intros q r H' H'0; case (le_or_lt (digit q) r).
intros H'1; case (le_lt_or_eq _ _ H'1); auto; intros H'2.
absurd (Zabs q < Zpower_nat n (digit q))%Z; auto with zarith.
apply Zle_not_lt; auto with zarith.
apply Zle_trans with (m := Zpower_nat n (pred r)); auto with zarith.
apply Zpower_nat_monotone_le.
generalize H'2; case r; auto with arith.
intros H'1.
absurd (Zpower_nat n (pred (digit q)) <= Zabs q)%Z; auto with zarith.
apply Zlt_not_le; auto with zarith.
apply Zlt_le_trans with (m := Zpower_nat n r); auto.
apply Zpower_nat_monotone_le.
generalize H'1; case (digit q); auto with arith.
apply digitLess; auto with zarith.
generalize H'1; case q; unfold digit in |- *; intros tmp; intros; red in |- *;
intros; try discriminate; Contradict tmp; auto with arith.
Qed.

(* Specifications: *)
Theorem digitO : digit 0 = 0.
unfold digit in |- *; simpl in |- *; auto with arith.