1179.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

5 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1179

References:

erdosproblems.com/1179
[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
[ErRe65] Erdős, P. and Rényi, A., Probabilistic methods in group theory. J. Analyse Math. (1965), 127-138.
[ErHa76] Erdős, P. and Hall, R. R., Probabilistic methods in group theory. II. Houston J. Math. (1976), 173-180.

Imports (1)

FormalConjecturesUtil

def Erdos1179.repCount $F_A(g) = \#\{ S\subseteq A : g = \sum_{x\in S}x\}$ line 44 source
Source documentation
$F_A(g) = \#\{ S\subseteq A : g = \sum_{x\in S}x\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Finset, Nat
Proof uses: Finset.card, Finset.filter, Eq, Finset.sum, Classical.propDecidable, Finset.powerset
1 instances, coercions and auxiliary constants left out.
```
noncomputable def repCount (A : Finset G) (g : G) : ℕ :=
…
```
def Erdos1179.IsBalanced A is ε-balanced: $\lvert F_A(g)-2^{|A|}/N\rvert \leq \epsilon 2^{|A|}/N$ for all g line 48 source
Source documentation
A is ε-balanced: $\lvert F_A(g)-2^{|A|}/N\rvert \leq \epsilon 2^{|A|}/N$ for all g.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Fintype, Real, Finset
Proof uses: LE.le, abs, HSub.hSub, Erdos1179.repCount, HDiv.hDiv, HPow.hPow, Nat, OfNat.ofNat, Finset.card, Fintype.card, HMul.hMul
18 instances, coercions and auxiliary constants left out.
```
def IsBalanced (ε : ℝ) (A : Finset G) : Prop :=
…
```
def Erdos1179.successProbability The probability that a uniformly random k-element subset of G is ε-balanced line 54 source
Source documentation
The probability that a uniformly random k-element subset of G is ε-balanced.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Fintype, Real, Nat
Proof uses: HDiv.hDiv, Finset.card, Finset, Finset.filter, Erdos1179.IsBalanced, Classical.propDecidable, Finset.powersetCard, Finset.univ
5 instances, coercions and auxiliary constants left out.
```
noncomputable def successProbability (G : Type*) [AddCommGroup G] [Fintype G] (ε : ℝ) (k : ℕ) :
    ℝ :=
…
```
theorem Erdos1179.erdos_1179 sorry line 85 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, Nat, And, Filter.Tendsto, HDiv.hDiv, Real.logb, Filter.atTop, nhds, AddCommGroup, Fintype, Fintype.card, Erdos1179.successProbability
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1179.erdos_1179.variants.lower_bound sorry line 97 source
depends on sorry what this means
Statement uses: AddCommGroup, Fintype, Real, LT.lt, OfNat.ofNat, Finset, Erdos1179.IsBalanced, LE.le, Nat, Fintype.card, HPow.hPow, Finset.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

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