intros p r H' H'0.
split.
apply FcanonicBound with (radix := radix); auto with float.
apply FNSuccCanonic; auto.
inversion H'; auto.
split.
case (Rle_or_lt (FNSucc b radix precision p) r); intros H'2; auto.
absurd (FNSucc b radix precision p <= p)%R.
apply Rlt_not_le.
unfold FtoRradix in |- *; apply FNSuccLt; auto.
inversion H'; auto.
elim H0; intros H'1 H'3; apply H'3; auto.
apply FcanonicBound with (radix := radix); auto with float.
apply Rlt_le; auto.
intros f H'2 H'3.
replace (FtoRradix f) with (FtoRradix (Fnormalize radix b precision f)).
unfold FtoRradix in |- *; apply FNSuccProp; auto.
inversion H'; auto.
apply FcanonicBound with (radix := radix); auto with float.
apply Rlt_le_trans with r; auto.
case (Rle_or_lt r p); auto.
intros H'4; Contradict H'0.
apply Rle_antisym; auto; apply isMin_inv1 with (1 := H'); auto.
rewrite FnormalizeCorrect; auto.
unfold FtoRradix in |- *; apply FnormalizeCorrect; auto.
Qed.
(* Find a minimun in a given list if it exists *)

Theorem MinExList :
forall (r : R) (L : list float),
(forall f : float, In f L -> (r < f)%R) \/
(exists min : float,
In min L /\
(min <= r)%R /\ (forall f : float, In f L -> (f <= r)%R -> (f <= min)%R)).