28.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

3 declarations, 1 contain sorry, 0 declared as axioms

Green's Open Problem 28

References:

Green, Ben. "100 open problems." (2024).
Mathoverflow/339137 asked by user Sil
MathStackexchange/3325163 asked by user Emmanuel Amiot

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Mathoverflow/339137.lean

def Green28.IsUniformOnSupport True if a PMF on $\mathbb{Z}$ is uniformly distributed on its support line 34 source
Source documentation
True if a PMF on $\mathbb{Z}$ is uniformly distributed on its support.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PMF, Int
Proof uses: Exists, Finset, Finset.Nonempty, Eq, PMF.uniformOfFinset
```
def IsUniformOnSupport (X : PMF ℤ) : Prop :=
…
```
def Green28.indepSum The discrete convolution of two PMFs on $\mathbb{Z}$, representing the distribution of the sum of two independent random variables line 41 source
Source documentation
The discrete convolution of two PMFs on $\mathbb{Z}$, representing the distribution of the sum of two independent random variables.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PMF, Int
Proof uses: Bind.bind, PMF.pure, HAdd.hAdd
4 instances, coercions and auxiliary constants left out.
```
noncomputable def indepSum (X Y : PMF ℤ) : PMF ℤ := do
…
```
theorem Green28.green_28 Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values, and such that $X + Y$ is uniformly distributed on its range sorry line 52 source
Source documentation
Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values, and such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly distributed on their ranges?

depends on sorry what this means
Statement uses: Iff, True, PMF, Int, And, Set.Finite, PMF.support, Green28.IsUniformOnSupport, Green28.indepSum
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_28 : answer(sorry) ↔
  ∀ (X Y : PMF ℤ), -- marginals, independence is built into indepSum
    X.support.Finite ∧ Y.support.Finite ∧ IsUniformOnSupport (indepSum X Y) →
      IsUniformOnSupport X ∧ IsUniformOnSupport Y := by
…
```
Mentions: Green28.indepSum, Green28.IsUniformOnSupport