intros p q H'.
replace (FtoRradix q) with (FtoRradix (Fnormalize radix b precision q)).
2: unfold FtoRradix in |- *; apply FnormalizeCorrect; auto.
case (Zle_or_lt (Fexp (Fnormalize radix b precision q)) (Fexp p)); intros H'1.
exists (Fnum p).
unfold FtoRradix in |- *; apply FSuccZleEq with (3 := pGivesBound); auto.
(* A min of a float is always represnetable with the same exposant *)
replace (Float (Fnum p) (Fexp p)) with p; [ idtac | case p ]; auto.
replace (FtoR radix (Fnormalize radix b precision q)) with (FtoR radix q);
[ idtac | rewrite FnormalizeCorrect ]; auto.
apply isMin_inv1 with (1 := H'); auto.
replace (FSucc b radix precision (Fnormalize radix b precision q)) with
(FNSucc b radix precision q); [ idtac | case p ];
auto.
replace (Float (Fnum p) (Fexp p)) with p; [ idtac | case p ]; auto.
case (Req_dec p q); intros Eq0.
unfold FtoRradix in Eq0; rewrite Eq0.
apply FNSuccLt; auto.
case (MinMax q p); auto.
intros H'2 H'3; elim H'3; intros H'4 H'5; clear H'3.
case H'4; auto.
intros H'0; absurd (p <= q)%R; rewrite H'0; auto.
apply Rlt_not_le; auto.
unfold FtoRradix in |- *; apply FNSuccLt; auto.
inversion H'.
elim H0; intros H'3 H'6; apply H'6; clear H0; auto.
rewrite <- H'0; auto with real.
exists
(Fnum
(Fshift radix (Zabs_nat (Fexp (Fnormalize radix b precision q) - Fexp p))
(Fnormalize radix b precision q))).
pattern (Fexp p) at 2 in |- *;
replace (Fexp p) with
(Fexp
(Fshift radix
(Zabs_nat (Fexp (Fnormalize radix b precision q) - Fexp p))
(Fnormalize radix b precision q))).
unfold FtoRradix in |- *;
rewrite <-
FshiftCorrect
with
(n :=
Zabs_nat (Fexp (Fnormalize radix b precision q) - Fexp p))
(x := Fnormalize radix b precision q).
case
(Fshift radix (Zabs_nat (Fexp (Fnormalize radix b precision q) - Fexp p))
(Fnormalize radix b precision q)); auto.
auto with arith.
simpl in |- *; rewrite inj_abs; auto with zarith.
Qed.

Theorem MaxBinade :
forall (r : R) (p : float),
Fbounded b p ->
(r <= p)%R -> (FNPred b radix precision p < r)%R -> isMax r p.