elim H; intros.
rewrite H0.
rewrite (plus_n_Sm (2 * q) 1).
apply (plus_even_even (2 * q) 2).
apply (even_intro (2 * q) q); auto with arith.
apply (even_intro 2 1).
auto with arith.
Qed.

(* Specifications: *)
Lemma pred_odd_even : forall a : nat, odd a -> even (pred a).
intros.