330.lean`

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L4DEL: Lean4 Domain Explanation Language

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

4 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 330

Reference: erdosproblems.com/330

Imports (1)

FormalConjecturesUtil

def Erdos330.Rep Rep A m h means m is a sum of exactly h elements of A, which is m ∈ h • A line 45 source
Source documentation
Rep A m h means m is a sum of exactly h elements of A, which is m ∈ h • A.
Exactly h rather than at most h, for two reasons. It is the notion Erdős states the problem with, via f_2(m), the number of solutions of m = a_i + a_j [Er80]. And it is the notion IsAsymptoticAddBasisOfOrder uses below, so the statement is not switching between two meanings of "representable" partway through.
Allowing fewer than h summands also breaks the statement outright: taking one summand makes every element of A count as represented, so UnrepWithout A n h would contain no element of A besides possibly n. Those are exactly the integers the problem is about. If a ∈ A with a ≠ n is only expressible as a = n + b, it cannot be represented without n and belongs in the set, but the one-element sum a itself would have excluded it.

standard axioms: propext what this means
Statement uses: Set, Nat
Proof uses: Membership.mem, HSMul.hSMul, Set.NSMul
5 instances, coercions and auxiliary constants left out.
```
def Rep (A : Set ℕ) (m h : ℕ) : Prop := m ∈ h • A
```
def Erdos330.UnrepWithout Integers not representable as a sum of exactly h elements of A while avoiding n line 49 source
Source documentation
Integers not representable as a sum of exactly h elements of A while avoiding n.

standard axioms: propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Not, Erdos330.Rep, SDiff.sdiff, Singleton.singleton
2 instances, coercions and auxiliary constants left out.
```
def UnrepWithout (A : Set ℕ) (n h : ℕ) : Set ℕ :=
…
```
def Erdos330.MinAsymptoticAddBasisOfOrder An asymptotic additive basis of order h is minimal when one cannot obtain an asymptotic additive basis by removing any element from it line 54 source
Source documentation
An asymptotic additive basis of order h is minimal when one cannot obtain an asymptotic additive basis by removing any element from it.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: And, Set.IsAsymptoticAddBasisOfOrder, Membership.mem, Not, SDiff.sdiff, Singleton.singleton
4 instances, coercions and auxiliary constants left out.
```
def MinAsymptoticAddBasisOfOrder (A : Set ℕ) (h : ℕ) : Prop :=
…
```
theorem Erdos330.erdos_330_statement sorry line 73 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Erdos330.MinAsymptoticAddBasisOfOrder, LT.lt, Real, OfNat.ofNat, Set.upperDensity, Set.univ, Membership.mem, Erdos330.UnrepWithout
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.