153.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, 1 contain sorry, 0 declared as axioms

Erdős Problem 153

#TODO: Formalize the corresponding conjecture for infinite Sidon sets.

References:

erdosproblems.com/153
[ESS94] Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number Theory (1994), 329-347.

Imports (1)

FormalConjecturesUtil

def Erdos153.f Define $f(n)$ to be the minimum of $\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2$ as $A$ ranges over all Sidon sets of size $n$, where $A+A=\{s_1<\cdots<s_t\}$ line 41 source
Source documentation
Define $f(n)$ to be the minimum of $\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2$ as $A$ ranges over all Sidon sets of size $n$, where $A+A=\{s_1<\cdots<s_t\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: iInf, Finset, Set.ofPred, And, Eq, Finset.card, IsSidon, OrderIso, Fin, HAdd.hAdd, Membership.mem, Set, Subtype, Finset.orderIsoOfFin, HDiv.hDiv, Finset.sum, Set.Ico, OfNat.ofNat, Finset.univ, HPow.hPow and 2 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
noncomputable def f (n : ℕ) : ℝ := ⨅ A : {A : Finset ℕ | A.card = n ∧ IsSidon (A : Set ℕ)},
…
```
Mentions: IsSidon
theorem Erdos153.erdos_153 Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$ sorry line 49 source
Source documentation
Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that $$\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty$$ as $\lvert A\rvert\to \infty$?

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, Real, Erdos153.f, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_153 : answer(sorry) ↔ Tendsto f atTop atTop := by
…
```
Mentions: Erdos153.f