intros p q H; case (le_or_lt (digit p) (digit q)); auto; intros H1;
Contradict H.
apply Zlt_not_le.
cut (p <> 0%Z); [ intros H2 | idtac ].
apply Zlt_le_trans with (2 := digitLess p H2).
cut (digit q <= pred (digit p)); [ intros H3 | idtac ].
apply Zlt_le_trans with (2 := Zpower_nat_monotone_le _ _ H3);
auto with zarith.
generalize H1; case (digit p); simpl in |- *; auto with arith.
generalize H1; case p; simpl in |- *; intros tmp; intros; red in |- *; intros;
try discriminate; Contradict tmp; auto with arith.
Qed.
Hint Resolve digit_monotone: arith.
(* if the number is not null so is the number of digits *)

Theorem digitNotZero : forall q : Z, q <> 0%Z -> 0 < digit q.
intros q H'.