14.lean`

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5 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 14

Reference: erdosproblems.com/14

Imports (1)

FormalConjecturesUtil

def Erdos14.nonUniqueSumCount Documented declaration: nonUniqueSumCount line 39 source
Source documentation
The number of integers in $\{1,\ldots,N\}$ which are not representable in exactly one way as the sum of two elements from $A$ (either because they are not representable at all, or because they are representable in more than one way).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Real
Proof uses: Set.ncard, SDiff.sdiff, Set.Icc, OfNat.ofNat, allUniqueSums
6 instances, coercions and auxiliary constants left out.
```
noncomputable def nonUniqueSumCount (A : Set ℕ) (N : ℕ) : ℝ :=
…
```
def Erdos14.almostSquareRoot line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: HPow.hPow, HSub.hSub, HDiv.hDiv, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.
def Erdos14.squareRoot line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Real.sqrt
2 instances, coercions and auxiliary constants left out.
theorem Erdos14.erdos_14.parts.i Let $A ⊆ \mathbb{N}$ sorry line 54 source
Source documentation
Let $A ⊆ \mathbb{N}$. Let $B ⊆ \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$. Is it true that for all $\epsilon > 0$ and large $N$, $|\{1,\ldots,N\} \setminus B| \gg_\epsilon N^{1/2 - \epsilon}$?

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Real, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Filter.atTop, Erdos14.almostSquareRoot, Erdos14.nonUniqueSumCount
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_14.parts.i :
    answer(sorry) ↔ ∀ A, ∀ ε > 0, nonUniqueSumCount A ≫ almostSquareRoot ε := by sorry
…
```
Mentions: Erdos14.nonUniqueSumCount, Erdos14.almostSquareRoot
theorem Erdos14.erdos_14.parts.ii Is it possible that $|\{1,\ldots,N\} \setminus B| = o(N^\frac{1}{2})$? sorry line 61 source
Source documentation
Is it possible that $|\{1,\ldots,N\} \setminus B| = o(N^\frac{1}{2})$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, Asymptotics.IsLittleO, Real, Filter.atTop, Erdos14.nonUniqueSumCount, Erdos14.squareRoot
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_14.parts.ii :
    answer(sorry) ↔ ∃ (A : Set ℕ), IsLittleO atTop (nonUniqueSumCount A) squareRoot := by
…
```
Mentions: Erdos14.nonUniqueSumCount, Erdos14.squareRoot