pattern r at 1 in |- *; replace r with (r + 0)%R; try ring.
replace (INR 2 * r)%R with (r + r)%R; simpl in |- *; try ring; auto with real.
Qed.

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