(** Positivity *)

(* Specifications: *)
Lemma pow_nonneg : forall a b, 0<=a -> 0<=a^b.
Lemma pow_pos_nonneg : forall a b, 0<a -> 0<=b -> 0<a^b.
(** Monotonicity *)
Lemma pow_lt_mono_l : forall a b c, 0<c -> 0<=a<b -> a^c < b^c.
Lemma pow_le_mono_l : forall a b c, 0<=a<=b -> a^c <= b^c.
Lemma pow_gt_1 : forall a b, 1<a -> (0<b <-> 1<a^b).
Lemma pow_lt_mono_r : forall a b c, 1<a -> 0<=c -> b<c -> a^b < a^c.
(** NB: since 0^0 > 0^1, the following result isn't valid with a=0 *)
Lemma pow_le_mono_r : forall a b c, 0<a -> b<=c -> a^b <= a^c.
Lemma pow_le_mono : forall a b c d, 0<a<=c -> b<=d ->
a^b <= c^d.
Lemma pow_lt_mono : forall a b c d, 0<a<c -> 0<b<d ->
a^b < c^d.