93.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 93

References:

erdosproblems.com/93
[Er46b] Erdős, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250.
[Er57] Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300.
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
[Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.
[Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
[Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
[Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
[Er97f] Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.
[Al63] Altman, E., On a problem of P. Erdős. Amer. Math. Monthly (1963), 148-157.

Imports (1)

FormalConjecturesUtil

theorem Erdos93.erdos_93 If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor \frac{n}{2}\rfloor$ distinct distances sorry line 60 source
Source documentation
If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor \frac{n}{2}\rfloor$ distinct distances.
Solved by Altman [Al63].

depends on sorry what this means
Statement uses: Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, EuclideanGeometry.ConvexIndep, LE.le, HDiv.hDiv, Finset.card, distinctDistances, ENNReal
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_93 (A : Finset ℝ²) (hA : ConvexIndep (A : Set ℝ²)) :
    A.card / 2 ≤ distinctDistances A := by
…
```
Mentions: distinctDistances