intros p H'.
case (Zle_lt_or_eq (Zsucc (Fnum p)) (Zpos (vNum b))); auto with float.
case (Zle_or_lt 0 (Fnum p)); intros H1.
rewrite <- (Zabs_eq (Fnum p)); auto with float zarith.
apply Zle_trans with 0%Z; auto with zarith.
intros H'0; exists (Float (Zsucc (Fnum p)) (Fexp p)); split; auto with float.
repeat split; simpl in |- *; auto with float.
case (Zle_or_lt 0 (Fnum p)); intros H1; auto with zarith.
rewrite Zabs_eq; auto with zarith.
apply Zlt_trans with (Zabs (Fnum p)); auto with float zarith.
repeat rewrite Zabs_eq_opp; auto with zarith.
intros H'0;
exists (Float (Zpower_nat radix (pred precision)) (Zsucc (Fexp p)));
split; auto.
repeat split; simpl in |- *; auto with zarith arith float.
rewrite pGivesBound.
rewrite Zabs_eq; auto with zarith.
rewrite H'0; rewrite pGivesBound.
pattern precision at 2 in |- *; replace precision with (1 + pred precision).
rewrite Zpower_nat_is_exp.
rewrite Zpower_nat_1.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite powerRZ_Zs; auto with real zarith.
rewrite Rmult_IZR; ring.
generalize precisionNotZero; case precision; simpl in |- *; auto with arith.
intros H'1; case H'1; auto.
Qed.

Theorem digitPredVNumiSPrecision :
digit radix (Zpred (Zpos (vNum b))) = precision.