898.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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 898

References:

erdosproblems.com/898
[Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.
Wikipedia

Imports (1)

FormalConjecturesUtil

theorem Erdos898.erdos_898 Documented declaration: erdos_898 sorry line 47 source
Source documentation
If $A,B,C\in \mathbb{R}^2$ form a triangle and $P$ is a point in the interior then, if $N$ is where the perpendicular from $P$ to $AB$ meets the triangle, and similarly for $M$ and $L$, $$ \overline{PA}+\overline{PB}+\overline{PC}\geq 2(\overline{PM}+\overline{PN}+\overline{PL}). $$
Conjectured by Erdős in 1932 (according to [Er82e]) and proved by Mordell soon afterwards, now known as the Erdős-Mordell inequality.

depends on sorry what this means
Statement uses: EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, AffineIndependent, ENNReal, Nat.succ, Matrix.vecCons, Matrix.vecEmpty, Membership.mem, Set, interior, ClosureOperator, convexHull, Insert.insert, Singleton.singleton, AffineSubspace, affineSpan, Submodule.IsOrtho and 5 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
122 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_898 (A B C P L M N : ℝ²) (hABC : AffineIndependent ℝ ![A, B, C])
    (hP : P ∈ interior (convexHull ℝ ({A, B, C} : Set ℝ²)))
    (hN : N ∈ line[ℝ, A, B]) (hPN : line[ℝ, P, N].direction ⟂ line[ℝ, A, B].direction)
    (hM : M ∈ line[ℝ, B, C]) (hPM : line[ℝ, P, M].direction ⟂ line[ℝ, B, C].direction)
    (hL : L ∈ line[ℝ, C, A]) (hPL : line[ℝ, P, L].direction ⟂ line[ℝ, C, A].direction) :
    dist P A + dist P B + dist P C ≥ 2 * (dist P M + dist P N + dist P L) := by
…
```