FormalConjectures/ErdosProblems/120.lean

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L4DEL
Lean4 Domain Explanation Language

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 120

Reference:

Imports (1)
  1. def Erdos120.Erdos120For There exists a set line 39 source

    Source documentation

    There exists a set $E \subseteq \mathbb{R}$, dependent on set $A \subseteq \mathbb{R}$, of positive measure which does not contain any set of the shape $a * A + b$ for some $a,b \in \mathbb{R}$ and $a \neq 0$?

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Set, Real

    Proof uses: Exists, And, MeasurableSet, LT.lt, ENNReal, OfNat.ofNat, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, Ne, Not, LE.le, Set.image, HAdd.hAdd, HMul.hMul

    16 instances, coercions and auxiliary constants left out.

    def Erdos120For (A : Set ℝ) : Prop := ∃ E : Set ℝ,
…
  2. theorem Erdos120.erdos_120 Let $A \subseteq \mathbb{R}$ be an infinite set sorry line 48 source

    Source documentation

    Let $A \subseteq \mathbb{R}$ be an infinite set. Must there be a set $E \subseteq \mathbb{R}$ of positive measure which does not contain any set of the shape $a * A + b$ for some $a,b \in \mathbb{R}$ and $a \neq 0$?

    depends on sorry what this means

    Statement uses: Iff, True, Set, Real, Set.Infinite, Erdos120.Erdos120For

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat

    1 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_120 : answer(sorry) ↔ ∀ A : Set ℝ, A.Infinite → Erdos120For A := by
…

    Mentions: Erdos120.Erdos120For

  3. theorem Erdos120.erdos_120.variants.finite_set Steinhaus [St20] has proved Erdős 120 to be false whenever $A$ is a finite set sorry line 55 source

    Source documentation

    Steinhaus [St20] has proved Erdős 120 to be false whenever $A$ is a finite set.

    depends on sorry what this means

    Statement uses: Set, Real, Set.Finite, Not, Erdos120.Erdos120For

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat

    1 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_120.variants.finite_set {A : Set ℝ} (h : A.Finite) : ¬ Erdos120For A := by
…

    Mentions: Erdos120.Erdos120For