rewrite (plus_comm x y).
rewrite (plus_comm x z).
rewrite plus_comm.
symmetry  in |- *.
rewrite plus_comm.
rewrite plus_permute.
auto with arith.
Qed. Hint Resolve plus_permute2.

Lemma mult_sym : forall a b : nat, a * b = b * a.
intros a b; elim a; simpl in |- *; auto with arith.