FormalConjectures/ErdosProblems/488.lean

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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

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 488

Reference: erdosproblems.com/488

Imports (1)
  1. theorem Erdos488.erdos_488 Documented declaration: erdos_488 sorry line 38 source

    Source documentation

    Let $A$ be a finite set and $$B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}.$$ Is it true that, for every $m>n\geq \max(A)$, $$\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?$$

    depends on sorry what this means

    Statement uses: Iff, True, Finset, Nat, Finset.Nonempty, Not, Membership.mem, OfNat.ofNat, GT.gt, LE.le, WithBot, Finset.max, LT.lt, Rat, HDiv.hDiv, Finset.card, Finset.filter, Set, Set.ofPred, And and 5 more

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    54 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_488 : answer(sorry) ↔ ∀ (A : Finset ℕ), A.Nonempty →
    -- These are needed for the reasons outlined here: https://github.com/google-deepmind/formal-conjectures/pull/256
    0 ∉ A → 1 ∉ A →
    letI B := {n ≥ 1 | ∃ a ∈ A, a ∣ n}
…