nat_of_atTop — Mathlib

English

If f1 = O_l f2 and c < 1, then f2 is bounded by a multiple of f2 − f1 after a scaling 1/(1−c).

Русский

Если f1 = O_l f2 и c < 1, то после масштабирования 1/(1−c) функция f2 ограничена относительнo разности f2−f1.

LaTeX

$$$\\displaystyle \\text{If } f_1 = O_l f_2, \\; c<1 \\Rightarrow f_2 = O_l \\Bigl(f_2 - f_1\\Bigr)\\cdot \\frac{1}{1-c}.$$$

Lean4

theorem nat_of_atTop {f : ℕ → E''} {g : ℕ → F''} (hfg : f =O[atTop] g) {l : Filter ℕ}
    (h : ∀ᶠ n in l, g n = 0 → f n = 0) : f =O[l] g :=
  by
  obtain ⟨C, hC_pos, hC⟩ := bound_of_isBigO_nat_atTop hfg
  refine isBigO_iff.mpr ⟨C, ?_⟩
  filter_upwards [h] with x h
  by_cases hf : f x = 0
  · simp [hf, hC_pos]
  exact hC fun a ↦ hf (h a)

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