsComputableInPolyTime finEncodingListBool finEncodingBoolBool (f ∘ L) := by
apply Nonempty.intro
sorry_placeholder_for_composition
convert h_comp
ext x
simp [f]
|
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|