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transcendental_gompertzConstant
Specification
The Gompertz constant $\delta$ is transcendental.
Lean 4 Statement
theorem transcendental_gompertzConstant : Transcendental ℚ gompertzConstant
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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 gompertzConstant := ?_
exact h
|
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"code": "have h : Transcendental \u211a gompertzConstant := ?_",
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|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
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|
|
| Gemini 3 Flash Agent v8 | Partial | 3 | Feb 17, 2026 |
intro h
unfold gompertzConstant at h
let f := fun (t : ℝ) => exp (-t) / (1 + t)
have h_int : gompertzConstant = ∫ (t : ℝ) in Set.Ioi (0 : ℝ), f t := by
unfold gompertzConstant
refine ?_
have h_subst : ∫ (t : ℝ) in Set.Ioi 0, f t = exp 1 * ∫ (u : ℝ) in Set.Ioi 1, exp (-u) / u := by
refine ?_
have h_trans : Transcendental ℚ (exp 1 * ∫ (u : ℝ) in Set.Ioi 1, exp (-u) / u) := by
refine ?_
have h_final : IsAlgebraic ℚ gompertzConstant := h
rw [h_int, h_subst] at h_final
exact h_trans h_final
|
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"proof_code": "intro h\nunfold gompertzConstant at h\nlet J := \u222b (t : \u211d) in (0 : \u211d)..\u221e, (1 / (1 + t)) * exp (-t)\nhave hJ : gompertzConstant = J := rfl\nrw [hJ] at h\nhave h_exp_int : J = \u222b (t : \u211d) in (0 : \u211d)..\u221e, exp (-t) / (1 + t) := by\n apply set_integral_congr\n \u00b7 exact measurableSet_Ioi 0\n \u00b7 intros t ht\n ring\nhave h_tr : Transcendental \u211a J := by\n refine ?_\nexact h_tr h",
"compiler_output_preview": "[path]:21:33: error: expected token\n'Transcendental.transcendental_gompertzConstant' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro h\nunfold gompertzConstant at h\nlet f := fun (t : \u211d) => exp (-t) / (1 + t)\nhave h_def : gompertzConstant = \u222b (t : \u211d) in Set.Ioi (0 : \u211d), f t := rfl\nrw [h_def] at h\nhave h_int : \u222b (t : \u211d) in Set.Ioi (0 : \u211d), exp (-t) / (1 + t) = exp 1 * (\u222b (u : \u211d) in Set.Ioi (1 : \u211d), exp (-u) / u) := by\n refine ?_\nrw [h_int] at h\nlet E1 := \u222b (u : \u211d) in Set.Ioi (1 : \u211d), exp (-u) / u\nhave h_E1 : Transcendental \u211a (exp 1 * E1) := by\n refine ?_\nexact h_E1 h",
"compiler_output_preview": "[path]:22:71: error: type mismatch\n rfl\nhas type\n ?m.790 = ?m.790 : Prop\nbut is expected to have type\n gompertzConstant = \u222b (t : \u211d) in Set.Ioi 0, f t : Prop\n'Transcendental.transcendental_gompertzConstant' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro h\nunfold gompertzConstant at h\nlet f := fun (t : \u211d) => exp (-t) / (1 + t)\nhave h_def : gompertzConstant = \u222b (t : \u211d) in Set.Ioi (0 : \u211d), f t := by\n unfold gompertzConstant\n refine ?_\nrw [h_def] at h\nhave h_subst : \u222b (t : \u211d) in Set.Ioi (0 : \u211d), exp (-t) / (1 + t) = exp 1 * \u222b (u : \u211d) in Set.Ioi (1 : \u211d), exp (-u) / u := by\n refine ?_\nrw [h_subst] at h\nlet E1 := \u222b (u : \u211d) in Set.Ioi (1 : \u211d), exp (-u) / u\nhave h_trans : Transcendental \u211a (exp 1 * E1) := by\n refine ?_\nexact h_trans h",
"compiler_output_preview": "[path]:22:71: error: unsolved goals\nh : IsAlgebraic \u211a (-rexp 1 * \u222b (t : \u211d) in Set.Ioi 1, rexp (-t) / t)\nf : \u211d \u2192 \u211d := fun t \u21a6 rexp (-t) / (1 + t)\n\u22a2 -rexp 1 * \u222b (t : \u211d) in Set.Ioi 1, rexp (-t) / t = \u222b (t : \u211d) in Set.Ioi 0, f t\n[path]:25:6: \n...<truncated>"
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"proof_code": "intro h\nunfold gompertzConstant at h\nlet f := fun (t : \u211d) => exp (-t) / (1 + t)\nhave h_int_def : gompertzConstant = \u222b (t : \u211d) in Set.Ioi (0 : \u211d), f t := rfl\nrw [h_int_def] at h\nhave h_subst : \u222b (t : \u211d) in Set.Ioi (0 : \u211d), exp (-t) / (1 + t) = exp 1 * \u222b (x : \u211d) in Set.Ioi (1 : \u211d), exp (-x) / x := by\n let g := fun (t : \u211d) => t + 1\n have h_meas : Measurable g := measurable_add_const 1\n refine ?_\nrw [h_subst] at h\nlet E1 := \u222b (x : \u211d) in Set.Ioi (1 : \u211d), exp (-x) / x\nhave h_E1_def : E1 = - real_ei (-1) := by\n refine ?_\nhave h_trans : Transcendental \u211a (exp 1 * E1) := by\n refine ?_\nexact h_trans h",
"compiler_output_preview": "[path]:22:75: error: type mismatch\n rfl\nhas type\n ?m.790 = ?m.790 : Prop\nbut is expected to have type\n gompertzConstant = \u222b (t : \u211d) in Set.Ioi 0, f t : Prop\n'Transcendental.transcendental_gompertzConstant' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro h\nunfold gompertzConstant at h\nlet f := fun (t : \u211d) => exp (-t) / (1 + t)\nhave h_int_def : gompertzConstant = \u222b (t : \u211d) in Set.Ioi (0 : \u211d), f t := by\n unfold gompertzConstant\n refine ?_\nrw [h_int_def] at h\nlet Ei := fun (x : \u211d) => \u222b (t : \u211d) in Set.Ioi x, exp (-t) / t\nhave h_subst : \u222b (t : \u211d) in Set.Ioi (0 : \u211d), f t = exp 1 * Ei 1 := by\n dsimp [Ei]\n refine ?_\nrw [h_subst] at h\nhave h_trans : Transcendental \u211a (exp 1 * Ei 1) := by\n refine ?_\nexact h_trans h",
"compiler_output_preview": "[path]:22:75: error: unsolved goals\nh : IsAlgebraic \u211a (-rexp 1 * \u222b (t : \u211d) in Set.Ioi 1, rexp (-t) / t)\nf : \u211d \u2192 \u211d := fun t \u21a6 rexp (-t) / (1 + t)\n\u22a2 -rexp 1 * \u222b (t : \u211d) in Set.Ioi 1, rexp (-t) / t = \u222b (t : \u211d) in Set.Ioi 0, f t\n[path]:25:6: \n...<truncated>"
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"proof_code": "intro h\nunfold gompertzConstant at h\nlet f := fun (t : \u211d) => exp (-t) / (1 + t)\nhave h_int : gompertzConstant = \u222b (t : \u211d) in Set.Ioi 0, f t := by\n unfold gompertzConstant\n refine ?_\nhave h_subst : \u222b (t : \u211d) in Set.Ioi 0, f t = exp 1 * \u222b (u : \u211d) in Set.Ioi 1, exp (-u) / u := by\n refine ?_\nhave h_trans : Transcendental \u211a (exp 1 * \u222b (u : \u211d) in Set.Ioi 1, exp (-u) / u) := by\n refine ?_\nrw [h_int, h_subst] at h\nexact h_trans h",
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"proof_code": "intro h\nunfold gompertzConstant at h\nlet f := fun (t : \u211d) => exp (-t) / (1 + t)\nhave h_int : gompertzConstant = \u222b (t : \u211d) in Set.Ioi (0 : \u211d), f t := by\n unfold gompertzConstant\n refine ?_\nhave h_subst : \u222b (t : \u211d) in Set.Ioi 0, f t = exp 1 * \u222b (u : \u211d) in Set.Ioi 1, exp (-u) / u := by\n refine ?_\nhave h_trans : Transcendental \u211a (exp 1 * \u222b (u : \u211d) in Set.Ioi 1, exp (-u) / u) := by\n refine ?_\nhave h_final : IsAlgebraic \u211a gompertzConstant := h\nrw [h_int, h_subst] at h_final\nexact h_trans h_final",
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"proof_code": "intro h\nunfold gompertzConstant at h\nhave h_def : gompertzConstant = \u222b (t : \u211d) in Set.Ioi 0, exp (-t) / (1 + t) := by\n unfold gompertzConstant\n refine ?_\nhave h_eq : \u222b (t : \u211d) in Set.Ioi 0, exp (-t) / (1 + t) = exp 1 * \u222b (u : \u211d) in Set.Ioi 1, exp (-u) / u := by\n refine ?_\nhave h_trans : Transcendental \u211a (exp 1 * \u222b (u : \u211d) in Set.Ioi 1, exp (-u) / u) := by\n refine ?_\nrw [h_def, h_eq] at h\nexact h_trans h",
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"stdout": "[path]:22:71: error: unsolved goals\nh : IsAlgebraic \u211a (-rexp 1 * \u222b (t : \u211d) in Set.Ioi 1, rexp (-t) / t)\nf : \u211d \u2192 \u211d := fun t \u21a6 rexp (-t) / (1 + t)\n\u22a2 -rexp 1 * \u222b (t : \u211d) in Set.Ioi 1, rexp (-t) / t = \u222b (t : \u211d) in Set.Ioi 0, f t\n[path]:25:95: error: unsolved goals\nh : IsAlgebraic \u211a (-rexp 1 * \u222b (t : \u211d) in Set.Ioi 1, rexp (-t) / t)\nf : \u211d \u2192 \u211d := fun t \u21a6 rexp (-t) / (1 + t)\nh_int : gompertzConstant = \u222b (t : \u211d) in Set.Ioi 0, f t\n\u22a2 \u222b (t : \u211d) in Set.Ioi 0, f t = rexp 1 * \u222b (u : \u211d) in Set.Ioi 1, rexp (-u) / u\n[path]:27:84: error: unsolved goals\nh : IsAlgebraic \u211a (-rexp 1 * \u222b (t : \u211d) in Set.Ioi 1, rexp (-t) / t)\nf : \u211d \u2192 \u211d := fun t \u21a6 rexp (-t) / (1 + t)\nh_int : gompertzConstant = \u222b (t : \u211d) in Set.Ioi 0, f t\nh_subst : \u222b (t : \u211d) in Set.Ioi 0, f t = rexp 1 * \u222b (u : \u211d) in Set.Ioi 1, rexp (-u) / u\n\u22a2 Transcendental \u211a (rexp 1 * \u222b (u : \u211d) in Set.Ioi 1, rexp (-u) / u)\n'Transcendental.transcendental_gompertzConstant' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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|
| Gemini 3 Flash Preview | Partial | 1 | Feb 14, 2026 |
unfold gompertzConstant
rw [exp_integral_exp_one_inv]
have h_e : Transcendental ℚ (exp 1) := transcendental_exp_one
by_contra h_alg
apply h_alg
ring_nf
apply Transcendental.algebraic_of_sub
· exact transcendental_exp_one
· simp
exact h_alg
|
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