auto with arith. simple induction b. simpl in |- *. auto with arith.
intros. apply gt_n_S.
apply H. generalize H1. simpl in |- *. elim plus_n_Sm. elim plus_n_Sm. intro.
cut (S (n + n) > S (n0 + n0)). intro. apply gt_S_n. exact H3.
apply gt_S_n. exact H2.
Qed.

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