29.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

2 declarations, 2 contain sorry, 0 declared as axioms

Ben Green's Open Problem 29

References:

Ben Green's Open Problem 29
[Gr12] Green, Ben. "What is... an approximate group." Notices Amer. Math. Soc 59.5 (2012): 655-656.
[Br13] Breuillard, Emmanuel, Ben Green, and Terence Tao. "Small doubling in groups." Erdős Centennial. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. 129-151.
[Sa10] Sanders, Tom. "On a nonabelian Balog–Szemerédi-type lemma." Journal of the Australian Mathematical Society 89.1 (2010): 127-132.
[CrSi10] Croot, Ernie, and Olof Sisask. "A probabilistic technique for finding almost-periods of convolutions." Geometric and functional analysis 20.6 (2010): 1367-1396.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

theorem Green29.green_29 sorry line 49 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, LT.lt, OfNat.ofNat, Group, DecidableEq, Finset, LE.le, IsApproximateSubgroup, HMul.hMul, HPow.hPow, Neg.neg, Finset.card, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green29.green_29.variant Such a conclusion is known with $|S| \gg_K |A|$ [Br13 Problem 6.5, CrSi10, Sa10] sorry line 60 source
Source documentation
Such a conclusion is known with $|S| \gg_K |A|$ [Br13 Problem 6.5, CrSi10, Sa10].

depends on sorry what this means
Statement uses: Real, LE.le, OfNat.ofNat, Exists, And, LT.lt, Group, DecidableEq, Finset, IsApproximateSubgroup, HMul.hMul, Finset.card, HPow.hPow, Nat
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 green_29.variant :
    ∀ K : ℝ, 1 ≤ K →
      ∃ c : ℝ, 0 < c ∧ -- Allow c to depend on K.
        ∀ {G : Type*} [Group G] [DecidableEq G] (A : Finset G),
          IsApproximateSubgroup K (A : Set G) →
            ∃ S : Finset G, -- No S ⊆ A requirement in this variant.
              c * (A.card : ℝ) ≤ (S.card : ℝ) ∧
              S ^ 8 ⊆ A ^ 4 := by
…
```
Mentions: c