Irrational.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

9 declarations, 9 contain sorry, 0 declared as axioms

Open questions on irrationality of numbers

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem Irrational.algebraicIndependent_e_pi Are $e$ and $\pi$ algebraically independent? sorry line 39 source
Source documentation
Are $e$ and $\pi$ algebraically independent?

depends on sorry what this means
Statement uses: Iff, True, AlgebraicIndependent, Fin, Nat.succ, OfNat.ofNat, Nat, Rat, Real, Matrix.vecCons, Real.exp, Real.pi, Matrix.vecEmpty
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem algebraicIndependent_e_pi :
    answer(sorry) ↔ AlgebraicIndependent ℚ ![e, π] := by
…
```
theorem Irrational.irrational_e_plus_pi Is $e + \pi$ irrational? sorry line 47 source
Source documentation
Is $e + \pi$ irrational?

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_e_plus_pi :
    answer(sorry) ↔ Irrational (e + π) := by
…
```
theorem Irrational.irrational_e_times_pi Is $e \pi$ irrational? sorry line 55 source
Source documentation
Is $e \pi$ irrational?

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_e_times_pi :
    answer(sorry) ↔ Irrational (e * π) := by
…
```
theorem Irrational.irrational_e_to_e Is $e ^ e$ irrational? sorry line 63 source
Source documentation
Is $e ^ e$ irrational?

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_e_to_e :
    answer(sorry) ↔ Irrational (e ^ e) := by
…
```
theorem Irrational.irrational_pi_to_e Is $\pi ^ e$ irrational? sorry line 71 source
Source documentation
Is $\pi ^ e$ irrational?

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_pi_to_e :
    answer(sorry) ↔ Irrational (π ^ e) := by
…
```
theorem Irrational.irrational_pi_to_pi Is $\pi ^ \pi$ irrational? sorry line 79 source
Source documentation
Is $\pi ^ \pi$ irrational?

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_pi_to_pi :
    answer(sorry) ↔ Irrational (π ^ π) := by
…
```
theorem Irrational.irrational_ln_pi Is $\ln(\pi)$ irrational? sorry line 87 source
Source documentation
Is $\ln(\pi)$ irrational?

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_ln_pi :
    answer(sorry) ↔ Irrational (log π) := by
…
```
theorem Irrational.irrational_eulerMascheroniConstant Is the Euler-Mascheroni constant $\gamma$ irrational? sorry line 95 source
Source documentation
Is the Euler-Mascheroni constant $\gamma$ irrational?

depends on sorry what this means
Statement uses: Iff, True, Irrational, Real.eulerMascheroniConstant
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_eulerMascheroniConstant :
    answer(sorry) ↔ Irrational eulerMascheroniConstant := by
…
```
theorem Irrational.irrational_catalanConstant Is the Catalan constant $$G = \sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \approx 0.91596$$ irrational? sorry line 103 source
Source documentation
Is the Catalan constant $$G = \sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \approx 0.91596$$ irrational?

depends on sorry what this means
Statement uses: Iff, True, Irrational, catalanConstant
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_catalanConstant :
    answer(sorry) ↔ Irrational catalanConstant := by
…
```
Mentions: catalanConstant