99.lean`

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L4DEL: Lean4 Domain Explanation Language

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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

Erdős Problem 99

References:

erdosproblems.com/99
[BeFo99] Bezdek, Andr'{a}s and Fodor, Ferenc, Minimal diameter of certain sets in the plane. J. Combin. Theory Ser. A (1999), 105-111.
[Er94b] Erd\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.

Imports (1)

FormalConjecturesUtil

def Erdos99.HasMinDist1 A set has minimum distance $1$ if all pairwise distances are at least $1$, and the minimum is achieved line 37 source
Source documentation
A set has minimum distance $1$ if all pairwise distances are at least $1$, and the minimum is achieved.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: And, Membership.mem, Ne, GE.ge, Dist.dist, ENNReal, Exists, Eq
16 instances, coercions and auxiliary constants left out.
```
def HasMinDist1 (A : Finset ℝ²) : Prop :=
…
```
def Erdos99.FormsEquilateralTriangle Three points form an equilateral triangle of side length 1 line 42 source
Source documentation
Three points form an equilateral triangle of side length 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: And, Eq, Dist.dist, ENNReal
13 instances, coercions and auxiliary constants left out.
```
def FormsEquilateralTriangle (p q r : ℝ²) : Prop :=
…
```
theorem Erdos99.erdos_99 For sufficiently large n, is it the case that any set of n points with minimum distance $1$ that minimizes diameter must contain an equilateral triangle of side length 1? sorry line 48 source
Source documentation
For sufficiently large n, is it the case that any set of n points with minimum distance $1$ that minimizes diameter must contain an equilateral triangle of side length 1?

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Eq, Finset.card, Erdos99.HasMinDist1, IsMinOn, Metric.diam, ENNReal, Set.ofPred, And, Exists, Membership.mem, Erdos99.FormsEquilateralTriangle and 1 more
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_99 :
    answer(sorry) ↔ ∀ᶠ n in Filter.atTop, ∀ A : Finset ℝ²,
      A.card = n → HasMinDist1 A →
      (IsMinOn (fun B: Finset ℝ² => diam (B : Set ℝ²)) {B : Finset ℝ² | B.card = n ∧ HasMinDist1 B} A) →
      ∃ᵉ (p ∈ A) (q ∈ A) (r ∈ A), FormsEquilateralTriangle p q r := by
…
```
Mentions: Erdos99.HasMinDist1, Erdos99.FormsEquilateralTriangle