Section pow.
Variable r : radix.
(** Well-used function called bpow for computing the power function #&beta;#^e *)
Definition bpow e :=
match e with
| Zpos p => IZR (Zpower_pos r p)
| Zneg p => Rinv (IZR (Zpower_pos r p))
| Z0 => 1%R
end.
(** Another well-used function for having the magnitude of a real number x to the base #&beta;# *)
Record mag_prop x := {
mag_val :> Z ;
_ : (x <> 0)%R -> (bpow (mag_val - 1)%Z <= Rabs x < bpow mag_val)%R
}.
Definition mag :
forall x : R, mag_prop x.

Lemma mag_mult :
forall x y,
(x <> 0)%R -> (y <> 0)%R ->
(mag x + mag y - 1 <= mag (x * y) <= mag x + mag y)%Z.