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

(* Specifications: *)
Lemma Req_constr_refl : forall x:R, Req_constr x x.
Lemma Req_constr_leibniz : forall x y:R, Req_constr x y -> x = y.