1125.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 1125

References:

erdosproblems.com/1125
[Er81b] Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35.
[Ke69] Kemperman, J. H. B., On the regularity of generalized convex functions. Trans. Amer. Math. Soc. (1969), 69-93.
[La84] Laczkovich, M., On Kemperman's inequality $2f(x)\leq f(x+h)+f(x+2h)$. Colloq. Math. (1984), 109-115.

Imports (1)

FormalConjecturesUtil

theorem Erdos1125.erdos_1125 Let $f:\mathbb{R}\to \mathbb{R}$ be such that $$2f(x) \leq f(x+h)+f(x+2h)$$ for every $x\in \mathbb{R}$ and $h>0$ sorry line 42 source
Source documentation
Let $f:\mathbb{R}\to \mathbb{R}$ be such that $$2f(x) \leq f(x+h)+f(x+2h)$$ for every $x\in \mathbb{R}$ and $h>0$. Must $f$ be monotonic?
A problem of Kemperman [Ke69], who proved it is true if $f$ is measurable. Erdős [Er81b] wrote 'if it were my problem I would offer $500 for it'. This was solved by Laczkovich [La84].

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, LE.le, HMul.hMul, Nat, HAdd.hAdd, Monotone
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1125 : answer(True) ↔ ∀ (f : ℝ → ℝ)
    (hf : ∀ x : ℝ, ∀ h : ℝ, h > 0 → 2 * f x ≤ f (x + h) + f (x + 2 * h)),
    Monotone f := by
…
```