(** ** Power of 2 open lower bounds for [2^z < q] and [2^z < Qabs q] *)

(** Note: the -2 is required cause of the Qlt limit.
    In case q is a power of two, the lower and upper bound must be a factor of 4 apart *)
Definition Qlowbound_ltabs_ZExp2 (q : Q) : Z := -2 + Qbound_ltabs_ZExp2 q.

(* Specifications: *)
Lemma Qlowbound_ltabs_ZExp2_inv: forall q : Q, q > 0 -> Qlowbound_ltabs_ZExp2 q = (- Qbound_ltabs_ZExp2 (/q))%Z.
Lemma Qlowbound_ltabs_ZExp2_opp: forall q : Q, (Qlowbound_ltabs_ZExp2 q = Qlowbound_ltabs_ZExp2 (-q))%Z.
Lemma Qlowbound_lt_ZExp2_spec : forall (q : Q) (Hqgt0 : q > 0), (2^(Qlowbound_ltabs_ZExp2 q) < q)%Q.
Lemma Qlowbound_ltabs_ZExp2_spec : forall (q : Q) (Hqgt0 : ~ q == 0), (2^(Qlowbound_ltabs_ZExp2 q) < Qabs q)%Q.