1089.lean`

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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, 2 contain sorry, 0 declared as axioms

Erdős Problem 1089

References:

erdosproblems.com/1089
[Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
[BBS83] Bannai, Eiichi and Bannai, Etsuko and Stanton, Dennis, An upper bound for the cardinality of an $s$-distance subset in real Euclidean space. II. Combinatorica (1983), 147-152.

Imports (1)

FormalConjecturesUtil

def Erdos1089.g Documented declaration: g line 42 source
Source documentation
g d n is the least m such that every m points in ℝ^d determine at least n distinct distances, i.e. the least m with n ≤ minimalDistinctDistances (EuclideanSpace ℝ (Fin d)) m.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, LE.le, minimalDistinctDistances, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal
11 instances, coercions and auxiliary constants left out.
```
noncomputable def g (d n : ℕ) : ℕ :=
…
```
theorem Erdos1089.erdos_1089 sorry line 59 source
depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Real, Filter.Tendsto, HDiv.hDiv, Erdos1089.g, HPow.hPow, HSub.hSub, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1089.erdos_1089.variants.bounds_and_limit sorry line 70 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, And, HAdd.hAdd, Nat.choose, HSub.hSub, Erdos1089.g, Filter.Tendsto, Real, HDiv.hDiv, HPow.hPow, Filter.atTop, nhds, Nat.factorial
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

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