intros P p H' H'0 H'1 H'2 x H'3; pattern x in |- *.
apply Z_lt_induction; auto.
intros x0 H'4 q H'5 H'6 H'7 H'8.
Casec H'8; intros H'8.
cut (FSucc b radix precision q <= p)%R; [ intros Rle1 | idtac ].
cut (P (FSucc b radix precision q)); [ intros P1 | idtac ].
rewrite <- (FPredSuc b radix precision) with (x := q); auto with arith.
apply H'2; auto with float arith.
apply Rle_lt_trans with (FtoRradix q); auto.
apply (FSuccLt b radix); auto with arith.
apply H'4 with (y := (Fweight p - Fweight (FSucc b radix precision q))%Z);
auto.
split.
cut (Fweight (FSucc b radix precision q) <= Fweight p)%Z; auto with zarith.
apply FweightZle; auto.
apply FSuccCanonic; auto with arith.
apply Rle_trans with (FtoRradix q); auto; apply Rlt_le.
apply (FSuccLt b radix); auto with arith.
rewrite H'5.
cut (Fweight q < Fweight (FSucc b radix precision q))%Z;
[ auto with zarith | idtac ].
apply FweightLt; auto with float.
apply (FSuccLt b radix); auto with arith.
apply FSuccCanonic; auto with arith.
apply Rle_trans with (FtoRradix q); auto; apply Rlt_le.
apply (FSuccLt b radix); auto with arith.
apply (FSuccProp b radix); auto with arith.
rewrite <-
(FcanonicUnique _ radixMoreThanOne b precision) with (p := p) (q := q);
auto with arith.
Qed.

Theorem FinductNeg :
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 q : float, Fcanonic radix b q -> (0 <= q)%R -> (q <= p)%R -> P q.