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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
Failed
--
Feb 17, 2026
unfold Transcendental
intro h
cases h with
| linarith =>
have h_poly : ∃ p : ℤ[X], p ≠ 0 ∧ (Polynomial.aeval (exp 1 * π)) p = 0 := by
rcases h with ⟨p, p_nonzero, p_roots⟩
use p
constructor
· exact p_nonzero
· exact p_roots
rcases h_poly with ⟨p, hp_nz, hp_root⟩
let f : ℂ → ℂ := fun z => exp z
have h_pi_e : IsTranscendental ℚ (exp 1 * π) := by
exact h
cases h_pi_e
View
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"proof_code": "unfold Transcendental\nintro h\ncases h with\n| linarith => \n have h_poly : \u2203 p : \u2124[X], p \u2260 0 \u2227 (Polynomial.aeval (exp 1 * \u03c0)) p = 0 := by\n rcases h with \u27e8p, p_nonzero, p_roots\u27e9\n use p\n constructor\n \u00b7 exact p_nonzero\n \u00b7 exact p_roots\n rcases h_poly with \u27e8p, hp_nz, hp_root\u27e9\n let f : \u2102 \u2192 \u2102 := fun z => exp z\n have h_pi_e : IsTranscendental \u211a (exp 1 * \u03c0) := by\n exact h\n cases h_pi_e",
"compiler_output_preview": "[path]:22:2: error: Invalid alternative name 'linarith': Expected 'intro'\n'Transcendental.exp_mul_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "intro h\ncases h with\n| intro p hp_nz hp_root =>\n have h_ne_zero : exp 1 * \u03c0 \u2260 0 := by\n apply ne_of_gt\n apply mul_pos\n \u00b7 exact exp_pos 1\n \u00b7 exact pi_pos\n let p_const := Polynomial.coeff p 0\n have : IsAlgebraic \u211a (exp 1 * \u03c0) := h\n let s := (exp 1 * \u03c0)\n have h_s : IsAlgebraic \u211a s := this\n let e := exp 1\n let pi := \u03c0\n -- Note: The transcendence of e*pi is a long-standing open problem.\n -- This proof attempt explores the structure of the algebraic property.\n induction p using Polynomial.induction_on\n case h_zero =>\n contradiction\n case h_add p q hp hq =>\n simp at hp_root\n repeat refine ?_\n case h_mono n a =>\n simp at hp_root\n repeat refine ?_",
"compiler_output_preview": "[path]:21:2: error: Too many variable names provided at alternative 'intro': 3 provided, but 2 expected\n[path]:28:40: error: unknown identifier 'h'\n[path]
},
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"proof_code": "intro h\nunfold IsAlgebraic at h\nrcases h with \u27e8p, hp_nz, hp_root\u27e9\nhave h_exp_pi : exp 1 * \u03c0 = Real.exp 1 * Real.pi := rfl\nrw [h_exp_pi] at hp_root\n-- The transcendence of e * \u03c0 is a famous unsolved problem in mathematics.\n-- There is no known proof that e * \u03c0 is transcendental.\n-- We represent the state of the goal to show completion progress.\ninduction p using Polynomial.induction_on\ncase h_C a =>\n simp at hp_root\n by_cases ha : a = 0\n \u00b7 subst ha\n simp at hp_nz\n \u00b7 exfalso\n exact hp_nz (Polynomial.C_inj.mpr ha)\ncase h_add p q hp hq =>\n refine ?_\ncase h_monomial n a ha_nz =>\n simp at hp_root\n have h_val : a * (Real.exp 1 * Real.pi) ^ n = 0 := hp_root\n have h_nz_pow : (Real.exp 1 * Real.pi) ^ n \u2260 0 := by\n apply pow_ne_zero\n apply mul_ne_zero\n \u00b7 exact ne_of_gt (Real.exp_pos 1)\n \u00b7 exact ne_of_gt Real.pi_pos\n have h_a_zero : (a : \u211d) = 0 := by\n apply (mul_eq_zero.mp h_val).resolve_right h_nz_pow\n norm_cast at h_a_zero\n contradiction",
"compiler_output_preview": "[path]:28:2: error: Case tag 'h_C' not found.\n\nAvailable tags: 'intro.intro.C', 'intro.intro.add', 'intro.intro.monomial'\n'Transcendental.exp_mul_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "intro h\nunfold IsAlgebraic at h\nrcases h with \u27e8p, hp_nz, hp_root\u27e9\ninduction p using Polynomial.induction_on\ncase intro.C a =>\n simp at hp_root\n norm_cast at hp_root\n rw [hp_root] at hp_nz\n simp at hp_nz\ncase intro.add p q hp hq =>\n refine ?_\ncase intro.monomial n a ha_nz =>\n simp at hp_root\n have h_ne_zero : exp 1 * \u03c0 \u2260 0 := by\n apply mul_ne_zero\n \u00b7 exact exp_ne_zero 1\n \u00b7 exact pi_ne_zero\n have h_pow_nz : (exp 1 * \u03c0) ^ n \u2260 0 := pow_ne_zero n h_ne_zero\n have h_a_zero : (a : \u211d) = 0 := by\n rw [mul_eq_zero] at hp_root\n exact hp_root.resolve_right h_pow_nz\n norm_cast at h_a_zero\n contradiction",
"compiler_output_preview": "[path]:28:27: error: unsolved goals\np q : Polynomial \u211a\nhp : p \u2260 0 \u2192 (Polynomial.aeval (rexp 1 * \u03c0)) p = 0 \u2192 False\nhq : q \u2260 0 \u2192 (Polynomial.aeval (rexp 1 * \u03c0)) q = 0 \u2192 False\nhp_nz : p + q \u2260 0\nhp_root : (Polynomial.aeval (rexp 1 * \u03c0)) (p + q) = 0\n\u22a2 False\n[path]
},
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"proof_code": "intro h\nunfold IsAlgebraic at h\nrcases h with \u27e8p, hp_nz, hp_root\u27e9\nhave h_s : exp 1 * \u03c0 = rexp 1 * Real.pi := rfl\nrw [h_s] at hp_root\ninduction p using Polynomial.induction_on\ncase intro.C a =>\n simp at hp_root\n norm_cast at hp_root\n subst hp_root\n simp at hp_nz\ncase intro.add p q hp hq =>\n refine ?_\ncase intro.monomial n a ha_nz =>\n simp at hp_root\n have h_ne_zero : rexp 1 * Real.pi \u2260 0 := by\n apply mul_ne_zero\n \u00b7 exact exp_ne_zero 1\n \u00b7 exact pi_ne_zero\n have h_pow_nz : (rexp 1 * Real.pi) ^ n \u2260 0 := pow_ne_zero n h_ne_zero\n have h_a_zero : (a : \u211d) = 0 := by\n apply (mul_eq_zero.mp hp_root).resolve_right h_pow_nz\n norm_cast at h_a_zero\n subst h_a_zero\n simp at ha_nz",
"compiler_output_preview": "[path]:30:27: error: unsolved goals\nh_s : rexp 1 * \u03c0 = rexp 1 * \u03c0\np q : Polynomial \u211a\nhp : p \u2260 0 \u2192 (Polynomial.aeval (rexp 1 * \u03c0)) p = 0 \u2192 False\nhq : q \u2260 0 \u2192 (Polynomial.aeval (rexp 1 * \u03c0)) q = 0 \u2192 False\nhp_nz : p + q \u2260 0\nhp_root : (Polynomial.aeval (rexp 1 * \u03c0)) (p + q) = 0\n\u22a2 False\n[path]
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"proof_code": "intro h\nunfold IsAlgebraic at h\nrcases h with \u27e8p, hp_nz, hp_root\u27e9\nlet e := exp 1\nlet pi := \u03c0\nlet s := e * pi\ninduction p using Polynomial.induction_on\ncase intro.C a =>\n simp at hp_root\n have ha : a = 0 := by\n norm_cast at hp_root\n subst ha\n simp at hp_nz\ncase intro.add p q hp hq =>\n simp at hp_root\n refine ?_\ncase intro.monomial n a ha_nz =>\n simp at hp_root\n have h_s_nz : s \u2260 0 := by\n apply mul_ne_zero\n \u00b7 exact exp_ne_zero 1\n \u00b7 exact pi_ne_zero\n have h_s_pow_nz : s ^ n \u2260 0 := pow_ne_zero n h_s_nz\n have ha : a = 0 := by\n norm_cast at hp_root\n exact (mul_eq_zero.mp hp_root).resolve_right h_s_pow_nz\n subst ha\n simp at hp_nz",
"compiler_output_preview": "[path]:32:27: error: unsolved goals\ne : \u211d := rexp 1\npi : \u211d := \u03c0\ns : \u211d := e * pi\np q : Polynomial \u211a\nhp : p \u2260 0 \u2192 (Polynomial.aeval (rexp 1 * \u03c0)) p = 0 \u2192 False\nhq : q \u2260 0 \u2192 (Polynomial.aeval (rexp 1 * \u03c0)) q = 0 \u2192 False\nhp_nz : p + q \u2260 0\nhp_root : (Polynomial.aeval (rexp 1 * \u03c0)) p + (Polynomial.aeval (rexp 1 * \u03c0)) q = 0\n\u22a2 False\n[path]
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"proof_code": "intro h\napply h\nlet e := rexp 1\nlet pi := Real.pi\nhave he : Transcendental \u211a e := transcendental_exp (ne_of_gt zero_lt_one) (IsAlgebraic.rat 1)\nhave hpi : Transcendental \u211a pi := transcendental_pi\nhave h_epi : IsAlgebraic \u211a (e * pi) := by\n refine ?_\nrcases h_epi with \u27e8p, hp_nz, hp_root\u27e9\ninduction p using Polynomial.induction_on\ncase intro.C a =>\n simp at hp_root\n norm_cast at hp_root\n subst hp_root\n exact hp_nz rfl\ncase intro.add p q hp hq =>\n refine ?_\ncase intro.monomial n a ha_nz =>\n simp at hp_root\n have h_nz : e * pi \u2260 0 := mul_ne_zero (exp_ne_zero 1) pi_ne_zero\n have h_pow : (e * pi) ^ n \u2260 0 := pow_ne_zero n h_nz\n have ha : a = 0 := by\n norm_cast at hp_root\n exact (mul_eq_zero.mp hp_root).resolve_right h_pow\n subst ha\n exact hp_nz rfl",
"compiler_output_preview": "[path]:20:2: error: tactic 'apply' failed, could not unify the type of `h`\n IsAlgebraic \u211a (rexp 1 * \u03c0)\nwith the goal\n False\nh : IsAlgebraic \u211a (rexp 1 * \u03c0)\n\u22a2 False\n'Transcendental.exp_mul_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"status": "failed",
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"banned_tokens": [],
"proof_code": "intro h\nlet e := rexp 1\nlet pi := Real.pi\nlet s := e * pi\nrcases h with \u27e8p, hp_nz, hp_root\u27e9\ninduction p using Polynomial.induction_on\ncase intro.C a =>\n simp at hp_root\n norm_cast at hp_root\n subst hp_root\n exact hp_nz rfl\ncase intro.monomial n a ha_nz =>\n simp at hp_root\n have h_s_nz : s \u2260 0 := mul_ne_zero (exp_ne_zero 1) pi_ne_zero\n have h_pow_nz : s ^ n \u2260 0 := pow_ne_zero n h_s_nz\n have ha : a = 0 := by\n norm_cast at hp_root\n exact (mul_eq_zero.mp hp_root).resolve_right h_pow_nz\n subst ha\n exact hp_nz (Polynomial.C_0.symm \u25b8 (Polynomial.monomial_zero n).symm)\ncase intro.add p q hp hq =>\n simp at hp_root\n refine ?_",
"compiler_output_preview": "[path]:29:16: error: Application type mismatch: In the application\n hp_nz rfl\nthe argument\n rfl\nhas type\n ?m.1921 = ?m.1921 : Prop\nbut is expected to have type\n Polynomial.C 0 = 0 : Prop\n[path]:36:6: error: no goals to be solved\n[path]
}
],
"verification": {
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"command": [
"[path]
"lean",
"[path]
],
"duration_s": 7.158774375915527,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"axioms_check_name": "exp_mul_pi_transcendental",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Transcendental__exp_mul_pi_transcendental",
"category": "research_open",
"endings": [
"end Transcendental"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 167,
"source_file": "FormalConjectures/Wikipedia/Transcendental.lean",
"title": "exp_mul_pi_transcendental"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:22:2: error: Invalid alternative name 'linarith': Expected 'intro'\n'Transcendental.exp_mul_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
Gemini 3 Flash Preview
Failed
--
Feb 14, 2026
unfold Transcendental
contrapose!
intro h
let e := Real.exp 1
let p := Real.pi
have h_prod : IsAlgebraic ℚ (e * p) := h
by_contra h_trans
apply h_trans
clear h_trans
exact h_prod
