1.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

1 declarations, 1 contain sorry, 0 declared as axioms

Ben Green's Open Problem 1

References:

[Ben Green's Open Problem 1](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.1 Problem 1)
[Be25] Bedert, Benjamin, Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series. arXiv:2502.08624 (2025).

Imports (1)

FormalConjecturesUtil

theorem Green1.green_1 Let $A$ be a set of $n$ positive integers sorry line 43 source
Source documentation
Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $\frac n 3 + Ω(n)$, where $Ω(n) → ∞$ as $n → ∞$?
Bedert [Be25] proved that every set of $n$ integers contains a sum-free subset of size at least $\frac n 3 + c \log\log n$ for some absolute constant $c > 0$, so the answer is yes.

L4DEL explanation
Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $\frac n 3 + Ω(n)$, where $Ω(n) → ∞$ as $n → ∞$? Bedert [Be25] proved that every set of $n$ integers contains a sum-free subset of size at least $\frac n 3 + c \log\log n$ for some absolute constant $c > 0$, so the answer is yes.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, Finset, Membership.mem, LT.lt, OfNat.ofNat, Eq, Finset.card, LE.le, IsSumFree, HAdd.hAdd, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_1 : answer(True) ↔ ∃ Ω : ℕ → ℝ, atTop.Tendsto Ω atTop ∧
     ∀ n, ∀ (A : Finset ℕ), (∀ a ∈ A, 0 < a) → A.card = n →
     ∃ (S : Finset ℕ), S ⊆ A ∧ IsSumFree (S : Set ℕ) ∧ ((n : ℝ) / 3) + Ω n ≤ S.card := by
…
```