intros w; elim w; unfold Qpositive_le in |- *; simpl in |- *.
intros w0 Hrec w'; case w'; simpl in |- *.
3: intros H; discriminate H.
intros; apply f_equal with (f := nR); auto.
intros w'0 H; discriminate H.
intros w0 Hrec w'; case w'; simpl in |- *.
3: intros H H1; discriminate H1.
intros w'1 H H1; discriminate H1.
intros; apply f_equal with (f := dL); auto.
intros w'; case w'; auto with *.
intros w'0 H H1; discriminate H1.
intros w'0 H; discriminate H.
Qed.

(* Specifications: *)
Theorem Qpositive_le_refl : forall w : Qpositive, Qpositive_le w w.
intros w; unfold Qpositive_le in |- *; elim w; simpl in |- *; auto.