elim plus_n_O; auto with arith.
elim plus_n_O; auto with arith.
auto with arith.
intros.
simpl in |- *.
apply H.
auto with arith.
apply lt_S_n.
replace (S (n + n0)) with (n + S n0).
auto with arith.
elim plus_comm; simpl in |- *; auto with arith.
Qed.

Lemma lt_neq_O_pred : forall n m : nat, S n < m -> pred (m - n) <> 0.
intros.