1062.lean`

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5 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1062

Reference: erdosproblems.com/1062

Imports (1)

FormalConjecturesUtil

def Erdos1062.ForkFree A set A of positive integers is fork-free if no element divides two distinct other elements of A line 35 source
Source documentation
A set A of positive integers is fork-free if no element divides two distinct other elements of A.

no axioms what this means
Statement uses: Set, Nat
Proof uses: Membership.mem, Set.Subsingleton, Set.ofPred, And, SDiff.sdiff, Singleton.singleton, Dvd.dvd
4 instances, coercions and auxiliary constants left out.
```
def ForkFree (A : Set ℕ) : Prop :=
…
```
def Erdos1062.f The extremal function from Erdős problem 1062: the largest size of a fork-free subset of {1,...,n} line 40 source
Source documentation
The extremal function from Erdős problem 1062: the largest size of a fork-free subset of {1,...,n}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.findGreatest, Exists, Set, And, LE.le, Set.Icc, OfNat.ofNat, Erdos1062.ForkFree, Eq, Set.ncard, Classical.propDecidable
3 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ :=
…
```
theorem Erdos1062.erdos_1062.parts.ii Erdős asked whether the limiting density f n / n exists and, if so, whether it is irrational sorry line 49 source
Source documentation
Erdős asked whether the limiting density f n / n exists and, if so, whether it is irrational.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, Filter.Tendsto, Nat, HDiv.hDiv, Erdos1062.f, Filter.atTop, nhds, Irrational
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1062.parts.ii :
    answer(sorry) ↔ ∃ l, Tendsto (fun n => (f n : ℝ) / n) atTop (𝓝 l) ∧ Irrational l := by
…
```
Mentions: Erdos1062.f
theorem Erdos1062.erdos_1062.variants.lower_bound The interval [⌊n/3⌋, n] is fork-free, and therefore f n is at least ⌈2n / 3⌉ line 55 source
Source documentation
The interval [⌊n/3⌋, n] is fork-free, and therefore f n is at least ⌈2n / 3⌉.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, Nat.ceil, Real, HDiv.hDiv, HMul.hMul, OfNat.ofNat, Erdos1062.f
Proof uses: Eq, rfl, Finset, Finset.Icc, HAdd.hAdd, Trans.trans, HSub.hSub, Eq.mpr, congrArg, propext, Nat.ceil_le, Nat.cast_sub, Decidable.byContradiction, Not, le_sub_iff_add_le, le_imp_le_of_le_of_le, add_le_add, le_refl, Nat.cast_div_le, le_of_eq and 88 more
131 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1062.variants.lower_bound (n : ℕ) : ⌈(2 * n / 3 : ℝ)⌉₊ ≤ f n := by
…
```
Mentions: Erdos1062.f
theorem Erdos1062.erdos_1062.variants.lebensold_bounds Lebensold proved that for large n, the function f n lies between 0.6725 n and 0.6736 n sorry line 82 source
Source documentation
Lebensold proved that for large n, the function f n lies between 0.6725 n and 0.6736 n.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, And, LE.le, Real, HMul.hMul, OfScientific.ofScientific, Bool.true, Erdos1062.f, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1062.variants.lebensold_bounds :
    ∀ᶠ n in atTop, (0.6725 : ℝ) * n ≤ f n ∧ f n ≤ (0.6736 : ℝ) * n := by
…
```
Mentions: Erdos1062.f