19.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

7 declarations, 1 contain sorry, 0 declared as axioms

Ben Green's Open Problem 19

References:

[Gr26] Ben Green's Open Problems
[FSS20] Fox, Jacob, et al. "Triforce and corners." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 169. No. 1. Cambridge University Press, 2020.
[Ma21] Mandache, Matei. "A variant of the Corners theorem." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 171. No. 3. Cambridge University Press, 2021.
[Ch11] Chu, Qing. "Multiple recurrence for two commuting transformations." Ergodic Theory and Dynamical Systems 31.3 (2011): 771-792.

Imports (1)

FormalConjecturesUtil

def Green19.IsCorner A corner in $A$ with common difference $d$ [FSS20] line 45 source
Source documentation
A corner in $A$ with common difference $d$ [FSS20].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Finset, Prod
Proof uses: And, Membership.mem, Prod.mk, HAdd.hAdd
7 instances, coercions and auxiliary constants left out.
```
def IsCorner (A : Finset (G × G)) (x y d : G) : Prop :=
…
```
def Green19.S From [FSS20]: given $A \subseteq G \times G$ and $d \in G$, let $$S_d(A) = \lbrace (x, y) \in G \times G : (x, y), (x + d, y), (x, y + d) \in A \rbrace$$ line 52 source
Source documentation
From [FSS20]: given $A \subseteq G \times G$ and $d \in G$, let $$S_d(A) = \lbrace (x, y) \in G \times G : (x, y), (x + d, y), (x, y + d) \in A \rbrace$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Fintype, Finset, Prod
Proof uses: Finset.filter, Green19.IsCorner, Prod.fst, Prod.snd, Classical.propDecidable, Finset.univ
1 instances, coercions and auxiliary constants left out.
```
noncomputable def S (d : G) (A : Finset (G × G)) : Finset (G × G) :=
…
```
def Green19.ValidExponent True if the given exponent satisfies Green's conditions [Gr26] line 62 source
Source documentation
True if the given exponent satisfies Green's conditions [Gr26].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Exists, And, GT.gt, OfNat.ofNat, LT.lt, Filter.Eventually, Nat, Finset, Prod, 𝔽₂, Fintype.card, Fin, ZMod, GE.ge, Finset.card, HMul.hMul, HPow.hPow, Ne, Green19.S, Filter.atTop
34 instances, coercions and auxiliary constants left out.
```
def ValidExponent (c : ℝ) : Prop :=
…
```
Mentions: c
def Green19.C The infimum of all valid exponents [Gr26] line 72 source
Source documentation
The infimum of all valid exponents [Gr26].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: InfSet.sInf, Set.ofPred, Green19.ValidExponent
1 instances, coercions and auxiliary constants left out.
```
noncomputable def C : ℝ := sInf {c | ValidExponent c}
…
```
Mentions: c, Green19.ValidExponent
theorem Green19.green_19 What is $C$, the infimum of all exponents $c$ for which the following is true, uniformly for $0 < \alpha < 1$? sorry line 83 source
Source documentation
What is $C$, the infimum of all exponents $c$ for which the following is true, uniformly for $0 < \alpha < 1$? Suppose that $A \subset \mathbb{F}_2^n \times \mathbb{F}_2^n$ is a set of density $\alpha$. Write $N := 2^n$. Then there is some $d \neq 0$ such that $A$ contains $\gg \alpha^c N^2$ corners $(x,y), (x,y+d), (x+d,y)$.
This question has been resolved by [FSS20], showing that $C = 4$.

depends on sorry what this means
Statement uses: Eq, Real, Green19.C, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_19 : C = 4 := by
…
```
Mentions: Green19.C
theorem Green19.green_19.lower [Ma21] showed that $3.13 \leq C$ line 88 source
Source documentation
[Ma21] showed that $3.13 \leq C$. This also follows from green_19.

depends on sorry what this means
Statement uses: GE.ge, Real, Green19.C, OfScientific.ofScientific, Bool.true
Proof uses: Eq.mpr, OfNat.ofNat, Nat, id, Eq, congrArg, Green19.green_19, Mathlib.Meta.NormNum.isRat_le_true, Int.ofNat, Mathlib.Meta.NormNum.IsNNRat.to_isRat, Mathlib.Meta.NormNum.isNNRat_ofScientific_of_true, Mathlib.Meta.NormNum.isNNRat_nnratCast, NNRat.divNat, HPow.hPow, Mathlib.Meta.NormNum.isNNRat_divNat, Mathlib.Meta.NormNum.IsNat.raw_refl, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.IsNatPowT.run and 12 more
49 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem green_19.lower : C >= 3.13 := by
…
```
Mentions: Green19.C
theorem Green19.green_19.upper [Ma21] showed that $C \leq 4$ line 94 source
Source documentation
[Ma21] showed that $C \leq 4$. This also follows from green_19.

depends on sorry what this means
Statement uses: LE.le, Real, Green19.C, OfNat.ofNat, Nat
Proof uses: Eq.mpr, id, Eq, congrArg, Green19.green_19, le_refl
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_19.upper : C <= 4 := by
…
```
Mentions: Green19.C