intros P H' p q H'0 H'1 H'2 H'3 H'4 r H'5 H'6 H'7.
replace (FtoRradix r) with (FtoRradix (Fnormalize radix b precision r));
[ idtac | apply (FnormalizeCorrect radix) ]; auto.
replace (FtoRradix p) with (FtoRradix (Fnormalize radix b precision p));
[ idtac | apply (FnormalizeCorrect radix) ]; auto.
apply
ExactMinusIntervalAux1 with (P := P) (q := Fnormalize radix b precision q);
auto; try repeat rewrite (FnormalizeCorrect radix);
auto; apply FnormalizeCanonic; auto with arith.
Qed.
(* Properties concerning LSB MSB *)

Theorem MSBroundLSB :
forall P : R -> float -> Prop,
RoundedModeP b radix P ->
forall f1 f2 : float,
P f1 f2 ->
~ is_Fzero (Fminus radix f1 f2) ->
(MSB radix (Fminus radix f1 f2) < LSB radix f2)%Z.