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Gemini 3 Flash Agent v8
Partial
1
Feb 17, 2026
by_cases h : Transcendental ℚ (exp 1 + π)
· exact h
· exfalso
-- Note: This is a known open problem in mathematics.
-- There is currently no proof that e + pi is transcendental.
-- We leave the goal as is to indicate the missing proof step.
apply h
refine ?_
done
View
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"proof_code": "intro h\nrw [Transcendental] at h\npush_neg at h\nobtain \u27e8p, hp1, hp2\u27e9 := h\nhave h_or : exp 1 + \u03c0 = exp 1 + \u03c0 := rfl\nhave h_e_pi : Transcendental \u211a (exp 1 + \u03c0) \u2228 \u00ac Transcendental \u211a (exp 1 + \u03c0) := Classical.em _\ncases h_e_pi with\n| h_trans => exact h_trans\n| h_not_trans =>\n refine ?_\n -- This is a well-known open problem in mathematics.\n -- Currently, it is not known whether e + pi is transcendental.\n -- We provide the structure to show that assuming it's algebraic (not transcendental)\n -- would lead to a contradiction, but the proof of such contradiction \n -- is currently beyond the capabilities of human mathematics.\n have h_alg : IsAlgebraic \u211a (exp 1 + \u03c0) := h_not_trans\n simp at h_alg\n -- The following goals are left open as this is an open conjecture.\n done",
"compiler_output_preview": "[path]:20:6: error: failed to rewrite using equation theorems for 'Transcendental'.\n'Transcendental.exp_add_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "intro h\nhave h_or : Transcendental \u211a (exp 1 + \u03c0) \u2228 \u00ac Transcendental \u211a (exp 1 + \u03c0) := Classical.em _\ncases h_or with\n| inl h_trans => exact h_trans\n| inr h_not_trans =>\n exfalso\n have h_alg : IsAlgebraic \u211a (exp 1 + \u03c0) := h_not_trans\n -- This is a famous open problem (Schanuel's conjecture implies this).\n -- Since we cannot prove it yet, we leave the contradiction as a goal.\n refine ?_\n done",
"compiler_output_preview": "[path]:22:19: error: type mismatch\n h_trans\nhas type\n Transcendental \u211a (rexp 1 + \u03c0) : Prop\nbut is expected to have type\n False : Prop\n[path]:25:46: error: type mismatch\n h_not_trans\nhas type\n \u00acTranscendental \u211a (rexp 1 + \u03c0) : Prop\nbut is expected\n...<truncated>"
},
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"proof_code": "have h_trans_or_alg : Transcendental \u211a (exp 1 + \u03c0) \u2228 IsAlgebraic \u211a (exp 1 + \u03c0) := by\n rw [transcendental_iff_not_isAlgebraic]\n exact em _\ncases h_trans_or_alg with\n| inl h_trans => exact h_trans\n| inr h_alg =>\n -- This is a famous open problem.\n -- Schanuel's conjecture implies the transcendence of e + pi.\n -- We leave the proof of this specific case open.\n refine ?_\n done",
"compiler_output_preview": "[path]:20:8: error: unknown identifier 'transcendental_iff_not_isAlgebraic'\n[path]:29:4: error: unsolved goals\ncase inr\nh_alg : IsAlgebraic \u211a (rexp 1 + \u03c0)\n\u22a2 Transcendental \u211a (rexp 1 + \u03c0)\n'Transcendental.exp_add_pi_transcendental' depends on axioms: [\n...<truncated>"
},
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"proof_code": "have h_trans_or_alg : Transcendental \u211a (exp 1 + \u03c0) \u2228 IsAlgebraic \u211a (exp 1 + \u03c0) := by\n rw [Algebra.transcendental_iff_not_isAlgebraic]\n exact Classical.em _\ncases h_trans_or_alg with\n| inl h_trans => exact h_trans\n| inr h_alg =>\n -- $e + \\pi$ is conjectured to be transcendental, but no proof currently exists.\n -- Schanuel's conjecture implies this, but it is unproven.\n refine ?_\n done",
"compiler_output_preview": "[path]:20:8: error: tactic 'rewrite' failed, did not find instance of the pattern in the target expression\n Algebra.Transcendental ?m.721 ?m.722\n\u22a2 Transcendental \u211a (rexp 1 + \u03c0) \u2228 IsAlgebraic \u211a (rexp 1 + \u03c0)\n[path]:28:4: error: unsolved goals\ncase inr\n...<truncated>"
},
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"proof_code": "have h_trans_or_alg : Transcendental \u211a (exp 1 + \u03c0) \u2228 IsAlgebraic \u211a (exp 1 + \u03c0) := by\n by_cases h : Transcendental \u211a (exp 1 + \u03c0)\n \u00b7 left; exact h\n \u00b7 right\n rw [\u2190 Algebra.transcendental_iff_not_isAlgebraic]\n exact h\ncases h_trans_or_alg with\n| inl h_trans => exact h_trans\n| inr h_alg =>\n -- $e + \\pi$ is conjectured to be transcendental, but no proof currently exists.\n -- This remains an open problem in transcendental number theory.\n refine ?_\n done",
"compiler_output_preview": "[path]:23:10: error: tactic 'rewrite' failed, did not find instance of the pattern in the target expression\n \u00acAlgebra.IsAlgebraic ?m.1099 ?m.1100\ncase neg.h\nh : \u00acTranscendental \u211a (rexp 1 + \u03c0)\n\u22a2 IsAlgebraic \u211a (rexp 1 + \u03c0)\n[path]:31:4: error: unsolved\n...<truncated>"
},
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"proof_code": "by_cases h : Transcendental \u211a (exp 1 + \u03c0)\n\u00b7 exact h\n\u00b7 exfalso\n -- Note: This is a known open problem in mathematics.\n -- There is currently no proof that e + pi is transcendental.\n -- We leave the goal as is to indicate the missing proof step.\n apply h\n refine ?_\n done",
"compiler_output_preview": "[path]:27:4: error: unsolved goals\ncase neg\nh : \u00acTranscendental \u211a (rexp 1 + \u03c0)\n\u22a2 Transcendental \u211a (rexp 1 + \u03c0)\n'Transcendental.exp_add_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "by_cases h : Transcendental \u211a (exp 1 + \u03c0)\n\u00b7 exact h\n\u00b7 -- This is a famous open conjecture; no proof is currently known.\n -- We leave the goal open as a partial result.\n refine ?_\n done",
"compiler_output_preview": "[path]:24:4: error: unsolved goals\ncase neg\nh : \u00acTranscendental \u211a (rexp 1 + \u03c0)\n\u22a2 Transcendental \u211a (rexp 1 + \u03c0)\n'Transcendental.exp_add_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"status": "failed",
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"proof_code": "by_cases h : Transcendental \u211a (exp 1 + \u03c0)\n\u00b7 exact h\n\u00b7 -- The transcendence of e + \u03c0 is a famous unsolved problem in number theory.\n -- It would follow from Schanuel's conjecture, which is also unproven.\n -- We include the cases and leave the open case as a partial proof.\n induction h\n exact ?_",
"compiler_output_preview": "[path]:24:4: error: tactic 'induction' failed, major premise type is not an inductive type\n Transcendental \u211a (rexp 1 + \u03c0) \u2192 False\n\nExplanation: the 'induction' tactic is for constructor-based reasoning as well as for applying custom induction principles with a 'using' clause or a registered '@[induction_eliminator]' theorem. The above type neither is an inductive type nor\n...<truncated>"
}
],
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"command": [
"[path]
"lean",
"[path]
],
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"lean_file_path": "[path]
"problem": {
"axioms_check_name": "exp_add_pi_transcendental",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Transcendental__exp_add_pi_transcendental",
"category": "research_open",
"endings": [
"end Transcendental"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 167,
"source_file": "FormalConjectures/Wikipedia/Transcendental.lean",
"title": "exp_add_pi_transcendental"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:27:4: error: unsolved goals\ncase neg\nh : \u00acTranscendental \u211a (rexp 1 + \u03c0)\n\u22a2 Transcendental \u211a (rexp 1 + \u03c0)\n'Transcendental.exp_add_pi_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
Gemini 3 Flash Preview
Failed
--
Feb 14, 2026
rw [Transcendental]
intro h
rcases h with ⟨p, hp_poly, hp_root⟩
set α := exp 1 + π
have h_exp : exp 1 = Real.exp 1 := rfl
have h_pi : π = Real.pi := rfl
by_contra h_alg
apply hp_poly
ext
simp
apply Polynomial.eq_zero_of_infinite_roots p
intro x
induction x
all_goals simp
ring_nf
apply Complex.exp_ne_zero
