FormalConjectures/ErdosProblems/41.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 41

Reference: erdosproblems.com/41

Imports (1)
Imported by 1 file
  1. def Erdos41.NtupleCondition NtupleCondition A n says that the sum of n elements of A determines the summands, counted with multiplicity and up to permutation line 40 source

    Source documentation

    NtupleCondition A n says that the sum of n elements of A determines the summands, counted with multiplicity and up to permutation.

    Multisets allow a summand to occur more than once, while multiset equality identifies precisely the trivial coincidences obtained by reordering the summands.

    standard axioms: propext what this means

    Statement uses: AddCommMonoid, Set, Nat

    Proof uses: Multiset, Membership.mem, Eq, Multiset.card, Multiset.sum

    2 instances, coercions and auxiliary constants left out.

    def NtupleCondition (A : Set α) (n : ℕ) : Prop :=
…
  2. theorem Erdos41.erdos_41 Let $A \subset \mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c \in A$ (aside from the trivial coincidences) sorry line 55 source

    Source documentation

    Let $A \subset \mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c \in A$ (aside from the trivial coincidences). Is it true that $$\liminf_{N \to \infty} \frac{\lvert A \cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?$$

    depends on sorry what this means

    Statement uses: Set, Nat, Erdos41.NtupleCondition, OfNat.ofNat, Set.Infinite, Eq, Real, Filter.liminf, HDiv.hDiv, Set.ncard, Inter.inter, Set.Icc, HPow.hPow, Filter.atTop

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

    40 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_41 (A : Set ℕ) (h_triple : NtupleCondition A 3) (h_infinite : A.Infinite) :
    Filter.atTop.liminf (fun N => (A ∩ Icc 1 N).ncard / (N : ℝ)^(1/3 : ℝ)) = 0 := by
…

    Mentions: Erdos41.NtupleCondition

  3. theorem Erdos41.erdos_41.variants.pairwise Erdős proved that if the pairwise sums sorry line 65 source

    Source documentation

    Erdős proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences, then $$\liminf_{N \to \infty} \frac{\lvert A \cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0.$$

    depends on sorry what this means

    Statement uses: Set, Nat, Erdos41.NtupleCondition, OfNat.ofNat, Set.Infinite, Eq, Real, Filter.liminf, HDiv.hDiv, Set.ncard, Inter.inter, Set.Icc, 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.

    theorem erdos_41.variants.pairwise (A : Set ℕ) (hA₂ : NtupleCondition A 2) (hA : A.Infinite) :
    Filter.atTop.liminf (fun N => (A ∩ Icc 1 N).ncard / (N : ℝ).sqrt) = 0 := by
…

    Mentions: Erdos41.NtupleCondition