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.

Theorem bpow_lt_bpow :
forall e1 e2,
(bpow (e1 - 1) < bpow e2)%R ->
(e1 <= e2)%Z.