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

Lemma gt_double_n_S : forall a b : nat, a > b -> a + a > S (b + b).
simple induction a. intros. absurd (0 > b). auto with arith.