58.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 58

Reference: [Ben Green's Open Problem 58](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 58)

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem Green58.green_58 Suppose $A, B ⊆ \{1, \dots, N\}$ both have size at least $N^{0.49}$ sorry line 36 source
Source documentation
Suppose $A, B ⊆ \{1, \dots, N\}$ both have size at least $N^{0.49}$. Must the sumset $A + B$ contain a composite number?

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Finset, LE.le, Finset.Icc, OfNat.ofNat, Real, HPow.hPow, OfScientific.ofScientific, Bool.true, Finset.card, Exists, And, Membership.mem, HAdd.hAdd, Nat.Composite, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_58 :
    answer(sorry) ↔
      ∀ᶠ (N : ℕ) in Filter.atTop, ∀ᵉ (A ⊆ Finset.Icc 1 N) (B ⊆ Finset.Icc 1 N),
        (N : ℝ) ^ (0.49 : ℝ) ≤ (A.card : ℝ) →
        (N : ℝ) ^ (0.49 : ℝ) ≤ (B.card : ℝ) →
        ∃ m ∈ (A + B), m.Composite := by
…
```