741.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, 4 contain sorry, 0 declared as axioms

Erdős Problem 741

References:

erdosproblems.com/741
[Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.

Imports (1)

FormalConjecturesUtil

theorem Erdos741.erdos_741.parts.i sorry line 45 source
depends on sorry what this means
Statement uses: Iff, True, Set, Nat, LT.lt, Real, OfNat.ofNat, Set.upperDensity, HAdd.hAdd, Set.univ, Exists, And, Eq, Union.union, Disjoint
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
35 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos741.erdos_741.variants.lower Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive lower density sorry line 56 source
Source documentation
Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive lower density. Can one always decompose $A=A_1\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive lower density?

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, LT.lt, Real, OfNat.ofNat, Set.lowerDensity, HAdd.hAdd, Set.univ, Exists, And, Eq, Union.union, Disjoint
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
35 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_741.variants.lower : answer(sorry) ↔ ∀ A : Set ℕ, 0 < lowerDensity (A + A) → ∃ A₁ A₂,
    A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ 0 < lowerDensity (A₁ + A₁)
    ∧ 0 < lowerDensity (A₂ + A₂) := by
…
```
theorem Erdos741.erdos_741.variants.exact_density sorry line 71 source
depends on sorry what this means
Statement uses: Iff, False, Set, Nat, Set.HasPosDensity, HAdd.hAdd, Set.univ, Exists, And, Eq, Union.union, Disjoint
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos741.erdos_741.parts.ii sorry line 85 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Set.IsAddBasisOfOrder, Union.union, Singleton.singleton, OfNat.ofNat, Eq, Disjoint, Not, IsSyndetic, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.