FormalConjectures/ErdosProblems/907.lean

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Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 907

References:

Imports (1)
  1. theorem Erdos907.erdos_907 Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$ sorry line 42 source

    Source documentation

    Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is it true that $$f=g+h$$ for some continuous $g$ and additive $h$ (i.e. $h(x+y)=h(x)+h(y)$)?

    A conjecture of Erdős from the early 1950s. Answered in the affirmative by de Bruijn [dB51].

    depends on sorry what this means

    Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Continuous, HSub.hSub, HAdd.hAdd, Exists, And, Eq

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

    23 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_907 : answer(True) ↔
    ∀ f : ℝ → ℝ, (∀ h : ℝ, 0 < h → Continuous fun x => f (x + h) - f x) →
      ∃ g a : ℝ → ℝ, Continuous g ∧ (∀ x y, a (x + y) = a x + a y) ∧ f = g + a := by
…