exact H.
simple induction b. simpl in |- *. intro.
apply gt_n_S. unfold gt in |- *. unfold lt in |- *. elim plus_n_Sm. auto with arith.
intros. simpl in |- *. apply gt_n_S.
elim plus_n_Sm. apply gt_n_S. elim plus_n_Sm. apply H. apply gt_S_n. exact H1.
Qed.

(* Specifications: *)
Lemma gt_double_S_n : forall a b : nat, a > b -> S (a + a) > b + b.
intros. apply gt_trans with (a + a). auto with arith. apply gt_double_inv. exact H.