347.lean`

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L4DEL: Lean4 Domain Explanation Language

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 347

Reference: erdosproblems.com/347

Imports (1)

FormalConjecturesUtil

theorem Erdos347.erdos_347 Is there a sequence sorry line 50 source
Source documentation
Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with $$\lim \frac{a_{n+1}}{a_n}=2$$ such that $$P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}$$ has density $1$ for every cofinite subsequence $A'$ of $A$?
This has been solved in the affirmative by ebarschkis in the comments (based on idea of Tao and van Doorn, also in the comments).
This was formalized in Lean by Barschkis using Aristotle.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, Monotone, Filter.Tendsto, Real, HDiv.hDiv, HAdd.hAdd, OfNat.ofNat, Filter.atTop, nhds, Set.Finite, Compl.compl, Set, Set.range, Set.HasDensity, subsetSums, Function.comp and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_347 :
    answer(True) ↔ ∃ a : ℕ → ℕ, (Monotone a) ∧
      (Tendsto (fun n ↦ (a (n + 1) : ℝ) / (a n : ℝ)) atTop (𝓝 2)) ∧
      (∀ ι : ℕ → ℕ, (range ι)ᶜ.Finite → HasDensity (𝓟 (range (a ∘ ι))) 1) := by
…
```