intros a; elim a; simpl in |- *; auto.
intros b; case b; simpl in |- *; auto.
intros n H b; case b; simpl in |- *.
intros H1; inversion H1.
intros n0 H0; apply H.
apply le_S_n; auto.
Qed.

Theorem le_bool_correct2 : forall a b : nat, b < a -> le_bool a b = false.
Proof using.