intros v r q; generalize r; elim q; clear r q; simpl in |- *.
intros p Rec r Hr; generalize (Zcompare_correct r v); case (r ?= v)%Z; auto.
intros H1 H2; rewrite <- H1.
apply Zle_lt_trans with (Zpower_nat n 0 * r)%Z; auto with zarith arith.
rewrite Zpower_nat_O; rewrite Zmult_1_l; auto with zarith.
intros H1 H2; rewrite digitAux1.
apply Rec.
apply Zlt_mult_ZERO; auto with zarith.
rewrite <- digitAux1; auto.
rewrite Zpower_nat_O; rewrite Zmult_1_l; auto with zarith.
intros p Rec r Hr; generalize (Zcompare_correct r v); case (r ?= v)%Z; auto.
intros H1 H2; rewrite <- H1.
apply Zle_lt_trans with (Zpower_nat n 0 * r)%Z; auto with zarith arith.
rewrite Zpower_nat_O; rewrite Zmult_1_l; auto with zarith.
intros H1 H2; rewrite digitAux1.
apply Rec.
apply Zlt_mult_ZERO; auto with zarith.
rewrite <- digitAux1; auto.
rewrite Zpower_nat_O; rewrite Zmult_1_l; auto with zarith.
auto.
Qed.

Theorem pos_length_pow :
forall p : positive, (Zpos p < Zpower_nat n (S (pos_length p)))%Z.