131.lean`

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

11 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 131

References:

erdosproblems.com/131
[ELRSS99] Erdős, P. and Lev, V. and Rauzy, G. and Sándor, C. and Sárközy, A., Greedy algorithm, arithmetic progressions, subset sums and divisibility. Discrete Math. (1999), 119-135.
[PhZa24] Pham, H. T. and Zakharov, D., Sharp bound for the Erdős–Straus non-averaging set problem. arXiv:2410.14624 (2024).

Imports (1)

FormalConjecturesUtil

def Erdos131.NonDividing A finite set $A$ of naturals is non-dividing if no $a \in A$ divides the sum of the elements of any nonempty subset $S \subseteq A \setminus \{a\}$ line 40 source
Source documentation
A finite set $A$ of naturals is non-dividing if no $a \in A$ divides the sum of the elements of any nonempty subset $S \subseteq A \setminus \{a\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Membership.mem, LE.le, Finset.erase, Finset.Nonempty, Not, Dvd.dvd, Finset.sum
8 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def NonDividing (A : Finset ℕ) : Prop :=
…
```
instance Erdos131.(instance at line 43) line 43 source
def Erdos131.F $F(N)$ is the maximal size of a non-dividing subset of $\{1, \dots, N\}$ line 47 source
Source documentation
$F(N)$ is the maximal size of a non-dividing subset of $\{1, \dots, N\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sup, Finset, Finset.filter, Erdos131.NonDividing, Finset.powerset, Finset.Icc, OfNat.ofNat, Finset.card
7 instances, coercions and auxiliary constants left out.
```
def F (N : ℕ) : ℕ :=
…
```
theorem Erdos131.exists_card_eq_F $F(N)$ is attained: some non-dividing $A \subseteq \{1, \dots, N\}$ has exactly $F(N)$ elements line 53 source
Source documentation
$F(N)$ is attained: some non-dividing $A \subseteq \{1, \dots, N\}$ has exactly $F(N)$ elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Erdos131.NonDividing, Eq, Finset.card, Erdos131.F
Proof uses: Membership.mem, Finset.filter, Finset.powerset, Finset.sup, Finset.exists_mem_eq_sup, Exists.intro, EmptyCollection.emptyCollection, of_eq_true, Eq.trans, True, congr, congrArg, Finset.erase, Finset.Nonempty, Not, Dvd.dvd, Finset.sum, forall_congr, False, implies_congr and 12 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem exists_card_eq_F (N : ℕ) :
    ∃ A ⊆ Finset.Icc 1 N, NonDividing A ∧ A.card = F N := by
…
```
Mentions: Erdos131.NonDividing, Erdos131.F
theorem Erdos131.card_le_F Every non-dividing $A \subseteq \{1, \dots, N\}$ has at most $F(N)$ elements line 63 source
Source documentation
Every non-dividing $A \subseteq \{1, \dots, N\}$ has at most $F(N)$ elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, LE.le, Finset.Icc, OfNat.ofNat, Erdos131.NonDividing, Finset.card, Erdos131.F
Proof uses: Finset.le_sup, Finset.filter, Finset.powerset, of_eq_true, Membership.mem, Eq.trans, And, True, congr, congrArg, eq_true, and_self
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem card_le_F {N : ℕ} {A : Finset ℕ} (hA : A ⊆ Finset.Icc 1 N) (hnd : NonDividing A) :
    A.card ≤ F N :=
…
```
Mentions: Erdos131.NonDividing, Erdos131.F
theorem Erdos131.F_le Trivial upper bound: $F(N) \le N$ line 69 source
Source documentation
Trivial upper bound: $F(N) \le N$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, Erdos131.F
Proof uses: Finset, And, Finset.Icc, OfNat.ofNat, Erdos131.NonDividing, Eq, Finset.card, Exists, Erdos131.exists_card_eq_F, Eq.mpr, id, congrFun', congrArg, Eq.symm, Eq.mp, Eq.trans, HSub.hSub, HAdd.hAdd, Nat.card_Icc, add_tsub_cancel_right and 1 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem F_le (N : ℕ) : F N ≤ N := by
…
```
Mentions: Erdos131.F
theorem Erdos131.F_mono $F$ is monotone line 75 source
Source documentation
$F$ is monotone.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Monotone, Nat, Erdos131.F
Proof uses: LE.le, Exists, Finset, And, Finset.Icc, OfNat.ofNat, Erdos131.NonDividing, Eq, Finset.card, Erdos131.exists_card_eq_F, Eq.rec, Erdos131.card_le_F, LE.le.trans, Finset.Icc_subset_Icc_right
8 instances, coercions and auxiliary constants left out.
```
theorem F_mono : Monotone F := fun _ _ hMN =>
…
```
Mentions: Erdos131.F
theorem Erdos131.one_le_F Trivial lower bound: for $N \ge 1$, the singleton $\{N\}$ is non-dividing, so $1 \le F(N)$ line 82 source
Source documentation
Trivial lower bound: for $N \ge 1$, the singleton $\{N\}$ is non-dividing, so $1 \le F(N)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Erdos131.F
Proof uses: Finset.card, Singleton.singleton, Finset, Erdos131.card_le_F, of_eq_true, Finset.Icc, Eq.trans, Membership.mem, True, And, congr, congrArg, eq_true, and_self, Finset.erase, Finset.Nonempty, Eq, EmptyCollection.emptyCollection, Eq.mp, Iff.mp and 11 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem one_le_F {N : ℕ} (hN : 1 ≤ N) : 1 ≤ F N := by
…
```
Mentions: Erdos131.F
theorem Erdos131.erdos_131.variants.pham_zakharov Every non-dividing set is non-averaging sorry line 93 source
Source documentation
Every non-dividing set is non-averaging. Pham and Zakharov [PhZa24] proved that a non-averaging subset of $\{1, \dots, N\}$ has size at most $N^{1/4 + o(1)}$, so $F(N) \le N^{1/4 + o(1)}$.

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, Filter.Eventually, LE.le, Erdos131.F, HPow.hPow, HAdd.hAdd, HDiv.hDiv
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_131.variants.pham_zakharov :
    ∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧
      ∀ᶠ N : ℕ in atTop, (F N : ℝ) ≤ (N : ℝ) ^ ((1 : ℝ) / 4 + f N) := by
…
```
Mentions: Erdos131.F
theorem Erdos131.erdos_131.variants.elrss Erdős, Lev, Rauzy, Sándor and Sárközy [ELRSS99] proved $F(N) < 3 N^{1/2} + 1$ sorry line 100 source
Source documentation
Erdős, Lev, Rauzy, Sándor and Sárközy [ELRSS99] proved $F(N) < 3 N^{1/2} + 1$.

depends on sorry what this means
Statement uses: Nat, LT.lt, Real, Erdos131.F, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Real.sqrt
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_131.variants.elrss (N : ℕ) : (F N : ℝ) < 3 * Real.sqrt N + 1 := by
…
```
Mentions: Erdos131.F
theorem Erdos131.erdos_131 Let $F(N)$ be the maximal size of $A \subseteq \{1, \dots, N\}$ such that no $a \in A$ divides the sum of any distinct elements of $A \setminus \{a\}$ line 112 source
Source documentation
Let $F(N)$ be the maximal size of $A \subseteq \{1, \dots, N\}$ such that no $a \in A$ divides the sum of any distinct elements of $A \setminus \{a\}$. Is it true that $F(N) > N^{1/2 - o(1)}$?
The answer is no, by the bound of Pham and Zakharov [PhZa24] (erdos_131.variants.pham_zakharov).

depends on sorry what this means
Statement uses: Iff, False, Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, Filter.Eventually, LT.lt, HPow.hPow, HSub.hSub, HDiv.hDiv, Erdos131.F
Proof uses: id, Iff.intro, False.elim, LE.le, HAdd.hAdd, Erdos131.erdos_131.variants.pham_zakharov, Filter.Tendsto.eventually, gt_mem_nhds, Mathlib.Meta.NormNum.isNNRat_lt_true, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Mathlib.Meta.NormNum.isNNRat_div, Mathlib.Meta.NormNum.isNNRat_mul, HMul.hMul, Inv.inv, Eq.refl, Nat.cast_one, Mathlib.Meta.NormNum.isNNRat_inv_pos, Mathlib.Meta.NormNum.instAtLeastTwo and 79 more
132 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_131 : answer(False) ↔
    ∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧
      ∀ᶠ N : ℕ in atTop, (N : ℝ) ^ ((1 : ℝ) / 2 - f N) < (F N : ℝ) := by
…
```
Mentions: Erdos131.F