253.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, 1 contain sorry, 0 declared as axioms

Erdős Problem 253

Reference: erdosproblems.com/253

Imports (1)

FormalConjecturesUtil

def Erdos253.RepresentsAPs The predicate that a : ℕ → ℕ is a strictly monotone sequence such that every infinite arithmetic progression contains infinitely many integers that are the sum of distinct line 35 source
Source documentation
The predicate that a : ℕ → ℕ is a strictly monotone sequence such that every infinite arithmetic progression contains infinitely many integers that are the sum of distinct $a_i$s.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, StrictMono, Set, Set.IsAPOfLength, Top.top, ENat, Set.Infinite, Inter.inter, subsetSums, Set.range
4 instances, coercions and auxiliary constants left out.
```
def RepresentsAPs (a : ℕ → ℕ) : Prop :=
…
```
theorem Erdos253.erdos_253 Let $a_1 < a_2 < \dotsc$ be an infinite sequence of positive integers such that $\frac{a_{i+1}}{a_i} \to 1$ sorry line 45 source
Source documentation
Let $a_1 < a_2 < \dotsc$ be an infinite sequence of positive integers such that $\frac{a_{i+1}}{a_i} \to 1$. If every arithmetic progression contains infinitely many integers which are the sum of distinct $a_i$ then every sufficiently large integer is the sum of distinct $a_i$.

depends on sorry what this means
Statement uses: Not, Nat, LT.lt, OfNat.ofNat, Erdos253.RepresentsAPs, Filter.Tendsto, Real, HDiv.hDiv, HAdd.hAdd, Filter.atTop, nhds, Membership.mem, Set, Filter, Filter.cofinite, subsetSums, Set.range
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_253 : ¬ ∀ a : ℕ → ℕ, 0 < a 0 →
    RepresentsAPs a → (Filter.atTop.Tendsto (fun n ↦ (a <| n + 1 : ℝ) / a n) (𝓝 1)) →
      subsetSums (Set.range a) ∈ Filter.cofinite := by
…
```
Mentions: Erdos253.RepresentsAPs, subsetSums