Definition Rplus : R -> R -> R.(* Extraction Rplus. *)

Definition Ropp : R -> R.Definition Rminus : R -> R -> R := fun x y => Rplus x (Ropp y).Definition Rlt : R -> R -> Set := fun x y => Rpos (Rminus y x).Definition Rle : R -> R -> Prop := fun x y => Rnonneg (Rminus y x).(* An alternative characterization of positivity upon R. *)

Definition Rpos_alt (x:R) := 
 {l:nat & { p:nat | forall n, (p<=n)%nat -> (1#2)^l <= cauchy x n}}.

(* Specifications: *)
Lemma Rpos_alt_1 : forall x:R, Rpos x -> Rpos_alt x.
Lemma Rpos_alt_2 : forall x, Rpos_alt x -> Rpos x.