rewrite boundRrOpp.
pattern r at 2 in |- *; rewrite <- (Ropp_involutive r).
unfold FtoRradix in |- *; rewrite Fopp_correct.
apply Ropp_lt_contravar; apply boundRCorrect1; auto.
apply Rle_lt_trans with 0%R; auto.
replace 0%R with (-0)%R; auto with real.
unfold FtoRradix in |- *; rewrite Fopp_correct.
apply Ropp_le_contravar.
unfold boundR, boundNat, FtoRradix, FtoR in |- *; simpl in |- *;
auto with real zarith.
rewrite Rmult_1_l; apply Rlt_le; auto with real zarith arith.
Qed.
(* A function that returns a list containing all the bounded smaller than a given real *)
Definition mBFloat (p : R) :=
map (fun p : Z * Z => Float (fst p) (snd p))
(mProd Z Z (Z * Z)
(mZlist (- pPred (vNum b)) (pPred (vNum b)))
(mZlist (- dExp b) (Fexp (boundR p)))).

Theorem mBFadic_correct1 :
forall (r : R) (q : float),
~ is_Fzero q ->
(Fopp (boundR r) < q)%R ->
(q < boundR r)%R -> Fbounded b q -> In q (mBFloat r).