Transcendental.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

22 declarations, 22 contain sorry, 0 declared as axioms

Open questions on transcendence of numbers

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem Transcendental.exp_add_pi_transcendental $e + \pi$ is transcendental sorry line 36 source
Source documentation
$e + \pi$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, HAdd.hAdd, Real.exp, OfNat.ofNat, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exp_add_pi_transcendental : Transcendental ℚ (exp 1 + π) := by
…
```
theorem Transcendental.exp_mul_pi_transcendental $e\pi$ is transcendental sorry line 43 source
Source documentation
$e\pi$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, HMul.hMul, Real.exp, OfNat.ofNat, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exp_mul_pi_transcendental : Transcendental ℚ (exp 1 * π) := by
…
```
theorem Transcendental.exp_pow_pi_sq_transcendental $e^{\pi^2}$ is transcendental sorry line 50 source
Source documentation
$e^{\pi^2}$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, Real.exp, HPow.hPow, Nat, Real.pi, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exp_pow_pi_sq_transcendental : Transcendental ℚ (exp (π ^ 2)) := by
…
```
theorem Transcendental.exp_exp_transcendental $e^e$ is transcendental sorry line 57 source
Source documentation
$e^e$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, Real.exp, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exp_exp_transcendental : Transcendental ℚ (exp (exp 1)) := by
…
```
theorem Transcendental.pi_pow_exp_transcendental $\pi^e$ is transcendental sorry line 64 source
Source documentation
$\pi^e$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, HPow.hPow, Real.pi, Real.exp, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pi_pow_exp_transcendental : Transcendental ℚ (π ^ (exp 1)) := by
…
```
theorem Transcendental.pi_pow_sqrt_two_transcendental $\pi^{\sqrt{2}}$ is transcendental sorry line 71 source
Source documentation
$\pi^{\sqrt{2}}$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, HPow.hPow, Real.pi, Real.sqrt, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pi_pow_sqrt_two_transcendental : Transcendental ℚ (π ^ √2) := by
…
```
theorem Transcendental.pi_pow_pi_transcendental $\pi^{\pi}$ is transcendental sorry line 78 source
Source documentation
$\pi^{\pi}$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, HPow.hPow, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pi_pow_pi_transcendental : Transcendental ℚ (π ^ π) := by
…
```
theorem Transcendental.pi_pow_pi_pow_pi_transcendental $\pi^{\pi^{\pi}}$ is transcendental sorry line 85 source
Source documentation
$\pi^{\pi^{\pi}}$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, HPow.hPow, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pi_pow_pi_pow_pi_transcendental : Transcendental ℚ (π ^ (π ^ π)) := by
…
```
theorem Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental $\pi^{\pi^{\pi^\pi}}$ is transcendental sorry line 92 source
Source documentation
$\pi^{\pi^{\pi^\pi}}$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, HPow.hPow, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pi_pow_pi_pow_pi_pow_pi_transcendental : Transcendental ℚ (π ^ (π ^ (π ^ π))) := by
…
```
theorem Transcendental.pi_pow_pi_pow_pi_pow_pi_not_integer $\pi^{\pi^{\pi^\pi}}$ is not an integer sorry line 105 source
Source documentation
$\pi^{\pi^{\pi^\pi}}$ is not an integer.
This would follow from $\pi^{\pi^{\pi^\pi}}$ being transcendental, but this formulation is of interest in its own right, as it could in principle be proven by direct computation.
Reference: YouTube

L4DEL explanation
$\pi^{\pi^{\pi^\pi}}$ is not an integer. This would follow from $\pi^{\pi^{\pi^\pi}}$ being transcendental, but this formulation is of interest in its own right, as it could in principle be proven by direct computation. Reference: YouTube
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Not, Exists, Int, Eq, Real, HPow.hPow, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pi_pow_pi_pow_pi_pow_pi_not_integer : ¬ ∃ (n : ℤ), π ^ π ^ π ^ π = n := by
…
```
theorem Transcendental.rlog_pi_transcendental $\log(\pi)$ is transcendental sorry line 112 source
Source documentation
$\log(\pi)$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, Real.log, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem rlog_pi_transcendental : Transcendental ℚ (log π) := by
…
```
theorem Transcendental.rlog_rlog_two_transcendental $\log(\log(2))$ is transcendental sorry line 119 source
Source documentation
$\log(\log(2))$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, Real.log, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem rlog_rlog_two_transcendental : Transcendental ℚ ((2 : ℝ).log.log) := by
…
```
theorem Transcendental.sin_exp_transcendental $\sin(e)$ is transcendental sorry line 126 source
Source documentation
$\sin(e)$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, Real.sin, Real.exp, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem sin_exp_transcendental : Transcendental ℚ (Real.sin (exp 1)) := by
…
```
theorem Transcendental.exp_add_pi_or_exp_add_mul_transcendental sorry line 135 source
depends on sorry what this means
Statement uses: Or, Transcendental, Rat, Real, HAdd.hAdd, Real.pi, Real.exp, OfNat.ofNat, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
35 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Transcendental.transcendental_eulerMascheroniConstant_or_gompertzConstant At least one of the Euler–Mascheroni constant $\gamma$ and the Gompertz constant $\delta$ is transcendental sorry line 147 source
Source documentation
At least one of the Euler–Mascheroni constant $\gamma$ and the Gompertz constant $\delta$ is transcendental.
[Ri12] Rivoal, T. (2012). On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant. Michigan Mathematical Journal, 61(2), 239–254.

depends on sorry what this means
Statement uses: Or, Transcendental, Rat, Real, Real.eulerMascheroniConstant, gompertzConstant
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem transcendental_eulerMascheroniConstant_or_gompertzConstant :
    Transcendental ℚ eulerMascheroniConstant ∨ Transcendental ℚ gompertzConstant := by
…
```
Mentions: gompertzConstant
theorem Transcendental.transcendental_catalanConstant The Catalan constant $G$ is transcendental sorry line 155 source
Source documentation
The Catalan constant $G$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, catalanConstant
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem transcendental_catalanConstant : Transcendental ℚ catalanConstant := by
…
```
Mentions: catalanConstant
theorem Transcendental.transcendental_gompertzConstant The Gompertz constant $\delta$ is transcendental sorry line 162 source
Source documentation
The Gompertz constant $\delta$ is transcendental.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, gompertzConstant
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem transcendental_gompertzConstant : Transcendental ℚ gompertzConstant := by
…
```
Mentions: gompertzConstant
theorem Transcendental.transcendental_gamma_one_div_two $\Gamma(1/2)$ is transcendental sorry line 171 source
Source documentation
$\Gamma(1/2)$ is transcendental.
[Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, Real.Gamma, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem transcendental_gamma_one_div_two : Transcendental ℚ (1 / 2 : ℝ).Gamma := by
…
```
theorem Transcendental.transcendental_gamma_one_div_three $\Gamma(1/3)$ is transcendental sorry line 180 source
Source documentation
$\Gamma(1/3)$ is transcendental.
[Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, Real.Gamma, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem transcendental_gamma_one_div_three : Transcendental ℚ (1 / 3 : ℝ).Gamma := by
…
```
theorem Transcendental.transcendental_gamma_one_div_four $\Gamma(1/4)$ is transcendental sorry line 189 source
Source documentation
$\Gamma(1/4)$ is transcendental.
[Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, Real.Gamma, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem transcendental_gamma_one_div_four : Transcendental ℚ (1 / 4 : ℝ).Gamma := by
…
```
theorem Transcendental.transcendental_gamma_one_div_six $\Gamma(1/6)$ is transcendental sorry line 198 source
Source documentation
$\Gamma(1/6)$ is transcendental.
[Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.

depends on sorry what this means
Statement uses: Transcendental, Rat, Real, Real.Gamma, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem transcendental_gamma_one_div_six : Transcendental ℚ (1 / 6 : ℝ).Gamma := by
…
```
theorem Transcendental.transcendental_gamma_one_div $\Gamma(1/n)$ for n ≥ 2 is transcendental sorry line 205 source
Source documentation
$\Gamma(1/n)$ for n ≥ 2 is transcendental.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Transcendental, Rat, Real, Real.Gamma, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem transcendental_gamma_one_div (n : ℕ) (hn : 2 ≤ n) : Transcendental ℚ (1 / n : ℝ).Gamma := by
…
```