(* This notation will be deprecated with a planned larger rework of Q lemma naming *)
Notation Qpower_pos := Qpower_0_le (only parsing).

(* Specifications: *)
Lemma Qpower_0_lt: forall (a : Q) (z : Z), 0 < a -> 0 < Qpower a z.
Lemma Qpower_1_lt_pos: forall (q : Q) (n : positive), (1<q)%Q -> (1 < q ^ (Z.pos n))%Q.
Lemma Qpower_1_lt: forall (q : Q) (n : Z), (1<q)%Q -> (0<n)%Z -> (1 < q ^ n)%Q.
Lemma Qpower_1_le_pos: forall (q : Q) (n : positive), (1<=q)%Q -> (1 <= q ^ (Z.pos n))%Q.
Lemma Qpower_1_le: forall (q : Q) (n : Z), (1<=q)%Q -> (0<=n)%Z -> (1 <= q ^ n)%Q.
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.
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.
Lemma Qpower_mult : forall a n m, a^(n*m) == (a^n)^m.
Lemma Qpower_decomp_pos: forall (p : positive) (a : Z) (b : positive), (a # b) ^ (Z.pos p) == a ^ (Z.pos p) # (b ^ p)%positive.
Lemma Qpower_decomp_neg_pos: forall (p a b: positive), (Z.pos a # b) ^ (Z.neg p) == (Z.pos b) ^ (Z.pos p) # (a ^ p)%positive.
Lemma Qpower_decomp_neg_neg: forall (p a b: positive), (Z.neg a # b) ^ (Z.neg p) == (Z.neg b) ^ (Z.pos p) # (a ^ p)%positive.
Lemma Qpower_lt_compat_l: forall (q : Q) (n m : Z), (n < m)%Z -> (1<q)%Q -> (q ^ n < q ^ m)%Q.
Lemma Qpower_le_compat_l: forall (q : Q) (n m : Z), (n <= m)%Z -> (1<=q)%Q -> (q ^ n <= q ^ m)%Q.
Lemma Qpower_lt_compat_l_inv: forall (q : Q) (n m : Z), (q ^ n < q ^ m)%Q -> (1<q)%Q -> (n < m)%Z.
Lemma Qpower_le_compat_l_inv: forall (q : Q) (n m : Z), (q ^ n <= q ^ m)%Q -> (1<q)%Q -> (n <= m)%Z.
Lemma Zpower_Qpower : forall (a n:Z), (0<=n)%Z -> inject_Z (a^n) == (inject_Z a)^n.
Lemma Qsqr_nonneg : forall a, 0 <= a^2.
Lemma Qarchimedean_power2_pos : forall q : Q, {p : positive | (q < Z.pos (2^p) # 1)%Q}.