intros P p H' H'0 H'1 H'2 q H'3 H'4.
apply FinductPosAux with (p := p) (x := (Fweight q - Fweight p)%Z); auto.
cut (Fweight p <= Fweight q)%Z; [ auto with zarith | idtac ].
apply FweightZle; auto with float.
Qed.

Theorem FinductNegAux :
forall (P : float -> Prop) (p : float),
(0 <= p)%R ->
Fcanonic radix b p ->
P p ->
(forall q : float,
Fcanonic radix b q ->
(0 < q)%R -> (q <= p)%R -> P q -> P (FPred b radix precision q)) ->
forall x : Z,
(0 <= x)%Z ->
forall q : float,
x = (Fweight p - Fweight q)%Z ->
Fcanonic radix b q -> (0 <= q)%R -> (q <= p)%R -> P q.