isBigO_congr — Mathlib

English

IsBigO is preserved under a homeomorphism: f = O[𝓝 b] g iff f ∘ e = O[𝓝 (e.symm b)] g ∘ e.

Русский

IsBigO сохраняется под гомеоморфизмом: f = O[𝓝 b] g эквивалентно f ∘ e = O[𝓝 (e.симм b)] g ∘ e.

LaTeX

$$$ f =O[\\mathcal{N} b] g \\iff (f \\circ e) =O[\\mathcal{N} (e.symm b)] (g \\circ e) $$$

Lean4

/-- Transfer `IsBigO` over a `Homeomorph`. -/
theorem isBigO_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 [IsBigO_def]
  exact exists_congr fun C => e.isBigOWith_congr

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