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pi_pow_pi_pow_pi_pow_pi_transcendental
Specification
$\pi^{\pi^{\pi^\pi}}$ is transcendental.
Lean 4 Statement
theorem pi_pow_pi_pow_pi_pow_pi_transcendental : Transcendental ℚ (π ^ (π ^ (π ^ π)))
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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 ^ (π ^ π))) := ?_
exact h
|
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},
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}
]
},
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"proof_code": "have h : Transcendental \u211a (\u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))) := ?_\n exact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772030057_8269.lean:18:89: error: unsolved goals\ncase refine_1\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n\n",
"compile_elapsed_s": 32.512,
"tactic_replay": {
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"first_failure_step": 1,
"goal_trajectory": [
1,
2
],
"max_goals": 2,
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"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 9,
"line": 19
},
"start": {
"character": 89,
"line": 17
}
},
"leanTags": [
1
],
"message": "unsolved goals\ncase refine_1\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)",
"range": {
"end": {
"character": 0,
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{
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"start": {
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},
"message": "'Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]",
"range": {
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},
"start": {
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"line": 21
}
},
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"source": "Lean 4"
}
],
"error_line": 17,
"error_char": 89,
"error_msg": "unsolved goals\ncase refine_1\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)",
"goal_before_error": null,
"full_timeline": [
{
"line": 18,
"code": "have h : Transcendental \u211a (\u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))) := ?_",
"goal": "```lean\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n```",
"goal_count": 1,
"ast_complexity": 46
},
{
"line": 19,
"code": "exact h",
"goal": "```lean\ncase refine_2\nh : Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n```\n---\n```lean\ncase refine_1\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n```",
"goal_count": 2,
"ast_complexity": 162
}
],
"tactic_count": 2,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "theorem pi_pow_pi_pow_pi_pow_pi_transcendental : Transcendental \u211a (\u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))) := by"
}
}
],
"verification": {
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"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772030090_2512.lean"
],
"comparator_passed": null,
"duration_s": 17.436927556991577,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "pi_pow_pi_pow_pi_pow_pi_transcendental",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Transcendental__pi_pow_pi_pow_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_pow_pi_pow_pi_transcendental"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772030090_2512.lean:18:89: error: unsolved goals\ncase refine_1\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n 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\"}",
"provider": "openrouter_primary",
"finish_reason": "",
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},
"reasoning_effort": "high",
"response_trace_count": 0
},
"response_trace": []
}
|
|
| Gemini 3 Flash Agent v8 | Partial | 1 | Feb 17, 2026 |
intro h
have h_not : ¬ IsAlgebraic ℚ (π ^ (π ^ (π ^ π))) := by
intro h_alg
refine ?_
apply h_not
exact h
|
View{
"agent_telemetry": {
"compilations": 8,
"iterations": 8,
"duration_s": 75.89,
"best_status": "partial",
"best_goals": 1,
"max_iterations": 8,
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"error": null,
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"response_trace": [
{
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"finish_reason": "stop",
"response_id": "gen-1771350471-bUTTgq0tc36ThTuI2gKV",
"api_elapsed_s": 1.83,
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{
"iteration": 1,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro h\nunfold Transcendental at h\nlet x := \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))\nhave hx : IsAlgebraic \u211a x := h\nrcases hx with \u27e8p, hp_nz, hp_root\u27e9\nhave h_poly : p.eval x = 0 := hp_root\nhave h_pi : Transcendental \u211a \u03c0 := pi_transcendental\ncases h_pi\nrefine ?_\nunfold Transcendental\nintro h_pi_alg\napply pi_transcendental\nexact h_pi_alg",
"compiler_output_preview": "[path]:20:9: error: tactic 'unfold' failed to unfold 'Transcendental' at\n IsAlgebraic \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n"
},
{
"iteration": 2,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771350480-s9Jyl9dfYeRUC6UH2doG",
"api_elapsed_s": 1.907,
"usage": {
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{
"iteration": 2,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
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"banned_tokens": [],
"proof_code": "intro h\nhave h_pi : Transcendental \u211a \u03c0 := pi_transcendental\napply h_pi\nrefine ?_\nclear h_pi\ninduction h using IsAlgebraic.induction_on\n\u00b7 constructor\n\u00b7 rename_i p hp_nz hp_root\n refine ?_\n\u00b7 rename_i x y hx hy ihx ihy\n refine ?_\n\u00b7 rename_i x y hx hy ihx ihy\n refine ?_\n\u00b7 rename_i x hx ihx\n refine ?_\n\u00b7 rename_i x hx ihx\n refine ?_",
"compiler_output_preview": "[path]:20:36: error: unknown identifier 'pi_transcendental'\n[path]:24:20: error: unknown constant 'IsAlgebraic.induction_on'\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental' depends on axioms: [propext,\n sorryAx,\n Cla\n...<truncated>"
},
{
"iteration": 3,
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"finish_reason": "stop",
"response_id": "gen-1771350490-DhbLEQ0EkpczKZ2r3LFW",
"api_elapsed_s": 1.707,
"usage": {
"prompt_tokens": 2421,
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},
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},
{
"iteration": 3,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro h\nunfold Transcendental at h\nlet val := \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))\nhave h_alg : IsAlgebraic \u211a val := h\nclear h\ncontrapose! h_alg\nrw [Transcendental] at h_alg\nrefine ?_\n\u00b7 refine ?_\n\u00b7 aesop",
"compiler_output_preview": "[path]:20:9: error: tactic 'unfold' failed to unfold 'Transcendental' at\n IsAlgebraic \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n"
},
{
"iteration": 4,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771350499-YuxV2I5IR5faov8TqjTO",
"api_elapsed_s": 1.26,
"usage": {
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"proof_code": "intro h\nhave h_not : \u00ac IsAlgebraic \u211a (\u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))) := by\n intro h_alg\n refine ?_\napply h_not\nexact h",
"compiler_output_preview": "[path]:20:54: error: unsolved goals\nh h_alg : IsAlgebraic \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n\u22a2 False\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n"
},
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"proof_code": "intro h\nhave h_pi : \u03c0 = Real.pi := rfl\nhave h_trans : \u00ac IsAlgebraic \u211a (\u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))) := by\n intro h_alg\n let x := \u03c0\n let y := x ^ x\n let z := x ^ y\n let w := x ^ z\n have : w = \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0)) := rfl\n have h_w : IsAlgebraic \u211a w := h_alg\n have : Transcendental \u211a \u03c0 := by\n rw [h_pi]\n exact Real.pi_transcendental\n have : \u00ac IsAlgebraic \u211a \u03c0 := this\n refine ?_\napply h_trans\nexact h",
"compiler_output_preview": "[path]:31:12: error: unknown constant 'Real.pi_transcendental'\n[path]:21:56: error: unsolved goals\nh : IsAlgebraic \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\nh_pi : \u03c0 = \u03c0\nh_alg : IsAlgebraic \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\nx : \u211d := \u03c0\ny : \u211d := x ^ x\nz : \u211d := x ^ y\n...<truncated>"
},
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"proof_code": "intro h\nhave h_not : \u00ac IsAlgebraic \u211a (\u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))) := by\n intro h_alg\n have h_pi_trans : Transcendental \u211a \u03c0 := Transcendental.pi_transcendental\n have h_pi_not_alg : \u00ac IsAlgebraic \u211a \u03c0 := h_pi_trans\n refine ?_\napply h_not\nexact h",
"compiler_output_preview": "[path]:22:44: error: unknown constant 'Transcendental.pi_transcendental'\n[path]:20:54: error: unsolved goals\nh h_alg : IsAlgebraic \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\nh_pi_trans : Transcendental \u211a \u03c0\nh_pi_not_alg : \u00acIsAlgebraic \u211a \u03c0\n\u22a2 False\n'Tr\n...<truncated>"
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"proof_code": "intro h\nhave h_not_alg : \u00ac IsAlgebraic \u211a (\u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))) := by\n intro h_alg\n let x := \u03c0\n let y := x ^ x\n let z := x ^ y\n let w := x ^ z\n have hw : w = \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0)) := rfl\n have : IsAlgebraic \u211a w := h_alg\n refine ?_\napply h_not_alg\nexact h",
"compiler_output_preview": "[path]:20:58: error: unsolved goals\nh h_alg : IsAlgebraic \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\nx : \u211d := \u03c0\ny : \u211d := x ^ x\nz : \u211d := x ^ y\nw : \u211d := x ^ z\nhw : w = \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0\nthis : IsAlgebraic \u211a w\n\u22a2 False\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n"
},
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"status": "partial",
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"proof_code": "intro h\nhave h_not_alg : \u00ac IsAlgebraic \u211a (\u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))) := by\n intro h_alg\n let x1 := \u03c0\n let x2 := x1 ^ x1\n let x3 := x1 ^ x2\n let x4 := x1 ^ x3\n have h_x4 : x4 = \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0)) := rfl\n rw [\u2190 h_x4] at h_alg\n have h_pi_not_alg : \u00ac IsAlgebraic \u211a \u03c0 := by\n intro h_pi_alg\n refine ?_\n refine ?_\napply h_not_alg\nexact h",
"compiler_output_preview": "[path]:28:45: error: unsolved goals\nh : IsAlgebraic \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\nx1 : \u211d := \u03c0\nx2 : \u211d := x1 ^ x1\nx3 : \u211d := x1 ^ x2\nx4 : \u211d := x1 ^ x3\nh_alg : IsAlgebraic \u211a x4\nh_x4 : x4 = \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0\nh_pi_alg : IsAlgebraic \u211a \u03c0\n\u22a2 False\n[path]
}
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"problem": {
"axioms_check_name": "pi_pow_pi_pow_pi_pow_pi_transcendental",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Transcendental__pi_pow_pi_pow_pi_pow_pi_transcendental",
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},
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"stderr": "",
"stdout": "[path]:20:54: error: unsolved goals\nh h_alg : IsAlgebraic \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n\u22a2 False\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n",
"timed_out": false
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}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
apply ne_of_gt
exact pi_gt_one
contrapose! h
apply transcendental_pi
exact h
|
View{
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"title": "pi_pow_pi_pow_pi_pow_pi_transcendental"
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"status": "failed",
"stderr": "",
"stdout": "[path]:20:4: error: unknown identifier 'exact'\n[path]:20:10: error: unknown identifier 'pi_gt_one'\n[path]:19:2: error: tactic 'apply' failed, could not unify the type of `ne_of_gt sorry sorry`\n False\nwith the goal\n Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n\u22a2 Transcendental \u211a (\u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0)\n",
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|