3.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 3

References:

[Ben Green's Open Problem 3](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.3 Problem 3)
[FGY26] Franchi, Leonardo and Gowers, W. T. and Yip, Fredy, Product-free subsets of $(0,1)$, arXiv:2607.06073, Theorem 1.1.

Imports (1)

FormalConjecturesUtil

theorem Green3.green_3 Suppose that $A \subset [0,1]$ is open and has measure greater than $\frac{1}{3}$ sorry line 41 source
Source documentation
Suppose that $A \subset [0,1]$ is open and has measure greater than $\frac{1}{3}$. Is there a solution to $xy = z$ with $x, y, z \in A$?
The answer is yes. [FGY26] prove that an open product-free subset of $(0,1)$ has measure strictly less than $\frac{1}{3}$.

depends on sorry what this means
Statement uses: Iff, True, Set, Real, IsOpen, LE.le, Set.Icc, OfNat.ofNat, GT.gt, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, HDiv.hDiv, Nat, Exists, And, Membership.mem, Eq, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
60 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_3 :
    answer(True) ↔ ∀ A : Set ℝ,
      IsOpen A → A ⊆ Icc 0 1 → volume A > 1/3 →
        ∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x * y = z := by
…
```