949.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 949

Reference: erdosproblems.com/949

Imports (1)

FormalConjecturesUtil

theorem Erdos949.erdos_949 Let $S \subseteq \mathbb{R}$ be a set containing no solutions to $a + b = c$ sorry line 40 source
Source documentation
Let $S \subseteq \mathbb{R}$ be a set containing no solutions to $a + b = c$. Must there be a set $A \subseteq \mathbb{R} \setminus S$ of cardinality continuum such that $A + A \subseteq \mathbb{R}\setminus S$?

depends on sorry what this means
Statement uses: Iff, True, Set, Real, Membership.mem, Not, HAdd.hAdd, Exists, And, LE.le, Compl.compl, Eq, Cardinal, Cardinal.mk, Cardinal.continuum
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_949 : answer(sorry) ↔
    ∀ S : Set ℝ, (∀ a ∈ S, ∀ b ∈ S, a + b ∉ S) → ∃ A ⊆ Sᶜ, #A = 𝔠 ∧ A + A ⊆ Sᶜ := by
…
```
theorem Erdos949.erdos_949.variants.sidon Let $S\sub \mathbb{R}$ be a Sidon set line 47 source
Source documentation
Let $S\sub \mathbb{R}$ be a Sidon set. Must there be a set $A\sub \mathbb{R}∖S$ of cardinality continuum such that $A + A \sub \mathbb{R}∖S$?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Iff, True, Set, Real, IsSidon, Exists, And, LE.le, Compl.compl, Eq, Cardinal, Cardinal.mk, Cardinal.continuum, HAdd.hAdd
Proof uses: id, Eq.mpr, Membership.mem, Eq.trans, congrArg, forall_congr, implies_congr, Eq.refl, funext, true_iff, LT.lt, Or, lt_or_eq_of_le, Eq.mp, Cardinal.mk_real, Cardinal.mk_set_le, Maximal, Set.ofPred, Not, zorn_subset and 155 more
192 instances, coercions and auxiliary constants left out.

Tactic trace: 57 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_949.variants.sidon : answer(True) ↔
    ∀ S : Set ℝ, IsSidon S → ∃ A ⊆ Sᶜ, #A = 𝔠 ∧ A + A ⊆ Sᶜ := by
…
```
Mentions: IsSidon