isBigOWith_principal — Mathlib

English

Same as the previous: for monotone u and nonzero m, expGrowthSup (fun n => u (m n)) = m · expGrowthSup u.

Русский

То же, что и ранее: для монотонной u и ненулевого m: expGrowthSup (n ↦ u(m n)) = m · expGrowthSup u.

LaTeX

$$$\expGrowthSup (\lambda n. u(m n)) = m \cdot \expGrowthSup u$$$

Lean4

@[simp]
theorem isBigOWith_principal {s : Set α} : IsBigOWith c (𝓟 s) f g ↔ ∀ x ∈ s, ‖f x‖ ≤ c * ‖g x‖ := by
  rw [IsBigOWith_def, eventually_principal]

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