frequently_frequently_nhds — Mathlib

English

For a predicate p, the two-step frequency is equivalent to the one-step frequency: frequently near y near x and frequently near x near y coincide.

Русский

Для предиката p частотность вокруг y в окрестности x с частотностью вокруг x в окрестности y эквивалентна частотности вокруг x в окрестности x.

LaTeX

$$$\\bigl(\\mathsf{Frequent}(y\\mapsto \\mathsf{Frequent}(x\\mapsto p(x))\\text{ near } 𝓝 y)\\bigr)\\iff \\mathsf{Frequent}(x\\mapsto p(x))\\text{ near } 𝓝 x$$$

Lean4

@[simp]
theorem frequently_frequently_nhds {p : X → Prop} : (∃ᶠ x' in 𝓝 x, ∃ᶠ x'' in 𝓝 x', p x'') ↔ ∃ᶠ x in 𝓝 x, p x :=
  by
  rw [← not_iff_not]
  simp only [not_frequently, eventually_eventually_nhds]

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