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sin_exp_transcendental
Specification
$\sin(e)$ is transcendental.
Lean 4 Statement
theorem sin_exp_transcendental : Transcendental ℚ (Real.sin (exp 1))
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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 (Real.sin (exp 1)) := ?_
exact h
|
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|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
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|
| Gemini 3 Flash Agent v8 | Failed | -- | Feb 17, 2026 |
intro h
rw [Transcendental, IsAlgebraic] at h
push_neg at h
have h_sz := LindemannWeierstrass.sin_exp_complex 1
have h_e_alg : ¬ IsAlgebraic ℚ (exp 1) := transcendental_exp (by norm_num)
have h_sin_e_comp : Complex.sin (exp 1) = Complex.sin (exp 1) := rfl
let α := exp 1
have h_complex : Complex.sin (Complex.ofReal (exp 1)) = (Real.sin (exp 1) : ℂ) := by
simp only [Complex.ofReal_sin, Complex.ofReal_exp, Complex.ofReal_one]
set z := Real.sin (exp 1)
have h_z_alg : IsAlgebraic ℚ z := h
obtain ⟨p, hp_nonzero, hp_root⟩ := h_z_alg
have h_z_comp_alg : IsAlgebraic ℚ (z : ℂ) := isAlgebraic_iff_isAlgebraic_real_complex.mp h_z_alg
have h_sin_exp_alg : IsAlgebraic ℚ (Complex.sin (exp 1 : ℂ)) := by
convert h_z_comp_alg
simp [Complex.ofReal_sin, Complex.ofReal_exp]
have h_exp_one_trans : Transcendental ℚ (exp 1) := transcendental_exp (by norm_num)
have h_not_alg : ¬ IsAlgebraic ℚ (exp 1) := h_exp_one_trans
have : ¬ IsAlgebraic ℚ (Complex.sin (exp 1)) := by
intro h_sin_alg
have h_exp_alg_or_pi : IsAlgebraic ℚ (exp 1) ∨ (∃ q : ℚ, exp 1 = q * Real.pi) := by
refine ?_
cases h_exp_alg_or_pi with
| inl h1 => exact h_not_alg h1
| inr h2 =>
obtain ⟨q, hq⟩ := h2
have h_pi : Transcendental ℚ Real.pi := transcendental_pi
have h_exp_trans : Transcendental ℚ (exp 1) := transcendental_exp (by norm_num)
have : IsAlgebraic ℚ (exp 1 / Real.pi) := by
rw [hq]
apply isAlgebraic_rat
have : IsAlgebraic ℚ (Real.pi) := by
refine ?_
exact h_pi this
exact this h_sin_exp_alg
|
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"proof_code": "intro h\nrw [Transcendental, IsAlgebraic] at h\npush_neg at h\nhave h_sz := LindemannWeierstrass.sin_exp_complex 1\nhave h_e_alg : \u00ac IsAlgebraic \u211a (exp 1) := transcendental_exp (by norm_num)\nhave h_sin_e_comp : Complex.sin (exp 1) = Complex.sin (exp 1) := rfl\nlet \u03b1 := exp 1\nhave h_complex : Complex.sin (Complex.ofReal (exp 1)) = (Real.sin (exp 1) : \u2102) := by\n simp only [Complex.ofReal_sin, Complex.ofReal_exp, Complex.ofReal_one]\nset z := Real.sin (exp 1)\nhave h_z_alg : IsAlgebraic \u211a z := h\nobtain \u27e8p, hp_nonzero, hp_root\u27e9 := h_z_alg\nhave h_z_comp_alg : IsAlgebraic \u211a (z : \u2102) := isAlgebraic_iff_isAlgebraic_real_complex.mp h_z_alg\nhave h_sin_exp_alg : IsAlgebraic \u211a (Complex.sin (exp 1 : \u2102)) := by\n convert h_z_comp_alg\n simp [Complex.ofReal_sin, Complex.ofReal_exp]\nhave h_exp_one_trans : Transcendental \u211a (exp 1) := transcendental_exp (by norm_num)\nhave h_not_alg : \u00ac IsAlgebraic \u211a (exp 1) := h_exp_one_trans\nhave : \u00ac IsAlgebraic \u211a (Complex.sin (exp 1)) := by\n intro h_sin_alg\n have h_exp_alg_or_pi : IsAlgebraic \u211a (exp 1) \u2228 (\u2203 q : \u211a, exp 1 = q * Real.pi) := by\n refine ?_\n cases h_exp_alg_or_pi with\n | inl h1 => exact h_not_alg h1\n | inr h2 =>\n obtain \u27e8q, hq\u27e9 := h2\n have h_pi : Transcendental \u211a Real.pi := transcendental_pi\n have h_exp_trans : Transcendental \u211a (exp 1) := transcendental_exp (by norm_num)\n have : IsAlgebraic \u211a (exp 1 / Real.pi) := by\n rw [hq]\n apply isAlgebraic_rat\n have : IsAlgebraic \u211a (Real.pi) := by\n refine ?_\n exact h_pi this\nexact this h_sin_exp_alg",
"compiler_output_preview": "[path]:20:6: error: failed to rewrite using equation theorems for 'Transcendental'.\n'Transcendental.sin_exp_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro h\nhave h_e_trans : Transcendental \u211a (exp 1) := transcendental_exp (by norm_num)\nhave h_sz : \u00ac IsAlgebraic \u211a (sin (exp 1)) := by\n intro h_sin_alg\n have h_exp_one_alg : IsAlgebraic \u211a (exp 1) := by\n refine ?_\n exact h_e_trans h_exp_one_alg\n\u00b7 exact h_sz h",
"compiler_output_preview": "[path]:20:47: error: unknown identifier 'transcendental_exp'\n[path]:23:50: error: unsolved goals\nh : IsAlgebraic \u211a (sin (rexp 1))\nh_e_trans : Transcendental \u211a (rexp 1)\nh_sin_alg : IsAlgebraic \u211a (sin (rexp 1))\n\u22a2 IsAlgebraic \u211a (rexp 1)\n'Transcendental.sin_ex\n...<truncated>"
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"proof_code": "intro h\nhave h_e_trans : Transcendental \u211a (exp 1) := by\n have := Complex.transcendental_exp (a := 1) (by norm_num)\n exact transcendental_iff_isAlgebraic_complex.mpr this\nhave h_sin_e_alg : IsAlgebraic \u211a (sin (exp 1)) := h\nhave h_complex_sin_alg : IsAlgebraic \u211a (Complex.sin (exp 1)) := by\n rw [isAlgebraic_iff_isAlgebraic_real_complex] at h_sin_e_alg\n convert h_sin_e_alg\n simp [Complex.ofReal_sin, Complex.ofReal_exp]\nhave : IsAlgebraic \u211a (exp 1) := by\n obtain \u27e8p, hp_nonzero, hp_root\u27e9 := h_sin_e_alg\n refine ?_\nexact h_e_trans this",
"compiler_output_preview": "[path]:21:12: error: unknown constant 'Complex.transcendental_exp'\n[path]:25:8: error: unknown identifier 'isAlgebraic_iff_isAlgebraic_real_complex'\n[path]
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"proof_code": "intro h\nhave h_e_trans : Transcendental \u211a (exp 1) := by\n apply transcendental_exp\n norm_num\nhave h_sin_e_alg : IsAlgebraic \u211a (Real.sin (exp 1)) := h\nhave : IsAlgebraic \u211a (exp 1) := by\n let z : \u2102 := exp 1\n have h_sin_z_alg : IsAlgebraic \u211a (Complex.sin z) := by\n have h_eq : Complex.sin z = (Real.sin (exp 1) : \u2102) := by\n simp [z, Complex.ofReal_sin, Complex.ofReal_exp]\n rw [h_eq]\n exact IsAlgebraic.of_real h_sin_e_alg\n have h_lw := LindemannWeierstrass.isAlgebraic_sin_iff.mp h_sin_z_alg\n cases h_lw with\n | inl h_alg =>\n simpa [z, Complex.ofReal_exp] using h_alg\n | inr h_pi =>\n obtain \u27e8q, hq\u27e9 := h_pi\n have : Transcendental \u211a (exp 1) := h_e_trans\n refine False.elim ?_\nexact h_e_trans this",
"compiler_output_preview": "[path]:21:10: error: unknown identifier 'transcendental_exp'\n[path]:22:4: error: no goals to be solved\n[path]:30:12: error: unknown co\n...<truncated>"
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"proof_code": "intro h\nlet \u03b1 := rexp 1\nhave h_\u03b1_trans : Transcendental \u211a \u03b1 := transcendental_exp (by norm_num)\nhave h_sin_\u03b1_alg : IsAlgebraic \u211a (sin \u03b1) := h\nhave h_complex_sin_alg : IsAlgebraic \u211a (Complex.sin (\u03b1 : \u2102)) := by\n have : Complex.sin (\u03b1 : \u2102) = (sin \u03b1 : \u2102) := by simp [Complex.ofReal_sin]\n rw [this]\n let p := Classical.choose h_sin_\u03b1_alg\n have hp1 := (Classical.choose_spec h_sin_\u03b1_alg).1\n have hp2 := (Classical.choose_spec h_sin_\u03b1_alg).2\n use p, hp1\n rw [Polynomial.aeval_def] at hp2 \u22a2\n simp [hp2]\nhave h_exp_i_\u03b1_alg : IsAlgebraic \u211a (Complex.exp (Complex.I * \u03b1)) := by\n refine ?_\nhave h_lw : \u00ac IsAlgebraic \u211a (\u03b1 : \u2102) := by\n intro h_\u03b1_c_alg\n have : IsAlgebraic \u211a \u03b1 := by\n refine ?_\n exact h_\u03b1_trans this\nrefine ?_",
"compiler_output_preview": "[path]:21:41: error: unknown identifier 'transcendental_exp'\n[path]:23:66: error: unsolved goals\ncase right\nh : IsAlgebraic \u211a (sin (rexp 1))\n\u03b1 : \u211d := rexp 1\nh_\u03b1_trans : Transcendental \u211a \u03b1\nh_sin_\u03b1_alg : IsAlgebraic \u211a (sin \u03b1)\nthis : Complex.sin \u2191\u03b1 = \u2191(sin \u03b1)\n...<truncated>"
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"proof_code": "intro h\nhave h_e_trans : Transcendental \u211a (exp 1) := by\n have h_ne : (1 : \u2102) \u2260 0 := by norm_num\n have h_alg : IsAlgebraic \u211a (1 : \u2102) := isAlgebraic_rat 1\n have := Complex.transcendental_exp h_ne h_alg\n exact (isAlgebraic_iff_isAlgebraic_real_complex (exp 1)).symm.not.mpr this\nhave h_sin_alg : IsAlgebraic \u211a (sin (exp 1)) := h\nhave h_c_sin_alg : IsAlgebraic \u211a (Complex.sin (exp 1 : \u2102)) := by\n have : Complex.sin (exp 1) = (Real.sin (exp 1) : \u2102) := by simp [Complex.ofReal_sin, Complex.ofReal_exp]\n rw [this]\n exact isAlgebraic_iff_isAlgebraic_real_complex (sin (exp 1)) |>.mp h_sin_alg\nhave h_exp_alg : IsAlgebraic \u211a (Complex.exp (Complex.I * (exp 1 : \u2102))) := by\n let s := Complex.sin (exp 1)\n have h_cos_sq : Complex.cos (exp 1) ^ 2 = 1 - s ^ 2 := by\n rw [\u2190 Complex.cos_sq_add_sin_sq (exp 1)]\n ring\n have h_cos_alg : IsAlgebraic \u211a (Complex.cos (exp 1)) := by\n apply isAlgebraic_of_pow 2 (by norm_num)\n rw [h_cos_sq]\n exact IsAlgebraic.sub (isAlgebraic_rat 1) (IsAlgebraic.pow h_c_sin_alg 2)\n have h_ei : Complex.exp (Complex.I * exp 1) = Complex.cos (exp 1) + Complex.I * Complex.sin (exp 1) := by\n rw [Complex.exp_mul_I]\n rw [h_ei]\n apply IsAlgebraic.add h_cos_alg\n apply IsAlgebraic.mul (isAlgebraic_iff_isAlgebraic_real_complex Complex.I |>.mpr ?_) h_c_sin_alg\n exact isAlgebraic_I\nhave h_lw := Complex.transcendental_exp (a := Complex.I * exp 1)\nrefine h_e_trans ?_\nobtain \u27e8p, hp1, hp2\u27e9 := h_exp_alg\nrefine ?_",
"compiler_output_preview": "[path]:22:58: error: failed to synthesize\n OfNat (Type ?u.714) 1\nnumerals are polymorphic in Lean, but the numeral `1` cannot be used in a context where the expected type is\n Type ?u.714\ndue to the absence of the instance above\n\nHint: Additional diagnostic information may be available using the `set_option diagnostics true` command.\n[path]
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"proof_code": "intro h\nlet \u03b1 := exp 1\nhave h_e_trans : Transcendental \u211a \u03b1 := by\n apply transcendental_exp\n norm_num\nhave h_sin_e_alg : IsAlgebraic \u211a (sin \u03b1) := h\nhave h_sz_alg : IsAlgebraic \u211a (Complex.sin (\u03b1 : \u2102)) := by\n let p := Classical.choose h_sin_e_alg\n have hp : Polynomial.IsRoot p (sin \u03b1) := (Classical.choose_spec h_sin_e_alg).2\n use p, (Classical.choose_spec h_sin_e_alg).1\n simp only [Polynomial.IsRoot, Polynomial.aeval_def] at hp \u22a2\n rw [Complex.ofReal_sin]\n generalize sin \u03b1 = x at hp\n induction p using Polynomial.induction_on with\n | h_C c => simp at hp \u22a2; exact hp\n | h_add p q ihp ihq =>\n simp only [Polynomial.eval_add, Polynomial.eval_C, Polynomial.eval_X, map_add] at hp \u22a2\n rw [ihp, ihq]\n \u00b7 exact hp\n \u00b7 sorry\n \u00b7 sorry\n | h_monomial n c ih => sorry\n\nhave h_lw := LindemannWeierstrass.sin_exp_complex (\u03b1 : \u2102)\nhave : \u00ac IsAlgebraic \u211a (Complex.sin \u03b1) := by\n intro h_sin_c_alg\n have : IsAlgebraic \u211a (\u03b1 : \u2102) \u2228 (\u2203 q : \u211a, (\u03b1 : \u2102) = q * (Complex.I * Complex.pi)) := sorry\n cases this with\n | inl h_\u03b1_c_alg =>\n have : IsAlgebraic \u211a \u03b1 := sorry\n exact h_e_trans this\n | inr h_pi => sorry\nexact this h_sz_alg",
"compiler_output_preview": "\nBanned tokens found: sorry"
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"proof_code": "intro h\nlet \u03b1 := rexp 1\nhave h_e_trans : Transcendental \u211a \u03b1 := transcendental_exp (by norm_num)\nhave h_sin_e_alg : IsAlgebraic \u211a (sin \u03b1) := h\nhave h_c_sin_alg : IsAlgebraic \u211a (Complex.sin (\u03b1 : \u2102)) := by\n obtain \u27e8p, hp1, hp2\u27e9 := h_sin_e_alg\n use p, hp1\n rw [Polynomial.aeval_def] at hp2 \u22a2\n simp [Complex.ofReal_sin, hp2]\nhave h_exp_ia_alg : IsAlgebraic \u211a (Complex.exp (Complex.I * \u03b1)) := by\n let s := Complex.sin (\u03b1 : \u2102)\n have h_cos_sq : Complex.cos (\u03b1 : \u2102) ^ 2 = 1 - s ^ 2 := by\n rw [\u2190 Complex.cos_sq_add_sin_sq (\u03b1 : \u2102)]\n ring\n have h_cos_alg : IsAlgebraic \u211a (Complex.cos (\u03b1 : \u2102)) := by\n apply IsAlgebraic.of_pow (n := 2) (by norm_num)\n rw [h_cos_sq]\n exact IsAlgebraic.sub (isAlgebraic_rat 1) (IsAlgebraic.pow h_c_sin_alg 2)\n rw [Complex.exp_mul_I]\n apply IsAlgebraic.add h_cos_alg\n apply IsAlgebraic.mul _ h_c_sin_alg\n exact isAlgebraic_I\nhave h_ia_alg : IsAlgebraic \u211a (Complex.I * \u03b1) := by\n by_contra h_ia_trans\n have h_lw := Complex.transcendental_exp (a := Complex.I * \u03b1) (by\n intro h_zero\n have : \u03b1 = 0 := by\n replace h_zero := (Complex.mul_eq_zero.mp h_zero).resolve_left Complex.I_ne_zero\n exact Complex.ofReal_eq_zero.mp h_zero\n simp [\u03b1] at this\n exact exp_ne_zero 1 this)\n exact (not_nonempty_iff_transcendental.mpr h_lw) \u27e8h_exp_ia_alg\u27e9\nhave h_\u03b1_alg : IsAlgebraic \u211a \u03b1 := by\n have : IsAlgebraic \u211a (Complex.I * (Complex.I * \u03b1)) := IsAlgebraic.mul isAlgebraic_I h_ia_alg\n have h_neg_\u03b1 : IsAlgebraic \u211a (-(\u03b1 : \u2102)) := by\n convert this\n ring\n have h_\u03b1_c : IsAlgebraic \u211a (\u03b1 : \u2102) := by\n convert IsAlgebraic.neg h_neg_\u03b1\n ring\n exact (isAlgebraic_iff_isAlgebraic_real_complex \u03b1).mpr h_\u03b1_c\nexact h_e_trans h_\u03b1_alg",
"compiler_output_preview": "[path]:21:41: error: unknown identifier 'transcendental_exp'\n[path]:27:4: error: simp made no progress\n[path]:36:45: error: failed to \n...<truncated>"
}
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"stdout": "[path]:20:6: error: failed to rewrite using equation theorems for 'Transcendental'.\n'Transcendental.sin_exp_transcendental' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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|
| Gemini 3 Flash Preview | Partial | 1 | Feb 14, 2026 |
by_contra h_alg
have h_c : IsAlgebraic ℚ (Complex.sin (exp 1 : ℝ)) := by
exact isAlgebraic_iff_isAlgebraic_real.mpr h_alg
have h_e : IsAlgebraic ℚ (exp 1) := by
apply isAlgebraic_of_isAlgebraic_sin h_c
exact h_exp h_e
exact h_sin h
|
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