Section operations.
Variable radix : Z.
Let FtoRradix := FtoR radix.
Local Coercion FtoRradix : float >-> R.
Hypothesis radixNotZero : (0 < radix)%Z.
Definition Fplus (x y : float) :=
Float
(Fnum x * Zpower_nat radix (Z.abs_nat (Fexp x - Z.min (Fexp x) (Fexp y))) +
Fnum y * Zpower_nat radix (Z.abs_nat (Fexp y - Z.min (Fexp x) (Fexp y))))
(Z.min (Fexp x) (Fexp y)).
intros x y; unfold Fplus, Fshift, FtoRradix, FtoR in |- *; simpl in |- *.
rewrite plus_IZR.
rewrite Rmult_comm; rewrite Rmult_plus_distr_l; auto.
repeat rewrite Rmult_IZR.
repeat rewrite (Rmult_comm (Fnum x)); repeat rewrite (Rmult_comm (Fnum y)).
repeat rewrite Zpower_nat_Z_powerRZ; auto.
repeat rewrite <- Rmult_assoc.
repeat rewrite <- powerRZ_add; auto with real zarith arith.
repeat rewrite inj_abs; auto with real zarith.
repeat rewrite Zplus_minus; auto.
Qed.
Definition Fopp (x : float) := Float (- Fnum x) (Fexp x).
unfold FtoRradix, FtoR, Fopp in |- *; simpl in |- *.
intros x.
rewrite Ropp_Ropp_IZR; auto with real.
Qed.
intros p; case p; unfold Fopp in |- *; simpl in |- *; auto.
intros; rewrite Z.opp_involutive; auto.
Qed.

Theorem Fzero_opp : forall f : float, ~ is_Fzero f -> ~ is_Fzero (Fopp f).