16.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

9 declarations, 4 contain sorry, 0 declared as axioms

Ben Green's Open Problem 16

References:

Ben Green's Open Problem 16
[Ruzsa](I. Z. Ruzsa, Solving a linear equation in a set of integers. I. Acta Arith. 65 (1993), no. 3, 259–282.)
[Schoen and Sisask](T. Schoen and O. Sisask, Roth’s theorem for four variables and additive structures in sums of sparse sets Forum of Mathematics, Sigma (2016), Vol. 4, e5, 28 pages.)
[Yufei Zhao](Via Personal Communication with Ben Green)

Imports (1)

FormalConjecturesUtil

def Green16.SolutionFree A set has no solution to $x + 3y = 2z + 2w$ in distinct elements line 37 source
Source documentation
A set has no solution to $x + 3y = 2z + 2w$ in distinct elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Membership.mem, List.Nodup, List.cons, List.nil, Ne, HAdd.hAdd, HMul.hMul, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def SolutionFree (A : Finset ℕ) : Prop :=
…
```
def Green16.f The maximum size of a solution-free subset of $[N]$ line 43 source
Source documentation
The maximum size of a solution-free subset of $[N]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Green16.SolutionFree, Eq, Finset.card
7 instances, coercions and auxiliary constants left out.
```
noncomputable def f (N : ℕ) : ℕ :=
…
```
theorem Green16.green_16 What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$? sorry line 48 source
Source documentation
What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?

depends on sorry what this means
Statement uses: Nat, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Green16.SolutionFree, Eq, Finset.card, sorryAx, Bool.false, MaximalFor
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_16 (N : ℕ) :
    ∃ A : Finset ℕ, A ⊆ Icc 1 N ∧ SolutionFree A ∧
      A.card = (answer(sorry) : ℕ → ℕ) N ∧
      MaximalFor (fun B => B ⊆ Icc 1 N ∧ SolutionFree B) Finset.card A := by
…
```
Mentions: Green16.SolutionFree
theorem Green16.green_16_lower_bound From [Ruzsa] $f(N) \gg N^{1/2}$ sorry line 56 source
Source documentation
From [Ruzsa] $f(N) \gg N^{1/2}$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HPow.hPow, HDiv.hDiv, OfNat.ofNat, Green16.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_16_lower_bound :
    (fun N ↦ (N : ℝ) ^ (1 / 2 : ℝ)) ≪ fun N ↦ (f N : ℝ) := by
…
```
Mentions: Green16.f
theorem Green16.green_16_upper_bound From [Schoen and Sisask] $f(N) \ll N \cdot e^{-c(\log N)^{1/7}}$ sorry line 62 source
Source documentation
From [Schoen and Sisask] $f(N) \ll N \cdot e^{-c(\log N)^{1/7}}$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, Green16.f, HMul.hMul, Real.exp, Neg.neg, HPow.hPow, Real.log, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_16_upper_bound :
    ∃ c > (0 : ℝ), (fun N ↦ (f N : ℝ)) ≪ fun N ↦ (N : ℝ) * exp (-c * (log N) ^ (1 / 7 : ℝ)) := by
…
```
Mentions: c, Green16.f
def Green16.IsTrivialZhaoSolution Documented declaration: IsTrivialZhaoSolution line 68 source
Source documentation
A solution $(x, y, z, x', y', z')$ of $x + 2y + 3z = x' + 2y' + 3z'$ is trivial if for every value $a$ the coefficients of the variables equal to $a$ have the same sum on both sides.

no axioms what this means
Statement uses: Nat
Proof uses: Eq, HAdd.hAdd, ite, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
```
def IsTrivialZhaoSolution (x y z x' y' z' : ℕ) : Prop :=
…
```
def Green16.ZhaoSolutionFree A set has no nontrivial solution to $x + 2y + 3z = x' + 2y' + 3z'$ line 73 source
Source documentation
A set has no nontrivial solution to $x + 2y + 3z = x' + 2y' + 3z'$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Membership.mem, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Green16.IsTrivialZhaoSolution
7 instances, coercions and auxiliary constants left out.
```
def ZhaoSolutionFree (A : Finset ℕ) : Prop :=
…
```
def Green16.g The maximum size of a Zhao-solution-free subset of $[N]$ line 78 source
Source documentation
The maximum size of a Zhao-solution-free subset of $[N]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Green16.ZhaoSolutionFree, Eq, Finset.card
7 instances, coercions and auxiliary constants left out.
```
noncomputable def g (N : ℕ) : ℕ :=
…
```
theorem Green16.zhao_question From [Yufei Zhao]: Is there a subset of $\{1, \ldots, N\}$ of size $N^{1/3 - o(1)}$ with no nontrivial solutions to $x + 2y + 3z = x' + 2y' + 3z'$? sorry line 84 source
Source documentation
From [Yufei Zhao]: Is there a subset of $\{1, \ldots, N\}$ of size $N^{1/3 - o(1)}$ with no nontrivial solutions to $x + 2y + 3z = x' + 2y' + 3z'$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, Filter.Eventually, GE.ge, Green16.g, HPow.hPow, HSub.hSub, 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 zhao_question :
    answer(sorry) ↔ ∃ h : ℕ → ℝ, Tendsto h atTop (𝓝 0) ∧
      ∀ᶠ N in atTop, (g N : ℝ) ≥ (N : ℝ) ^ (1 / 3 - h N) := by
…
```
Mentions: Green16.g