Definition radix2 := Build_radix 2 (refl_equal _).
Variable r : radix.

(* Specifications: *)
Theorem radix_gt_0 : (0 < r)%Z.
Theorem radix_gt_1 : (1 < r)%Z.
Theorem Zpower_gt_1 :
forall p,
(0 < p)%Z ->
(1 < Zpower r p)%Z.
Theorem Zpower_gt_0 :
forall p,
(0 <= p)%Z ->
(0 < Zpower r p)%Z.
Theorem Zpower_ge_0 :
forall e,
(0 <= Zpower r e)%Z.
Theorem Zpower_le :
forall e1 e2, (e1 <= e2)%Z ->
(Zpower r e1 <= Zpower r e2)%Z.
Theorem Zpower_lt :
forall e1 e2, (0 <= e2)%Z -> (e1 < e2)%Z ->
(Zpower r e1 < Zpower r e2)%Z.
Theorem Zpower_lt_Zpower :
forall e1 e2,
(Zpower r (e1 - 1) < Zpower r e2)%Z ->
(e1 <= e2)%Z.
Theorem radix_gt_0 : (0 < r)%Z.
Theorem radix_gt_1 : (1 < r)%Z.
Theorem Zpower_gt_1 :
forall p,
(0 < p)%Z ->
(1 < Zpower r p)%Z.
Theorem Zpower_gt_0 :
forall p,
(0 <= p)%Z ->
(0 < Zpower r p)%Z.
Theorem Zpower_ge_0 :
forall e,
(0 <= Zpower r e)%Z.
Theorem Zpower_le :
forall e1 e2, (e1 <= e2)%Z ->
(Zpower r e1 <= Zpower r e2)%Z.
Theorem Zpower_lt :
forall e1 e2, (0 <= e2)%Z -> (e1 < e2)%Z ->
(Zpower r e1 < Zpower r e2)%Z.
Theorem Zpower_lt_Zpower :
forall e1 e2,
(Zpower r (e1 - 1) < Zpower r e2)%Z ->
(e1 <= e2)%Z.