101.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

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 101

Reference: erdosproblems.com/101

Imports (1)

FormalConjecturesUtil

def Erdos101.linesWithPointsFor The set of lines in $\mathbb{R}^2$ containing exactly $k$ points from a given set $S$ line 37 source
Source documentation
The set of lines in $\mathbb{R}^2$ containing exactly $k$ points from a given set $S$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, AffineSubspace, ENNReal
Proof uses: Set.ofPred, Exists, And, Membership.mem, Ne, Eq, affineSpan, Insert.insert, Singleton.singleton, Set.ncard, Inter.inter
58 instances, coercions and auxiliary constants left out.
```
noncomputable def linesWithPointsFor (k : ℕ) (S : Set ℝ²) : Set (AffineSubspace ℝ ℝ²) :=
…
```
def Erdos101.numLinesWithFourPointMax The maximum number of lines containing exactly $4$ points among all sets $S$ of $n$ points in $\mathbb{R}^2$ satisfying the condition that no five points are collinear line 45 source
Source documentation
The maximum number of lines containing exactly $4$ points among all sets $S$ of $n$ points in $\mathbb{R}^2$ satisfying the condition that no five points are collinear.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Eq, Set.ncard, Set.Finite, EuclideanGeometry.NonCollinearFor, ENNReal, AffineSubspace, Erdos101.linesWithPointsFor
34 instances, coercions and auxiliary constants left out.
```
noncomputable def numLinesWithFourPointMax (n : ℕ) : ℕ :=
…
```
theorem Erdos101.erdos_101 Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$ sorry line 54 source
Source documentation
Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.

depends on sorry what this means
Statement uses: Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos101.numLinesWithFourPointMax, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_101 : (fun n => (numLinesWithFourPointMax n : ℝ)) =o[atTop] (fun n => (n : ℝ)^2) := by
…
```
Mentions: Erdos101.numLinesWithFourPointMax