95.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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 95

References:

erdosproblems.com/95
[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.
[Er97c] Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67.
[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.
[GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erdős distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.

Imports (1)

FormalConjecturesUtil

theorem Erdos95.erdos_95 sorry line 59 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, And, Finset, EuclideanSpace, Fin, Nat, LE.le, Finset.sum, distanceSet, ENNReal, HPow.hPow, distanceMultiplicity, HMul.hMul, Finset.card, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
56 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos95.erdos_95.variants.guth_katz Guth and Katz [GuKa15] proved the upper bound $\sum_i f(u_i)^2 \ll n^3\log n$ sorry line 66 source
Source documentation
Guth and Katz [GuKa15] proved the upper bound $\sum_i f(u_i)^2 \ll n^3\log n$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Finset, EuclideanSpace, Fin, Nat, LE.le, Finset.card, Finset.sum, distanceSet, ENNReal, HPow.hPow, distanceMultiplicity, HMul.hMul, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_95.variants.guth_katz : ∃ C : ℝ, 0 < C ∧ ∀ P : Finset ℝ², 2 ≤ P.card →
    ∑ u ∈ distanceSet P, (distanceMultiplicity P u : ℝ) ^ 2 ≤
      C * (P.card : ℝ) ^ 3 * Real.log P.card := by
…
```
Mentions: distanceSet, distanceMultiplicity