18.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

4 declarations, 2 contain sorry, 0 declared as axioms

Ben Green's Open Problem 18

Reference:

[Gr26] Ben Green's Open Problem 18
[Au16] Austin, Tim. "Ajtai–Szemerédi theorems over quasirandom groups." Recent trends in combinatorics. Cham: Springer International Publishing, 2016. 453-484.
[So13] Solymosi, Jozsef. "Roth-type theorems in finite groups." European Journal of Combinatorics 34.8 (2013): 1454-1458.
[Go01] Gowers, William T. "A new proof of Szemerédi's theorem." Geometric & Functional Analysis GAFA 11.3 (2001): 465-588.

Imports (1)

FormalConjecturesUtil

def Green18.numNaiveCorners The number of triples $(x, y, g)$ in $G^3$ such that $g \neq e$, and $(x, y), (gx, y), (x, gy)$ are all in $A$ line 47 source
Source documentation
The number of triples $(x, y, g)$ in $G^3$ such that $g \neq e$, and $(x, y), (gx, y), (x, gy)$ are all in $A$. These are called "naive corners" by [Au16].
Note: the shortened formulation from [Gr26] does not mention $g \neq e$, but this is the original statement from [Au16], which ensure non-trivial corners. Note however that [Au16] use more generally compact groups and not just finite discrete groups.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, DecidableEq, Finset, Prod, Nat
Proof uses: Finset.card, Finset.filter, Green18.numNaiveCorners.match_1, And, Ne, OfNat.ofNat, Membership.mem, Prod.mk, HMul.hMul, Prod.snd, Prod.fst, Eq, Finset.univ
18 instances, coercions and auxiliary constants left out.
```
def numNaiveCorners {G : Type*} [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)) : ℕ :=
…
```
theorem Green18.green_18 Suppose that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$ sorry line 60 source
Source documentation
Suppose that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Is it true that there are $\gg_\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$?
Note: A is taken as $\alpha$-dense, i.e. $|A| \ge \alpha |G|^2$ [Au16, Question 2]

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Exists, And, Nat, Group, Fintype, DecidableEq, Finset, Prod, GE.ge, Fintype.card, Finset.card, HMul.hMul, HPow.hPow, Green18.numNaiveCorners
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_18 : answer(sorry) ↔
    ∀ α > 0, ∃ c > 0, ∃ m₀ : ℕ,
      ∀ (G : Type*) [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)),
      Fintype.card G ≥ m₀ →
      (A.card : ℝ) ≥ α * (Fintype.card G) ^ 2 →
      (numNaiveCorners A : ℝ) ≥ c * (Fintype.card G) ^ 3 := by
…
```
Mentions: c, Green18.numNaiveCorners
def Green18.numBmzCorners The number of triples $(x, y, g)$ in $G^3$ such that $g \neq e$, and $(x, y), (xg, y), (x, gy)$ are all in $A$ line 72 source
Source documentation
The number of triples $(x, y, g)$ in $G^3$ such that $g \neq e$, and $(x, y), (xg, y), (x, gy)$ are all in $A$. These are called "BMZ corners" by [Au16].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, DecidableEq, Finset, Prod, Nat
Proof uses: Finset.card, Finset.filter, Green18.numNaiveCorners.match_1, And, Ne, OfNat.ofNat, Membership.mem, Prod.mk, HMul.hMul, Prod.snd, Prod.fst, Eq, Finset.univ
18 instances, coercions and auxiliary constants left out.
```
def numBmzCorners {G : Type*} [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)) : ℕ :=
…
```
theorem Green18.green_18.bmz_corners [So13] proved this is true for "BMZ corners" sorry line 82 source
Source documentation
[So13] proved this is true for "BMZ corners". Follows from the proof of Theorem 2.1, p.1456-1457.

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Exists, And, Nat, Group, Fintype, DecidableEq, Finset, Prod, GE.ge, Fintype.card, Finset.card, HMul.hMul, HPow.hPow, Green18.numBmzCorners
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_18.bmz_corners : ∀ α > 0, ∃ c > 0, ∃ m₀ : ℕ,
    ∀ (G : Type*) [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)),
    Fintype.card G ≥ m₀ →
    (A.card : ℝ) ≥ α * (Fintype.card G) ^ 2 →
    (numBmzCorners A : ℝ) ≥ c * (Fintype.card G) ^ 3 := by
…
```
Mentions: c, Green18.numBmzCorners