326.lean`

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 326

References:

erdosproblems.com/326
[ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980), p. 47.
[Ca57] Cassels, J. W. S., Über Basen der natürlichen Zahlenreihe. Abh. Math. Sem. Univ. Hamburg (1957), 247-257.

Imports (1)

FormalConjecturesUtil

theorem Erdos326.erdos_326 Does there exist sorry line 52 source
Source documentation
Does there exist $A = \{a_1 < a_2 < \cdots\} \subset \mathbb{N}$ which is a minimal basis of order $2$ (i.e. every large integer is the sum of $2$ elements from $A$, and no proper subset of $A$ has this property), such that $$\lim_{k\to\infty} \frac{a_k}{k^2} = c$$ for some $c \neq 0$?
Erdős and Graham conjectured a negative answer to this question [ErGr80].
"Minimal basis of order $2$" is formalised as Minimal for the predicate Set.IsAsymptoticAddBasisOfOrder · 2 on sets of naturals ordered by inclusion.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, StrictMono, Minimal, Set, Set.IsAsymptoticAddBasisOfOrder, OfNat.ofNat, Set.range, Real, Ne, Filter.Tendsto, HDiv.hDiv, HPow.hPow, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_326 : answer(sorry) ↔ ∃ (a : ℕ → ℕ), StrictMono a ∧
    Minimal (fun A : Set ℕ ↦ A.IsAsymptoticAddBasisOfOrder 2) (Set.range a) ∧
      ∃ (c : ℝ), c ≠ 0 ∧ Tendsto (fun n ↦ (a n : ℝ) / n ^ 2) atTop (𝓝 c) := by
…
```
Mentions: c
theorem Erdos326.erdos_326.variants.eq Erdős originally asked this for any basis (not necessarily minimal); such a basis was constructed by Cassels [Ca57] sorry line 64 source
Source documentation
Erdős originally asked this for any basis (not necessarily minimal); such a basis was constructed by Cassels [Ca57].

depends on sorry what this means
Statement uses: Exists, Nat, StrictMono, Set.IsAddBasisOfOrder, Set.range, OfNat.ofNat, Real, LT.lt, Filter.Tendsto, HDiv.hDiv, HPow.hPow, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_326.variants.eq :
    ∃ (a : ℕ → ℕ) (_ : StrictMono a) (_ : Set.range a |>.IsAddBasisOfOrder 2) (x : ℝ) (_ : 0 < x),
      Tendsto (fun n ↦ (a n : ℝ) / n ^ 2) atTop (𝓝 x) := by
…
```