isLittleO_congr — Mathlib

English

Transfer IsBigO across an OpenPartialHomeomorph: IsBigO at b for f,g is equivalent to IsBigO at e.symm(b) for f∘e, g∘e.

Русский

Переносим IsBigO через OpenPartialHomeomorph: IsBigO на b для f,g эквивалентно IsBigO на e^{-1}(b) для f∘e и g∘e.

LaTeX

$$$ IsBigO (\\mathcal{N} b) f g \\iff IsBigO (\\mathcal{N} (e.symm.toFun' b)) (f \\circ e) (g \\circ e) $$$

Lean4

/-- Transfer `IsLittleO` over a `Homeomorph`. -/
theorem isLittleO_congr (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} : f =o[𝓝 b] g ↔ (f ∘ e) =o[𝓝 (e.symm b)] (g ∘ e) :=
  by
  simp only [IsLittleO_def]
  exact forall₂_congr fun c _hc => e.isBigOWith_congr

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