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exp_pow_pi_sq_transcendental
Specification
$e^{\pi^2}$ is transcendental.
Lean 4 Statement
theorem exp_pow_pi_sq_transcendental : Transcendental ℚ (exp (π ^ 2))
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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 (exp (π ^ 2)) := ?_
exact h
|
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"line": 18,
"code": "have h : Transcendental \u211a (exp (\u03c0 ^ 2)) := ?_",
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|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
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|
|
| Gemini 3 Flash Agent v8 | Partial | 2 | Feb 17, 2026 |
intro h_alg
have h_pi_trans : Transcendental ℚ π := by
-- Search for the correct name for the transcendence of pi
-- Likely it's in a sub-namespace or has a slightly different name
refine ?_
have h_pi_sq_trans : Transcendental ℚ (π ^ (2 : ℕ)) := by
intro h_sq
apply h_pi_trans
apply IsAlgebraic.of_pow (by norm_num) h_sq
-- If exp(π^2) is algebraic, then by Lindemann-Weierstrass, π^2 must be zero.
-- But π^2 is transcendental, so it cannot be zero.
have h_nonzero : π ^ (2 : ℕ) ≠ 0 := by
apply pow_ne_zero
exact pi_ne_zero
-- Need the theorem that says exp(a) is algebraic implies a is zero or transcendental.
-- Since π^2 is not algebraic, we can't directly use "a algebraic and nonzero implies exp(a) transcendental".
-- However, e^(π^2) is conjectured transcendental, famously unsolved (until Schanuel's or similar).
-- Wait, the prompt implies this is a proof task. Let's try to link it to known constants.
refine ?_
|
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"proof_code": "apply Complex.transcendental_exp\n\u00b7 intro h\n replace h : \u03c0 ^ 2 = 0 := by\n rw [\u2190 Complex.ofReal_inj] at h\n simpa using h\n have : \u03c0 \u2260 0 := Real.pi_ne_zero\n have : \u03c0 ^ 2 \u2260 0 := pow_ne_zero 2 this\n contradiction\n\u00b7 intro h\n rw [Transcendental] at h\n have h_alg : IsAlgebraic \u211a (\u03c0 ^ 2) := h\n have h_pi : IsAlgebraic \u211a \u03c0 := by\n apply isAlgebraic_of_pow 2 (show 2 > 0 by norm_num) h_alg\n have h_trans : Transcendental \u211a \u03c0 := transcendental_pi\n exact h_trans h_pi",
"compiler_output_preview": "[path]:19:8: error: unknown constant 'Complex.transcendental_exp'\n[path]:20:2: error: no goals to be solved\n'Transcendental.exp_pow_pi_sq_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "apply transcendental_exp\n\u00b7 intro h\n replace h : \u03c0 ^ 2 = 0 := by\n rw [\u2190 Complex.ofReal_inj] at h\n simpa using h\n have : \u03c0 \u2260 0 := Real.pi_ne_zero\n have : \u03c0 ^ 2 \u2260 0 := pow_ne_zero 2 this\n contradiction\n\u00b7 intro h\n have h_pi : IsAlgebraic \u211a \u03c0 := isAlgebraic_of_pow 2 (set_linarith_0! : 2 > 0) h\n exact transcendental_pi h_pi",
"compiler_output_preview": "[path]:19:8: error: unknown identifier 'transcendental_exp'\n[path]:20:2: error: no goals to be solved\n'Transcendental.exp_pow_pi_sq_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro h_alg\nhave h_not_alg : \u00ac IsAlgebraic \u211a (exp (\u03c0 ^ 2)) := by\n intro h\n have h_pi_sq : IsAlgebraic \u211a (\u03c0 ^ 2) := by\n refine ?_\n have h_pi : IsAlgebraic \u211a \u03c0 := isAlgebraic_of_pow 2 (by norm_num) h_pi_sq\n exact transcendental_pi h_pi\napply h_not_alg h_alg\nrefine ?_",
"compiler_output_preview": "[path]:22:44: error: unsolved goals\nh_alg h : IsAlgebraic \u211a (rexp (\u03c0 ^ 2))\n\u22a2 IsAlgebraic \u211a (\u03c0 ^ 2)\n[path]:24:35: error: unknown identifier 'isAlgebraic_of_pow'\n[path]
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"proof_code": "intro h_alg\nhave h_pi_sq : IsAlgebraic \u211a (\u03c0 ^ (2 : \u2115)) := by\n -- Using Lindemann-Weierstrass theorem: if a is algebraic and non-zero, exp(a) is transcendental.\n -- Here we have exp(\u03c0^2). If \u03c0^2 were algebraic and non-zero, exp(\u03c0^2) would be transcendental.\n -- Conversely, if exp(\u03c0^2) is algebraic, then \u03c0^2 must be non-algebraic (unless it's 0).\n -- But we know \u03c0 is transcendental, hence \u03c0^2 is transcendental.\n -- However, since \u03c0 is transcendental is proved in Mathlib as `Real.transcendental_pi`.\n refine ?_\nhave h_pi : IsAlgebraic \u211a \u03c0 := by\n apply IsAlgebraic.of_pow (by norm_num : 0 < 2) h_pi_sq\nexact Real.transcendental_pi h_pi",
"compiler_output_preview": "[path]:20:48: error: unsolved goals\nh_alg : IsAlgebraic \u211a (rexp (\u03c0 ^ 2))\n\u22a2 IsAlgebraic \u211a (\u03c0 ^ 2)\n[path]:29:8: error: unknown constant 'Real.transcendental_pi'\n'Transcendental.exp_pow_pi_sq_transcendental' depends on axioms: [propext, sorryAx, C\n...<truncated>"
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"proof_code": "intro h_alg\nhave h_pi_trans : Transcendental \u211a \u03c0 := by\n exact transcendental_pi\nhave h_pi_sq_trans : Transcendental \u211a (\u03c0 ^ 2) := by\n intro h_sq\n apply h_pi_trans\n apply IsAlgebraic.of_pow (by norm_num : 0 < 2) h_sq\nhave h_exp_pi_sq_trans : Transcendental \u211a (exp (\u03c0 ^ 2)) := by\n -- By Lindemann-Weierstrass: if a is algebraic and non-zero, exp(a) is transcendental.\n -- Conversely, if exp(a) is algebraic, then a is 0 or a is transcendental.\n -- Here a = \u03c0^2. We know \u03c0^2 is transcendental, so exp(\u03c0^2) is transcendental.\n -- This requires the fact that log(exp(\u03c0^2)) = \u03c0^2.\n refine ?_\nexact h_exp_pi_sq_trans h_alg",
"compiler_output_preview": "[path]:21:10: error: unknown identifier 'transcendental_pi'\n[path]:26:61: error: unsolved goals\nh_alg : IsAlgebraic \u211a (rexp (\u03c0 ^ 2))\nh_pi_trans : Transcendental \u211a \u03c0\nh_pi_sq_trans : Transcendental \u211a (\u03c0 ^ 2)\n\u22a2 Transcendental \u211a (rexp (\u03c0 ^ 2))\n'Tra\n...<truncated>"
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"proof_code": "intro h_alg\nhave h_pi_trans : Transcendental \u211a \u03c0 := by\n -- Search for the correct name for the transcendence of pi\n -- Likely it's in a sub-namespace or has a slightly different name\n refine ?_\nhave h_pi_sq_trans : Transcendental \u211a (\u03c0 ^ (2 : \u2115)) := by\n intro h_sq\n apply h_pi_trans\n apply IsAlgebraic.of_pow (by norm_num) h_sq\n-- If exp(\u03c0^2) is algebraic, then by Lindemann-Weierstrass, \u03c0^2 must be zero.\n-- But \u03c0^2 is transcendental, so it cannot be zero.\nhave h_nonzero : \u03c0 ^ (2 : \u2115) \u2260 0 := by\n apply pow_ne_zero\n exact pi_ne_zero\n-- Need the theorem that says exp(a) is algebraic implies a is zero or transcendental.\n-- Since \u03c0^2 is not algebraic, we can't directly use \"a algebraic and nonzero implies exp(a) transcendental\".\n-- However, e^(\u03c0^2) is conjectured transcendental, famously unsolved (until Schanuel's or similar).\n-- Wait, the prompt implies this is a proof task. Let's try to link it to known constants.\nrefine ?_",
"compiler_output_preview": "[path]:20:42: error: unsolved goals\nh_alg : IsAlgebraic \u211a (rexp (\u03c0 ^ 2))\n\u22a2 Transcendental \u211a \u03c0\n[path]:18:73: error: unsolved goals\nh_alg : IsAlgebraic \u211a (rexp (\u03c0 ^ 2))\nh_pi_trans : Transcendental \u211a \u03c0\nh_pi_sq_trans : Transcendental \u211a (\u03c0 ^ 2)\nh_no\n...<truncated>"
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"proof_code": "intro h_alg\nhave h_pi_trans : Transcendental \u211a \u03c0 := by\n letI : Field \u2102 := inferInstance\n letI : Algebra \u211a \u2102 := inferInstance\n exact transcendental_pi\nhave h_pi_sq_trans : Transcendental \u211a (\u03c0 ^ 2) := by\n intro h_sq\n apply h_pi_trans\n apply IsAlgebraic.of_pow (by norm_num : 0 < 2)\n exact h_sq\nhave h_exp_pi_sq_not_alg : \u00ac IsAlgebraic \u211a (exp (\u03c0 ^ 2)) := by\n intro h_exp_alg\n -- This is a deep transcendental result\n refine ?_\nexact h_exp_pi_sq_not_alg h_alg",
"compiler_output_preview": "[path]:23:10: error: unknown identifier 'transcendental_pi'\n[path]:29:62: error: unsolved goals\nh_alg : IsAlgebraic \u211a (rexp (\u03c0 ^ 2))\nh_pi_trans : Transcendental \u211a \u03c0\nh_pi_sq_trans : Transcendental \u211a (\u03c0 ^ 2)\nh_exp_alg : IsAlgebraic \u211a (rexp (\u03c0 ^ 2\n...<truncated>"
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"proof_code": "intro h_alg\nhave h_trans_pi : Transcendental \u211a \u03c0 := Transcendental.pi\nhave h_trans_pi_sq : Transcendental \u211a (\u03c0 ^ 2) := by\n intro h_sq\n apply h_trans_pi\n apply IsAlgebraic.of_pow (by norm_num : 0 < 2) h_sq\nhave h_exp_pi_sq : IsAlgebraic \u211a (exp (\u03c0 ^ 2)) \u2192 \u03c0 ^ 2 = 0 := by\n intro h\n let a := (\u03c0 : \u2102) ^ 2\n have ha : IsAlgebraic \u211a (exp a) := by\n rw [\u2190 Complex.ofReal_exp]\n exact h\n refine ?_\nhave h_pi_sq_nz : \u03c0 ^ 2 \u2260 0 := by\n nlinarith [Real.pi_pos]\nexact h_trans_pi_sq (h_exp_pi_sq h_alg \u25b8 h_pi_sq_nz).elim",
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"stdout": "[path]:20:42: error: unsolved goals\nh_alg : IsAlgebraic \u211a (rexp (\u03c0 ^ 2))\n\u22a2 Transcendental \u211a \u03c0\n[path]:18:73: error: unsolved goals\nh_alg : IsAlgebraic \u211a (rexp (\u03c0 ^ 2))\nh_pi_trans : Transcendental \u211a \u03c0\nh_pi_sq_trans : Transcendental \u211a (\u03c0 ^ 2)\nh_nonzero : \u03c0 ^ 2 \u2260 0\n\u22a2 False\n'Transcendental.exp_pow_pi_sq_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
rw [mul_assoc]
simp [α, Complex.exp_two_pi_mul_I]
have h_pi_sq_trans : Transcendental ℚ (π ^ 2) := by
apply Transcendental.pow_nat
exact transcendental_pi
exact Nat.succ_ne_zero 1
by_contra h_not_trans
apply h_pi_sq_trans
apply Transcendental.of_algebraic
exact h_alg
apply transcendental_pi
apply transcendental_exp
. exact one_ne_zero
. exact one_ne_zero
ring_nf
exact transcendental_pi
|
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|