(** This file gives definitions of equivalent (g ~ f) and dominant (g
= o(f)). This is used for defining differentiability on a
[NormedModule]. *)

Definition is_domin {T} {Ku Kv : AbsRing}
  {U : NormedModule Ku} {V : NormedModule Kv}
  (F : (T -> Prop) -> Prop) (f : T -> U) (g : T -> V) :=
  forall eps : posreal, F (fun x => norm (g x) <= eps * norm (f x)).Definition is_equiv {T} {K : AbsRing} {V : NormedModule K}
  (F : (T -> Prop) -> Prop) (f g : T -> V) :=
  is_domin F g (fun x => minus (g x) (f x)).

Lemma equiv_mult : forall {T} {F : (T -> Prop) -> Prop} {FF : Filter F} (f1 f2 g1 g2 : T -> R), is_equiv F f1 g1 -> is_equiv F f2 g2 -> is_equiv F (fun x => f1 x * f2 x) (fun x => g1 x * g2 x).