(* The constructive equality on R. *)
Definition Req_constr (x y : R) : Prop
:= (x < y -> False) /\ (y < x -> False).

(* Not RealField.RTheory, because it uses Leibniz equality. *)
Lemma Rring : ring_theory (IQR 0) (IQR 1) Rplus Rmult
(fun x y : R => (x + - y)%R) Ropp Req_constr.