229.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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 229

References:

erdosproblems.com/229
[BaSc72] Barth, K. F. and Schneider, W. J., On a problem of Erd\H{o}s concerning the zeros of the derivatives of an entire function. Proc. Amer. Math. Soc. (1972), 229--232.
[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.

Imports (1)

FormalConjecturesUtil

theorem Erdos229.erdos_229 Let $(S_n)_{n \ge 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point sorry line 46 source
Source documentation
Let $(S_n)_{n \ge 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n \ge 1$, there exists some $k_n \ge 0$ such that $f^{(k_n)}(z) = 0$ for all $z \in S_n$.
This is Problem 2.30 in [Ha74], where it is attributed to Erdős.
Solved in the affirmative by Barth and Schneider [BaSc72].
This was formalized in Lean by Alexeev using Aristotle.

depends on sorry what this means
Statement uses: Iff, True, Nat, Set, Complex, Eq, derivedSet, EmptyCollection.emptyCollection, Exists, And, Transcendental, Polynomial, Polynomial.algebraPi, Differentiable, GE.ge, OfNat.ofNat, Membership.mem, iteratedDeriv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
66 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_229 :
    letI := Polynomial.algebraPi ℂ ℂ ℂ
…
```
theorem Erdos229.theorem_1 Let $\{S_k\}$ be any sequence of sets in the complex plane, each of which has no finite limit point sorry line 59 source
Source documentation
Let $\{S_k\}$ be any sequence of sets in the complex plane, each of which has no finite limit point. Then there exists a sequence $\{n_k\}$ of positive integers and a transcendental entire function $f(z)$ such that $f^{(n_k)}(z) = 0$ if $z \in S_k$.

depends on sorry what this means
Statement uses: Nat, Set, Complex, Eq, derivedSet, EmptyCollection.emptyCollection, Exists, And, Differentiable, Transcendental, Polynomial, Polynomial.algebraPi, LT.lt, OfNat.ofNat, Membership.mem, iteratedDeriv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
66 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem theorem_1
    {S : ℕ → Set ℂ}
    (h : ∀ (k), derivedSet (S k) = ∅) :
  letI := Polynomial.algebraPi ℂ ℂ ℂ
…
```