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.
Casec H'7; intros H'7.
cut (p <= FPred b radix precision q)%R; [ intros Rl1 | idtac ].
cut (P (FPred b radix precision q)); [ intros P1 | idtac ].
rewrite <- (FSucPred b radix precision) with (x := q); auto with arith.
apply H'2; auto with float arith.
apply H'4 with (y := (Fweight (FPred b radix precision q) - Fweight p)%Z);
auto.
split.
cut (Fweight p <= Fweight (FPred b radix precision q))%Z; auto with zarith.
apply FweightZle; auto.
apply FPredCanonic; auto with arith.
rewrite H'5.
cut (Fweight (FPred b radix precision q) < Fweight q)%Z;
[ auto with zarith | idtac ].
apply FweightLt; auto with float.
apply (R0RltRlePred b radix precision); auto.
apply Rle_lt_trans with (FtoRradix p); auto.
apply (FPredLt b radix precision); auto.
apply (FPredCanonic b radix precision); auto with arith.
apply (FPredProp b radix precision); auto with arith.
rewrite <-
(FcanonicUnique _ radixMoreThanOne b precision) with (p := p) (q := q);
auto with arith.
Qed.

Theorem FinductPos :
forall (P : float -> Prop) (p : float),
(0 <= p)%R ->
Fcanonic radix b p ->
P p ->
(forall q : float,
Fcanonic radix b q -> (p <= q)%R -> P q -> P (FSucc b radix precision q)) ->
forall q : float, Fcanonic radix b q -> (p <= q)%R -> P q.