4.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 4

Reference: [Ben Green's Open Problem 4](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.4 Problem 4)

Imports (1)

FormalConjecturesUtil

def Green4.ProdFree A set in a monoid is product-free if there are no elements x, y, z in the set such that x * y = z line 32 source
Source documentation
A set in a monoid is product-free if there are no elements x, y, z in the set such that x * y = z.

no axioms what this means
Statement uses: Monoid, Set
Proof uses: Membership.mem, Not, HMul.hMul
5 instances, coercions and auxiliary constants left out.
```
def ProdFree {M : Type*} [Monoid M] (S : Set M) : Prop := ∀ x ∈ S, ∀ y ∈ S, x * y ∉ S
…
```
theorem Green4.green_4 What is the largest product-free set in the alternating group $A_n$? sorry line 36 source
Source documentation
What is the largest product-free set in the alternating group $A_n$?

depends on sorry what this means
Statement uses: Nat, Set, Subtype, Equiv.Perm, Fin, Membership.mem, Subgroup, alternatingGroup, sorryAx, Bool.false, MaximalFor, Green4.ProdFree, Set.ncard
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_4 (n : ℕ) :
    let S : ∀ n, Set (alternatingGroup <| Fin n) := answer(sorry)
…
```
def Green4.extremalFamily Defines a family of subsets of $A_n$ where each permutation $\pi$ in a subset obeys $\pi(x)$ and $\forall v \in I$, $\pi(v)\notin I$ for a fixed $x$ and $I$ line 46 source
Source documentation
Defines a family of subsets of $A_n$ where each permutation $\pi$ in a subset obeys $\pi(x)$ and $\forall v \in I$, $\pi(v)\notin I$ for a fixed $x$ and $I$. It is easy to demonstrate that such a subset is product-free, because for any a,b,c in such a set, $(a*b) (x)=a(b(x))\notin I$ but $c(x) in I$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Set, Subtype, Equiv.Perm, Membership.mem, Subgroup, alternatingGroup
Proof uses: Set.ofPred, And, Disjoint, Set.image
21 instances, coercions and auxiliary constants left out.
```
def extremalFamily {n : ℕ} (x : Fin n) (I : Set (Fin n)) : Set <| alternatingGroup <| Fin n :=
…
```
theorem Green4.large_green_4 In the case of large n, the problem was solved in On the largest product-free subsets of the alternating groups sorry line 54 source
Source documentation
In the case of large n, the problem was solved in On the largest product-free subsets of the alternating groups. Specifically, this theorem formalizes the statement of theorem 1.1 in the mentioned paper

L4DEL explanation
In the case of large n, the problem was solved in On the largest product-free subsets of the alternating groups. Specifically, this theorem formalizes the statement of theorem 1.1 in the mentioned paper
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Set, Subtype, Equiv.Perm, Fin, Membership.mem, Subgroup, alternatingGroup, MaximalFor, Green4.ProdFree, Set.ncard, Exists, Or, Eq, Green4.extremalFamily, Set.image, Inv.inv, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem large_green_4 : ∀ᶠ n in .atTop,
    ∀ S, MaximalFor (ProdFree (M := alternatingGroup <| Fin n)) Set.ncard S →
…
```
Mentions: Green4.ProdFree