sub_iff_right — Mathlib

English

For any f1, f2, g, l, the statement f1 - f2 = OWith c l g is equivalent to f1 = OWith c l g given f2 = OWith 0? (context dependent).

Русский

Для любых f1, f2, g, l формулировка f1 - f2 = OWith c l g эквивалентна f1 = OWith c l g в зависимости от контекста.

LaTeX

$$$ (h_2 : f_2 =O[l] g) \\Rightarrow (f_1 - f_2) =O[l] g \\Leftrightarrow (f_1 =O[l] g) $$$

Lean4

theorem sub_iff_right (h₁ : f₁ =O[l] g) : (fun x => f₁ x - f₂ x) =O[l] g ↔ (f₂ =O[l] g) :=
  ⟨fun h ↦ h₁.sub h |>.congr (fun _ ↦ sub_sub_self ..) (fun _ ↦ rfl), fun h ↦ h₁.sub h⟩

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