intros n' Hrec b; case b; auto.
Qed.
Remark expand_mult2 : forall x : nat, 2 * x = x + x.
intros x; ring.
Qed.

Theorem lt_neq : forall x y : nat, x < y -> x <> y.
unfold not in |- *; intros x y H H1; elim (lt_irrefl x);
pattern x at 2 in |- *; rewrite H1; auto.