2.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

6 declarations, 5 contain sorry, 0 declared as axioms

Ben Green's Open Problem 2

References:

[Gr24] Green, Ben. "100 open problems." (2024).
[Er65] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII, pages 181–189. Amer. Math. Soc., Providence, R.I., 1965.
[Sa21] Sanders, Tom. "The Erdős–Moser Sum-free Set Problem." Canadian Journal of Mathematics 73.1 (2021): 63-107.
[Ru05] I. Z. Ruzsa, Sum-avoiding subsets. Ramanujan J., 9 (2005) (1-2):77–82.
[Ch71] S. L. G. Choi. On a combinatorial problem in number theory. Proc. London Math. Soc. (3), 23:629–642, 1971. doi:10.1112/plms/s3-23.4.629.
[BSS00] A. Baltz, T. Schoen, and A. Srivastav. Probabilistic construction of small strongly sum-free sets via large Sidon sets. Colloq. Math., 86(2):171–176, 2000. doi:10.4064/cm-86-2-171-176.

Imports (1)

FormalConjecturesUtil

def Green2.maxRestrictedSumAvoidingSubsetSize We define the construction from [Sa21, p1] as $M(A) := \max \{|S| : S \subseteq A \text{ and } (S \hat{+} S) \cap A = \varnothing \}$ line 50 source
Source documentation
We define the construction from [Sa21, p1] as $M(A) := \max \{|S| : S \subseteq A \text{ and } (S \hat{+} S) \cap A = \varnothing \}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Int, Nat
Proof uses: Finset.sup, Finset.filter, Disjoint, Finset.restrictedSumset, Finset.powerset, Finset.card
8 instances, coercions and auxiliary constants left out.
```
def maxRestrictedSumAvoidingSubsetSize (A : Finset ℤ) : ℕ :=
…
```
theorem Green2.green_2 Let $A \subset \mathbf{Z}$ be a set of $n$ integers sorry line 58 source
Source documentation
Let $A \subset \mathbf{Z}$ be a set of $n$ integers. Is there a set $S \subset A$ of size $(\log n)^{100}$ such that the restricted sumset$S \hat{+} S$ is disjoint from $A$?

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Finset, Int, Eq, Finset.card, GE.ge, Real, Green2.maxRestrictedSumAvoidingSubsetSize, HPow.hPow, Real.log, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_2 : answer(sorry) ↔
    ∀ᶠ n : ℕ in atTop, ∀ A : Finset ℤ, A.card = n →
      (maxRestrictedSumAvoidingSubsetSize A : ℝ) ≥ (Real.log n) ^ 100 := by
…
```
Mentions: Green2.maxRestrictedSumAvoidingSubsetSize
theorem Green2.green_2_lower_bound_sanders From [Sa21] it is known that there is always such an S with $|S| \gt (\log |A|)^{1+c}$ sorry line 67 source
Source documentation
From [Sa21] it is known that there is always such an S with $|S| \gt (\log |A|)^{1+c}$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, Finset, Int, Eq, Finset.card, GE.ge, Green2.maxRestrictedSumAvoidingSubsetSize, HPow.hPow, Real.log, HAdd.hAdd, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_2_lower_bound_sanders :
    ∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, ∀ A : Finset ℤ, A.card = n →
      maxRestrictedSumAvoidingSubsetSize A ≥ Real.log n ^ (1 + c) := by
…
```
Mentions: c, Green2.maxRestrictedSumAvoidingSubsetSize
theorem Green2.green_2_upper_bound_erdos From [Er65] it is known that $M(A) \le \frac{1}{3}|A| + O(1)$ sorry line 76 source
Source documentation
From [Er65] it is known that $M(A) \le \frac{1}{3}|A| + O(1)$.

depends on sorry what this means
Statement uses: Exists, Real, Filter.Eventually, Nat, Finset, Int, And, Eq, Finset.card, LE.le, Green2.maxRestrictedSumAvoidingSubsetSize, HAdd.hAdd, HDiv.hDiv, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_2_upper_bound_erdos :
    ∃ C : ℝ, ∀ᶠ n in atTop, ∃ A : Finset ℤ, A.card = n ∧
      (maxRestrictedSumAvoidingSubsetSize A : ℝ) ≤ A.card / 3 + C := by
…
```
Mentions: Green2.maxRestrictedSumAvoidingSubsetSize
theorem Green2.green_2_upper_bound_choi From [Ch71] it is known that $M(A) \le |A|^{2/5 + o(1)}$ sorry line 90 source
Source documentation
From [Ch71] it is known that $M(A) \le |A|^{2/5 + o(1)}$.

depends on sorry what this means
Statement uses: Exists, Nat, Real, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, Filter.Eventually, Finset, Int, And, Eq, Finset.card, LE.le, Green2.maxRestrictedSumAvoidingSubsetSize, HPow.hPow, HAdd.hAdd, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_2_upper_bound_choi :
    ∃ (o : ℕ → ℝ) (_ : Tendsto o atTop (𝓝 0)), ∀ᶠ n in atTop, ∃ A : Finset ℤ, A.card = n ∧
      (maxRestrictedSumAvoidingSubsetSize A : ℝ) ≤ A.card ^ (2 / 5 + o A.card) := by
…
```
Mentions: Green2.maxRestrictedSumAvoidingSubsetSize
theorem Green2.green_2_upper_bound_ruzsa From [Ru05] the best-known upper bound is $|S| \lt e^{C \sqrt{\log |A|}}$ sorry line 99 source
Source documentation
From [Ru05] the best-known upper bound is $|S| \lt e^{C \sqrt{\log |A|}}$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, Finset, Int, Eq, Finset.card, LT.lt, Green2.maxRestrictedSumAvoidingSubsetSize, Real.exp, HMul.hMul, Real.sqrt, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_2_upper_bound_ruzsa :
    ∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∃ A : Finset ℤ, A.card = n ∧
      (maxRestrictedSumAvoidingSubsetSize A : ℝ) < Real.exp (C * Real.sqrt (Real.log A.card)) := by
…
```
Mentions: Green2.maxRestrictedSumAvoidingSubsetSize