sub_iff_left — Mathlib

English

If h1 = IsBigOWith c1 l f1 g and h2 = f2 = o_l g with c1 < c2, then f1 - f2 is in IsBigOWith c2 l g.

Русский

Если h1 = IsBigOWith c1 l f1 g и h2 = f2 = o_l g с c1 < c2, тогда f1 - f2 ∈ IsBigOWith c2 l g.

LaTeX

$$$ (h_1 : IsBigOWith c_1 l f_1 g) \\land (h_2 : f_2 =o[l] g) \\land (c_1 < c_2) \\Rightarrow IsBigOWith c_2 l (f_1 - f_2) g $$$

Lean4

theorem sub_iff_left (h₂ : f₂ =O[l] g) : (fun x => f₁ x - f₂ x) =O[l] g ↔ (f₁ =O[l] g) :=
  ⟨fun h ↦ h.add h₂ |>.congr (fun _ ↦ sub_add_cancel ..) (fun _ ↦ rfl), fun h ↦ h.sub h₂⟩

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