53.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 53

References:

erdosproblems.com/53
[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[ErSz83] Erdős, P. and Szemerédi, E., On sums and products of integers. Studies in pure mathematics (1983), 213-218.
[Er91] Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.
[Er97] Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.
[Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
[Ch03] Chang, M.-C., The Erdős-Szemerédi problem on sum set and product set. Annals of Math. (2003), 939-957.

Imports (1)

FormalConjecturesUtil

def Erdos53.sumsAndProducts The integers which are either the sum or the product of (one or more) distinct elements of A line 46 source
Source documentation
The integers which are either the sum or the product of (one or more) distinct elements of A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Int
Proof uses: Union.union, Finset.image, Finset.sum, Finset.erase, Finset.powerset, EmptyCollection.emptyCollection, Finset.prod
6 instances, coercions and auxiliary constants left out.
```
def sumsAndProducts (A : Finset ℤ) : Finset ℤ :=
…
```
theorem Erdos53.erdos_53 sorry line 67 source
depends on sorry what this means
Statement uses: Iff, True, Nat, Exists, Finset, Int, LE.le, Finset.card, HPow.hPow, Erdos53.sumsAndProducts
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos53.erdos_53.variants.upper Erdős and Szemerédi [ErSz83] proved that there exist arbitrarily large sets sorry line 79 source
Source documentation
Erdős and Szemerédi [ErSz83] proved that there exist arbitrarily large sets $A$ such that the number of integers which are the sum or product of distinct elements of $A$ is at most $$\exp\left(c \frac{(\log \lvert A\rvert)^2}{\log\log\lvert A\rvert}\right)$$ for some constant $c>0$. (erdosproblems.com multiplies by $\log\log\lvert A\rvert$; [ErSz83, Theorem 2] divides by it.)

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, Finset, Int, LE.le, Finset.card, Erdos53.sumsAndProducts, Real.exp, HDiv.hDiv, HMul.hMul, HPow.hPow, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_53.variants.upper : ∃ c : ℝ, 0 < c ∧ ∀ N : ℕ, ∃ A : Finset ℤ, N ≤ A.card ∧
    ((sumsAndProducts A).card : ℝ) ≤
      Real.exp (c * Real.log A.card ^ 2 / Real.log (Real.log A.card)) := by
…
```
Mentions: c, Erdos53.sumsAndProducts