Definition CReal_mult_seq (x y : CReal) :=
  (fun n : Z => seq x (n - scale y - 1)%Z
              * seq y (n - scale x - 1)%Z).Definition CReal_mult_scale (x y : CReal) : Z :=
  x.(scale) + y.(scale).Local Ltac simplify_Qpower_exponent :=
  match goal with |- context [(_ ^ ?a)%Q] => ring_simplify a end.Local Ltac simplify_Qabs :=
  match goal with |- context [(Qabs ?a)%Q] => ring_simplify a end.Local Ltac simplify_Qabs_in H :=
  match type of H with context [(Qabs ?a)%Q] => ring_simplify a in H end.Local Ltac field_simplify_Qabs :=
  match goal with |- context [(Qabs ?a)%Q] => field_simplify a end.Local Ltac pose_Qabs_pos :=
  match goal with |- context [(Qabs ?a)%Q] => pose proof Qabs_nonneg a end.Local Ltac simplify_Qle :=
  match goal with |- (?l <= ?r)%Q => ring_simplify l; ring_simplify r end.Local Ltac simplify_Qle_in H :=
  match type of H with (?l <= ?r)%Q => ring_simplify l in H; ring_simplify r in H end.Local Ltac simplify_Qlt :=
  match goal with |- (?l < ?r)%Q => ring_simplify l; ring_simplify r end.Local Ltac simplify_Qlt_in H :=
  match type of H with (?l < ?r)%Q => ring_simplify l in H; ring_simplify r in H end.Local Ltac simplify_seq_idx :=
  match goal with |- context [seq ?x ?n] => progress ring_simplify n end.

Local Lemma Weaken_Qle_QpowerRemSubExp: forall (q : Q) (n m : Z), (m >= 0)%Z -> (q <= 2^(n-m))%Q -> (q <= 2^n)%Q.