(** ** Power of two open upper bound [q < 2^z] and [Qabs q < 2^z] *)

(** Compute a z such that q<2^z.
    z shall be close to as small as possible, but we need a compromise between
    the tighness of the bound and the computation speed and proof complexity.
    Looking just at the log2 of the numerator and denominator, this is a tight bound
    except when the numerator is a power of 2 and the denomintor is not.
    E.g. this return 4/5 < 2^1 instead of 4/5< 2^0.
    If q==0, we return -1000, because as binary integer this has just 10 bits but
    2^-1000 should be smaller than almost any number in practice.
    If numbers are much smaller, computations might be inefficient. *)

Definition Qbound_lt_ZExp2 (q : Q) : Z :=
  match Qnum q with
  (* The -1000 is a compromise between a tight Archimedian and avoiding too large numbers *)
  | Z0 => -1000
  | Zneg p => 0
  | Zpos p => Z.pos_sub (Pos.succ (Pos_log2floor_plus1 p)) (Pos_log2floor_plus1 (Qden q))
  end.Remark Qbound_lt_ZExp2_test_1 : Qbound_lt_ZExp2 (4#4) = 1%Z.Remark Qbound_lt_ZExp2_test_2 : Qbound_lt_ZExp2 (5#4) = 1%Z.Remark Qbound_lt_ZExp2_test_3 : Qbound_lt_ZExp2 (4#5) = 1%Z.Remark Qbound_lt_ZExp2_test_4 : Qbound_lt_ZExp2 (7#5) = 1%Z.

Lemma Qbound_lt_ZExp2_spec : forall (q : Q), (q < 2^(Qbound_lt_ZExp2 q))%Q.