FormalConjectures/ErdosProblems/143.lean

GitHub source file

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 143

Reference: erdosproblems.com/143

Imports (1)
  1. def Erdos143.WellSeparatedSet Let $A \subseteq (1, \infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k \geq 1$ we have $|kx - y| \geq 1$ line 37 source

    Source documentation

    Let $A \subseteq (1, \infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k \geq 1$ we have $|kx - y| \geq 1$.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Set, Real

    Proof uses: And, LE.le, Set.Ioi, OfNat.ofNat, Set.Infinite, Set.Countable, Membership.mem, Ne, Nat, GE.ge, abs, HSub.hSub, HMul.hMul

    16 instances, coercions and auxiliary constants left out.

    def WellSeparatedSet (A : Set ℝ) : Prop :=
…
  2. theorem Erdos143.erdos_143.parts.i Does this imply that $$ \liminf \frac{|A \cap [1,x]|}{x} = 0? $$ sorry line 48 source

    Source documentation

    Does this imply that $$ \liminf \frac{|A \cap [1,x]|}{x} = 0? $$

    depends on sorry what this means

    Statement uses: Iff, True, Set, Real, Erdos143.WellSeparatedSet, Eq, Filter.liminf, HDiv.hDiv, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Filter.atTop

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

    27 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_143.parts.i : answer(sorry) ↔ ∀ (A : Set ℝ), WellSeparatedSet A →
    liminf (fun x => (A ∩ (Set.Icc 1 x)).ncard / x) atTop = 0 := by
…

    Mentions: Erdos143.WellSeparatedSet

  3. theorem Erdos143.erdos_143.parts.ii Or $$ \sum_{x \in A} \frac{1}{x \log x} < \infty, $$ sorry line 59 source

    Source documentation

    Or $$ \sum_{x \in A} \frac{1}{x \log x} < \infty, $$

    depends on sorry what this means

    Statement uses: Set, Real, Erdos143.WellSeparatedSet, Summable, HDiv.hDiv, OfNat.ofNat, HMul.hMul, Membership.mem, Real.log, SummationFilter.unconditional

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

    29 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_143.parts.ii (A : Set ℝ) (h : WellSeparatedSet A) :
    Summable fun (x : A) ↦ 1 / (x * Real.log x) := by
…

    Mentions: Erdos143.WellSeparatedSet