intros m n f; generalize m; clear m; elim n; simpl in |- *; intros.
rewrite (plus_comm m); auto.
rewrite H; repeat (rewrite (plus_comm m); simpl in |- *); auto with arith.
Qed.

Lemma t_sum_Svars :
forall (n k : nat) (f : nat -> nat),
sum_nm k n f = sum_nm k (S n) (fun i : nat => f (pred i)).