2.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 2

Reference: erdosproblems.com/2

Erdős asked whether the smallest modulus in a distinct covering system can be arbitrarily large. Hough proved that the answer is no, and Balister, Bollobás, Morris, Sahasrabudhe, and Tiba later gave a simpler proof with an improved explicit upper bound.

Imports (1)

FormalConjecturesUtil

theorem Erdos2.erdos_2 Can the smallest modulus of a covering system be arbitrarily large? sorry line 41 source
Source documentation
Can the smallest modulus of a covering system be arbitrarily large?
This problem has a negative answer: there is a universal bound on the least modulus of any distinct covering system.

depends on sorry what this means
Statement uses: Iff, False, Nat, Exists, StrictCoveringSystem, Int, CoveringSystem.ι, StrictCoveringSystem.toCoveringSystem, And, Eq, Ideal, CoveringSystem.moduli, Ideal.span, Singleton.singleton, Set, LT.lt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_2 :
    answer(False) ↔
      ∀ B : ℕ, ∃ c : StrictCoveringSystem ℤ, ∀ i, ∃ m : ℕ,
        c.moduli i = Ideal.span {(m : ℤ)} ∧ B < m := by
…
```
Mentions: c, StrictCoveringSystem