15.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

2 declarations, 2 contain sorry, 0 declared as axioms

Green's Open Problem 15

References:

[Gr24] Green, Ben. "100 open problems." (2024).
[BJP14] T. Brown, V. Jungić and A. Poelstra, "On double 3-term arithmetic progressions", Integers 14 (2014), Paper No. A43.
[CCS14] J. Cassaigne, J. D. Currie, L. Schaeffer and J. Shallit, "Avoidance of additive cubes and related results", Adv. in Appl. Math. 56 (2014), 25–66.

Imports (1)

FormalConjecturesUtil

theorem Green15.green_15 Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{N}\} \subseteq \mathbb{Z}^2$ is free of 3-term progressions? sorry line 43 source
Source documentation
Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{N}\} \subseteq \mathbb{Z}^2$ is free of 3-term progressions?

depends on sorry what this means
Statement uses: Iff, True, Exists, NNReal, Nat, Int, And, LipschitzWith, Set.IsAPOfLengthFree, Prod, Set.ofPred, Eq, Prod.mk, OfNat.ofNat, ENat
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_15 :
    answer(sorry) ↔ ∃ K : ℝ≥0, ∃ f : ℕ → ℤ, LipschitzWith K f ∧
      IsAPOfLengthFree {((n, f n) : ℤ × ℤ) | (n : ℕ)} 3 := by
…
```
theorem Green15.green_15_ap4 The answer is YES for 4-term progressions [BJP14] sorry line 52 source
Source documentation
The answer is YES for 4-term progressions [BJP14].

depends on sorry what this means
Statement uses: Exists, NNReal, Nat, Int, And, LipschitzWith, Set.IsAPOfLengthFree, Prod, Set.ofPred, Eq, Prod.mk, OfNat.ofNat, ENat
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_15_ap4 :
    ∃ K : ℝ≥0, ∃ f : ℕ → ℤ, LipschitzWith K f ∧
      IsAPOfLengthFree  {((n, f n) : ℤ × ℤ) | (n : ℕ)} 4 := by
…
```