FormalConjectures/ErdosProblems/440.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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 440

References:

Imports (1)
  1. def Erdos440.count For a strictly increasing sequence a of positive integers, count a x is the number of indices i with lcm(aᵢ, aᵢ₊₁) ≤ x line 39 source

    Source documentation

    For a strictly increasing sequence a of positive integers, count a x is the number of indices i with lcm(aᵢ, aᵢ₊₁) ≤ x.

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

    Statement uses: Nat

    Proof uses: Set.ncard, Set.ofPred, LE.le, Nat.lcm, HAdd.hAdd, OfNat.ofNat

    4 instances, coercions and auxiliary constants left out.

    noncomputable def count (a : ℕ → ℕ) (x : ℕ) : ℕ :=
…
  2. theorem Erdos440.erdos_440.parts.i sorry line 51 source

    depends on sorry what this means

    Statement uses: Iff, True, Nat, StrictMono, LT.lt, OfNat.ofNat, Asymptotics.IsBigO, Real, Filter.atTop, Erdos440.count, Real.sqrt

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

    12 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  3. theorem Erdos440.erdos_440.parts.ii sorry line 66 source

    depends on sorry what this means

    Statement uses: IsGreatest, Real, Set.ofPred, Exists, Nat, And, StrictMono, LT.lt, OfNat.ofNat, Eq, Filter.liminf, HDiv.hDiv, Erdos440.count, Real.sqrt, Filter.atTop

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

    26 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  4. def Erdos440.erdosSzemerediConstant The Erdős–Szemerédi constant $c=\sum_{n\geq 1}\frac{1}{n^{1/2}(n+1)}\approx 1.86$ (summed here over $n = m + 1$, $m \geq 0$) line 73 source

    Source documentation

    The Erdős–Szemerédi constant $c=\sum_{n\geq 1}\frac{1}{n^{1/2}(n+1)}\approx 1.86$ (summed here over $n = m + 1$, $m \geq 0$).

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

    Statement uses: Real

    Proof uses: tsum, Nat, HDiv.hDiv, OfNat.ofNat, HMul.hMul, Real.sqrt, HAdd.hAdd, SummationFilter.unconditional

    17 instances, coercions and auxiliary constants left out.

    noncomputable def erdosSzemerediConstant : ℝ := ∑' m : ℕ, 1 / (√(m + 1 : ℝ) * (m + 2))
…
  5. theorem Erdos440.erdos_440.variants.erdos_szemeredi sorry line 82 source

    depends on sorry what this means

    Statement uses: IsGreatest, Real, Set.ofPred, Exists, Nat, And, StrictMono, LT.lt, OfNat.ofNat, Eq, Filter.limsup, HDiv.hDiv, Erdos440.count, Real.sqrt, Filter.atTop, Erdos440.erdosSzemerediConstant

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

    22 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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