481.lean`

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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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 481

References:

erdosproblems.com/481
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980), p.96.
[Kl82] Klarner, David A., A sufficient condition for certain semigroups to be free. J. Algebra (1982), 140-148.
[KoTa22] Kolpakov, Alexander and Talambutsa, Alexey, On free semigroups of affine maps on the real line. Proc. Amer. Math. Soc. (2022), 2301-2307.

Imports (1)

FormalConjecturesUtil

def Erdos481.T Documented declaration: T line 41 source
Source documentation
For a finite sequence of $n$ (not necessarily distinct) integers $A = (x_1,\ldots,x_n)$, the sequence T a b A of length $rn$ given by $(a_ix_j+b_i)_{1\leq j\leq n, 1\leq i\leq r}$.

no axioms what this means
Statement uses: Nat, Fin, List
Proof uses: List.flatMap, List.map, HAdd.hAdd, HMul.hMul, List.finRange
4 instances, coercions and auxiliary constants left out.
```
def T {r : ℕ} (a b : Fin r → ℕ) (A : List ℕ) : List ℕ :=
…
```
theorem Erdos481.erdos_481 Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$ sorry line 57 source
Source documentation
Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by $$(a_ix_j+b_i)_{1\leq j\leq n, 1\leq i\leq r}.$$ Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.
This is true. This appears to have first been shown by Klarner [Kl82], with a generalisation given by Kolpakov and Talambutsa [KoTa22]. Essentially the same proof was found independently by Barreto in the comment section.

depends on sorry what this means
Statement uses: Nat, Fin, LT.lt, OfNat.ofNat, Real, Finset.sum, Finset.univ, HDiv.hDiv, Exists, Not, List.Nodup, Nat.iterate, List, Erdos481.T, List.cons, List.nil
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_481 {r : ℕ} (a b : Fin r → ℕ) (ha : ∀ i, 0 < a i)
    (hab : 1 < ∑ i, (1 : ℝ) / a i) :
    ∃ k, ¬ ((T a b)^[k] [1]).Nodup := by
…
```
Mentions: Erdos481.T