(** ** Qpower and multiplication, exponent addition *)

(* Specifications: *)
Lemma Qmult_power : forall a b n, (a*b)^n == a^n*b^n.
Lemma Qpower_plus : forall a n m, ~a==0 -> a^(n+m) == a^n*a^m.
Lemma Qpower_plus' : forall a n m, (n+m <> 0)%Z -> a^(n+m) == a^n*a^m.
(** ** Qpower and inversion, division, exponent subtraction *)
Lemma Qinv_power : forall a n, (/a)^n == /a^n.
Lemma Qdiv_power : forall a b n, (a/b)^n == (a^n/b^n).
Lemma Qinv_power_n : forall n p, (1#p)^n == /(inject_Z (Zpos p))^n.
Lemma Qpower_opp : forall a n, a^(-n) == /a^n.
Lemma Qpower_minus: forall (a : Q) (n m : Z),
~ a == 0 -> a ^ (n - m) == a ^ n / a ^ m.
Lemma Qpower_minus_pos: forall (a b : positive) (n m : Z),
(Z.pos a#b) ^ (n - m) == (Z.pos a#b) ^ n * (Z.pos b#a) ^ m.
Lemma Qpower_minus_neg: forall (a b : positive) (n m : Z),
(Z.neg a#b) ^ (n - m) == (Z.neg a#b) ^ n * (Z.neg b#a) ^ m.