881.lean`

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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 881

Reference: erdosproblems.com/881

Imports (1)

FormalConjecturesUtil

def Erdos881.IsMinimalAsymptoticAddBasisOfOrder We interpret "additive basis of order k" as an asymptotic additive basis of order k, using the predicate Set.IsAsymptoticAddBasisOfOrder from additive combinatorics line 41 source
Source documentation
We interpret "additive basis of order k" as an asymptotic additive basis of order k, using the predicate Set.IsAsymptoticAddBasisOfOrder from additive combinatorics.
A minimal additive basis of order k is a set A such that
- A is an asymptotic additive basis of order k, and
- for every infinite subset B ⊆ A, the complement A \ B is not an asymptotic additive basis of order k.
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: And, Set.IsAsymptoticAddBasisOfOrder, LE.le, Set.Infinite, Not, SDiff.sdiff
3 instances, coercions and auxiliary constants left out.
```
def IsMinimalAsymptoticAddBasisOfOrder (k : ℕ) (A : Set ℕ) : Prop :=
…
```
theorem Erdos881.erdos_881 Let A ⊂ ℕ be an additive basis of order k which is minimal in the sense that if B ⊂ A is any infinite set, then A \ B is not a basis of order k sorry line 53 source
Source documentation
Let A ⊂ ℕ be an additive basis of order k which is minimal in the sense that if B ⊂ A is any infinite set, then A \ B is not a basis of order k.
Must there exist an infinite B ⊂ A such that A \ B is an additive basis of order k + 1?

depends on sorry what this means
Statement uses: Iff, True, Nat, Set, Erdos881.IsMinimalAsymptoticAddBasisOfOrder, Exists, And, LE.le, Set.Infinite, Set.IsAsymptoticAddBasisOfOrder, SDiff.sdiff, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_881 :
    answer(sorry) ↔ ∀ (k : ℕ) (A : Set ℕ),
      IsMinimalAsymptoticAddBasisOfOrder k A →
        ∃ (B : Set ℕ), B ⊆ A ∧ B.Infinite ∧
          (A \ B).IsAsymptoticAddBasisOfOrder (k + 1) := by
…
```
Mentions: Erdos881.IsMinimalAsymptoticAddBasisOfOrder