replace (INR 2 * r)%R with (r + r)%R; [ idtac | simpl in |- *; ring ].
pattern r at 1 in |- *; replace r with (r + 0)%R; [ idtac | ring ].
apply Rplus_le_compat_l; auto.
Qed.

Theorem Rltdouble : forall r : R, (0 < r)%R -> (r < INR 2 * r)%R.
intros r H'.