right_le_sub_of_lt_one — Mathlib

English

If f1 = O_l f2 and c < 1, then there is a bound with a shifted difference bound: f2 ≤ (1/(1−c))(f2−f1).

Русский

Если f1 = O_l f2 и c < 1, существует don bound: f2 ≤ (1/(1−c))(f2−f1).

LaTeX

$$$\\displaystyle \\text{If } f_1 = O_l f_2 \\text{ and } c<1, \\; f_2 = O_l \\bigl(f_2 - f_1\\bigr) \\cdot \\frac{1}{1-c}.$$$

Lean4

theorem right_le_sub_of_lt_one {f₁ f₂ : α → E'} (h : IsBigOWith c l f₁ f₂) (hc : c < 1) :
    IsBigOWith (1 / (1 - c)) l f₂ fun x => f₂ x - f₁ x :=
  IsBigOWith.of_bound <|
    mem_of_superset h.bound fun x hx => by
      simp only [mem_setOf_eq] at hx ⊢
      rw [mul_comm, one_div, ← div_eq_mul_inv, le_div_iff₀, mul_sub, mul_one, mul_comm]
      · exact le_trans (sub_le_sub_left hx _) (norm_sub_norm_le _ _)
      · exact sub_pos.2 hc

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