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pi_pow_pi_transcendental
Specification
$\pi^{\pi}$ is transcendental.
Lean 4 Statement
theorem pi_pow_pi_transcendental : Transcendental ℚ (π ^ π)
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Evaluation Results
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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 : Transcendental ℚ (π ^ π) := ?_
exact h
|
View{
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"duration_s": 104.46,
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"query": "transcendental_pi_pow_pi",
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"event": "search_mathlib",
"query": "transcendental_pi",
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"query": "Real.transcendental",
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},
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"query": "pi transcendental",
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},
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"compiler_output_preview": "/app/asi_portal/verifier/quick_1772030078_4167.lean:18:63: error: unsolved goals\ncase refine_1\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0)\n'Transcendental.pi_pow_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"goal_before_error": null,
"full_timeline": [
{
"line": 18,
"code": "have h : Transcendental \u211a (\u03c0 ^ \u03c0) := ?_",
"goal": "```lean\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0)\n```",
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"code": "exact h",
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}
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"error_code": "theorem pi_pow_pi_transcendental : Transcendental \u211a (\u03c0 ^ \u03c0) := by"
}
},
{
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"query": "rpow_pi transcendental",
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},
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"proof_code": "have h : Transcendental \u211a (\u03c0 ^ \u03c0) := by\n simpa using Real.transcendental_pi_pow_pi\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772030132_5002.lean:20:16: error(lean.unknownIdentifier): Unknown constant `Real.transcendental_pi_pow_pi`\n'Transcendental.pi_pow_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"goal_before_error": "```lean\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0)\n```",
"full_timeline": [
{
"line": 18,
"code": "have h : Transcendental \u211a (\u03c0 ^ \u03c0) := by",
"goal": "```lean\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0)\n```",
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"goal": "```lean\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0)\n```",
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"code": "exact h",
"goal": "```lean\nh : Transcendental \u211a (\u03c0 ^ \u03c0)\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0)\n```",
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"error_code": "simpa using Real.transcendental_pi_pow_pi"
}
},
{
"iteration": 10,
"event": "search_mathlib",
"query": "pi_transcendental",
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}
],
"verification": {
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"command": [
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"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "pi_pow_pi_transcendental",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Transcendental__pi_pow_pi_transcendental",
"category": "research_open",
"endings": [
"end Transcendental"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 167,
"source_file": "FormalConjectures/Wikipedia/Transcendental.lean",
"title": "pi_pow_pi_transcendental"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772030163_7453.lean:18:63: error: unsolved goals\ncase refine_1\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0)\n'Transcendental.pi_pow_pi_transcendental' 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{
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"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 | Partial | 2 | Feb 17, 2026 |
intro h_alg
unfold IsAlgebraic at h_alg
obtain ⟨p, p_nz, p_ev⟩ := h_alg
suffices False from this.elim
refine ?_
case _ =>
have h_pi : Transcendental ℚ π
· refine ?_
have h_exp : IsAlgebraic ℚ (π ^ π)
· exact ⟨p, p_nz, p_ev⟩
refine ?_
|
View{
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"proof_code": "intro h\nrw [Transcendental, IsAlgebraic] at h\npush_neg at h\nspecialize h (\u03c0 ^ \u03c0)\napply h\nexact rfl",
"compiler_output_preview": "[path]:20:6: error: failed to rewrite using equation theorems for 'Transcendental'.\n'Transcendental.pi_pow_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
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"proof_code": "unfold Transcendental at h\nunfold IsAlgebraic at h\nobtain \u27e8p, p_nonzero, p_roots\u27e9 := h\nhave h_trans : Transcendental \u211a \u03c0 := Real.pi_transcendental\nhave h_exp_trans : \u00ac IsAlgebraic \u211a (\u03c0 ^ \u03c0)\n\u00b7 intro h_alg\n refine ?_\n\u00b7 exact h_exp_trans h",
"compiler_output_preview": "[path]:19:27: error: unknown identifier 'h'\n'Transcendental.pi_pow_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "intro h_alg\nunfold IsAlgebraic at h_alg\nobtain \u27e8p, p_nonzero, p_roots\u27e9 := h_alg\nhave h_pi : Transcendental \u211a \u03c0 := Real.pi_transcendental\nhave h_not_alg : \u00ac IsAlgebraic \u211a (\u03c0 ^ \u03c0) := by\n intro h_invalid\n have : IsAlgebraic \u211a \u03c0 := by\n refine ?_\n exact h_pi this\nexact h_not_alg \u27e8p, p_nonzero, p_roots\u27e9",
"compiler_output_preview": "[path]:22:36: error: unknown constant 'Real.pi_transcendental'\n[path]:25:30: error: unsolved goals\np : Polynomial \u211a\np_nonzero : p \u2260 0\np_roots : (Polynomial.aeval (\u03c0 ^ \u03c0)) p = 0\nh_pi : Transcendental \u211a \u03c0\nh_invalid : IsAlgebraic \u211a (\u03c0 ^ \u03c0)\n\u22a2 IsAlgebraic \u211a\n...<truncated>"
},
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"proof_code": "intro h_alg\nobtain \u27e8p, p_nz, p_ev\u27e9 := h_alg\nhave h_pi_trans : Transcendental \u211a \u03c0 := Transcendental.pi_transcendental\nhave : IsAlgebraic \u211a (\u03c0 ^ \u03c0) := \u27e8p, p_nz, p_ev\u27e9\nrefine ?_\ncase _ =>\n have h_pi_pi := GelfondSchneider ?_ ?_ ?_\n \u00b7 exact ?_\n \u00b7 exact ?_\n \u00b7 exact ?_\n \u00b7 exact ?_",
"compiler_output_preview": "[path]:21:42: error: unknown constant 'Transcendental.pi_transcendental'\n[path]:25:20: error: unknown identifier 'GelfondSchneider'\n'Transcendental.pi_pow_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "intro h_alg\nunfold IsAlgebraic at h_alg\nobtain \u27e8p, p_nz, p_ev\u27e9 := h_alg\nsuffices False from this.elim\nrefine ?_\ncase _ =>\n have h_pi : Transcendental \u211a \u03c0\n \u00b7 refine ?_\n have h_exp : IsAlgebraic \u211a (\u03c0 ^ \u03c0)\n \u00b7 exact \u27e8p, p_nz, p_ev\u27e9\n refine ?_",
"compiler_output_preview": "[path]:26:4: error: unsolved goals\ncase h_pi\np : Polynomial \u211a\np_nz : p \u2260 0\np_ev : (Polynomial.aeval (\u03c0 ^ \u03c0)) p = 0\n\u22a2 Transcendental \u211a \u03c0\n[path]:24:9: error: unsolved goals\np : Polynomial \u211a\np_nz : p \u2260 0\np_ev : (Polynomial.aeval (\u03c0 ^ \u03c0)) p = 0\nh_pi : Tran\n...<truncated>"
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"proof_code": "intro h_alg\nobtain \u27e8p, p_nz, p_ev\u27e9 := h_alg\nsuffices False from this.elim\nhave h_pi_base : Transcendental \u211a \u03c0 := by\n rw [Transcendental]\n intro h_pi_alg\n obtain \u27e8q, q_nz, q_ev\u27e9 := h_pi_alg\n refine ?_\nhave h_pi_pow : IsAlgebraic \u211a (\u03c0 ^ \u03c0) := \u27e8p, p_nz, p_ev\u27e9\nrefine ?_",
"compiler_output_preview": "[path]:22:41: error: unsolved goals\ncase intro.intro\np : Polynomial \u211a\np_nz : p \u2260 0\np_ev : (Polynomial.aeval (\u03c0 ^ \u03c0)) p = 0\nq : Polynomial \u211a\nq_nz : q \u2260 0\nq_ev : (Polynomial.aeval \u03c0) q = 0\n\u22a2 False\n[path]:18:63: error: unsolved goals\ncase intro.intro\np : \n...<truncated>"
},
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"proof_code": "intro h_alg\nobtain \u27e8p, p_nz, p_ev\u27e9 := h_alg\nhave h_pi : Transcendental \u211a \u03c0\n\u00b7 intro h_pi_alg\n obtain \u27e8q, q_nz, q_ev\u27e9 := h_pi_alg\n refine ?_\nhave h_ge : \u03c0 > 0 := Real.pi_pos\nhave h_ne : \u03c0 \u2260 1 := by\n have : 3 < \u03c0 := pi_gt_three\n linarith\nhave h_log : Complex.log (Complex.ofReal \u03c0) \u2260 0 := by\n rw [Complex.log_ne_zero_iff]\n push_neg\n constructor\n \u00b7 rw [Complex.ofReal_ne_zero]\n exact Real.pi_ne_zero\n \u00b7 intro h_one\n injection h_one with h_eq\n exact h_ne h_eq\nrefine ?_",
"compiler_output_preview": "[path]:22:2: error: unsolved goals\ncase h_pi.intro.intro\np : Polynomial \u211a\np_nz : p \u2260 0\np_ev : (Polynomial.aeval (\u03c0 ^ \u03c0)) p = 0\nq : Polynomial \u211a\nq_nz : q \u2260 0\nq_ev : (Polynomial.aeval \u03c0) q = 0\n\u22a2 False\n[path]:30:8: error: unknown constant 'Complex.log_ne_\n...<truncated>"
},
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"proof_code": "intro h_alg\nobtain \u27e8p, p_nz, p_ev\u27e9 := h_alg\nhave h_pos : \u03c0 > 0 := pi_pos\nhave h_ne_one : \u03c0 \u2260 1 := by\n have : 3 < \u03c0 := pi_gt_three\n linarith\nhave h_pi_trans : Transcendental \u211a \u03c0 := by\n intro h_pi_alg\n obtain \u27e8q, q_nz, q_ev\u27e9 := h_pi_alg\n refine ?_\nhave h1 : IsAlgebraic \u211a (\u03c0 ^ \u03c0) := \u27e8p, p_nz, p_ev\u27e9\nrefine ?_\ncase _ =>\n obtain \u27e8r, r_nz, r_ev\u27e9 := h1\n refine ?_",
"compiler_output_preview": "[path]:25:42: error: unsolved goals\ncase intro.intro\np : Polynomial \u211a\np_nz : p \u2260 0\np_ev : (Polynomial.aeval (\u03c0 ^ \u03c0)) p = 0\nh_pos : \u03c0 > 0\nh_ne_one : \u03c0 \u2260 1\nq : Polynomial \u211a\nq_nz : q \u2260 0\nq_ev : (Polynomial.aeval \u03c0) q = 0\n\u22a2 False\n[path]:31:9: error: unsolv\n...<truncated>"
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"source_file": "FormalConjectures/Wikipedia/Transcendental.lean",
"title": "pi_pow_pi_transcendental"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:26:4: error: unsolved goals\ncase h_pi\np : Polynomial \u211a\np_nz : p \u2260 0\np_ev : (Polynomial.aeval (\u03c0 ^ \u03c0)) p = 0\n\u22a2 Transcendental \u211a \u03c0\n[path]:24:9: error: unsolved goals\np : Polynomial \u211a\np_nz : p \u2260 0\np_ev : (Polynomial.aeval (\u03c0 ^ \u03c0)) p = 0\nh_pi : Transcendental \u211a \u03c0\nh_exp : IsAlgebraic \u211a (\u03c0 ^ \u03c0)\n\u22a2 False\n'Transcendental.pi_pow_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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}
|
| Gemini 3 Flash Preview | Partial | 1 | Feb 14, 2026 |
rw [rpow_def_of_pos h_pi_pos]
have h_not_alg : ¬ IsAlgebraic ℚ π := transcendental_pi
have h_alg_pow : IsAlgebraic ℚ (π ^ π) := by assumption
aesop
· exfalso
aesop
|
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"stdout": "[path]:19:22: error: unknown identifier 'h_pi_pos'\n[path]:18:63: error: unsolved goals\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0)\n[path]:20:4: error: unexpected token 'have'; expected command\n",
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|