Definition CReal_abs (x : CReal) : CReal :=
{|
  seq := CReal_abs_seq x;
  scale := CReal_abs_scale x;
  cauchy := CReal_abs_cauchy x;
  bound := CReal_abs_bound x
|}.

(* Specifications: *)
Lemma CRealLt_RQ_from_single_dist : forall (r : CReal) (q : Q) (n :Z), (2^n < q - seq r n)%Q -> r < inject_Q q.
Lemma CRealLe_0R_to_single_dist : forall (x : CReal) (n : Z), 0 <= x -> (-(2^n) <= seq x n)%Q.
Lemma CReal_abs_right : forall x : CReal, 0 <= x -> CReal_abs x == x.
Lemma CReal_le_abs : forall x : CReal, x <= CReal_abs x.
Lemma CReal_abs_pos : forall x : CReal, 0 <= CReal_abs x.
Lemma CReal_abs_opp : forall x : CReal, CReal_abs (-x) == CReal_abs x.
Lemma CReal_abs_left : forall x : CReal, x <= 0 -> CReal_abs x == -x.
Lemma CReal_abs_appart_0 : forall x : CReal, 0 < CReal_abs x -> x # 0.