1126.lean`

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L4DEL: Lean4 Domain Explanation Language

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1126

References:

erdosproblems.com/1126
[Er60c] Erdős, P., Problem 310. Colloq. Math., 311.
[dB66] de Bruijn, N. G., On almost additive functions. Colloq. Math. (1966), 59-63.
[Ju65] Jurkat, Wolfgang B., On Cauchy's functional equation. Proc. Amer. Math. Soc. (1965), 683-686.

Imports (1)

FormalConjecturesUtil

theorem Erdos1126.erdos_1126 Documented declaration: erdos_1126 sorry line 46 source
Source documentation
If $$f(x+y)=f(x)+f(y)$$ for almost all $x,y\in \mathbb{R}$ then there exists a function $g$ such that $$g(x+y)=g(x)+g(y)$$ for all $x,y\in\mathbb{R}$ such that $f(x)=g(x)$ for almost all $x$.
Proved independently by de Bruijn [dB66] and Jurkat [Ju65].

depends on sorry what this means
Statement uses: Iff, True, Real, Filter.Eventually, Prod, Eq, HAdd.hAdd, Prod.fst, Prod.snd, MeasureTheory.ae, MeasureTheory.Measure, MeasureTheory.Measure.prod, MeasureTheory.MeasureSpace.volume, Exists, And
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_1126 : answer(True) ↔ ∀
    (f : ℝ → ℝ)
    (h :
      ∀ᵐ (p : ℝ × ℝ) ∂(volume.prod volume),
        f (p.1 + p.2) = f p.1 + f p.2),
    ∃ h : ℝ → ℝ,
      (∀ x y, h (x + y) = h x + h y) ∧ (∀ᵐ x ∂volume, f x = h x) := by
…
```