intros f1 f2 H' H'0 H'1 H'2.
apply Zle_antisym.
2: apply MSB_monotone; auto.
2: repeat rewrite Fabs_correct1; auto with arith.
2: apply isMin_inv1 with (1 := H'0).
2: apply RleRoundedR0 with (P := isMin b radix) (r := FtoRradix f1); auto.
2: apply MinRoundedModeP with (precision := precision); auto.
case (Zle_or_lt (MSB radix f1) (MSB radix f2)); auto.
intros H'3; absurd (Float 1%nat (Zsucc (MSB radix f2)) <= f2)%R.
apply Rgt_not_le.
red in |- *; unfold FtoRradix in |- *; rewrite <- Fabs_correct1;
auto with float arith.
apply abs_lt_MSB; auto.
apply RleRoundedR0 with (P := isMin b radix) (r := FtoRradix f1);
auto with float.
apply MinRoundedModeP with (precision := precision); auto.
case H'0.
intros H'4 H'5; elim H'5; intros H'6 H'7; apply H'7; clear H'5; auto.
apply (FboundedOne _ radixMoreThanOne b precision); auto with arith.
apply Zle_trans with (Fexp f2).
case H'4; auto.
apply Zle_trans with (MSB radix f2); auto with zarith.
apply Fexp_le_MSB; auto.
apply Rle_trans with (FtoR radix (Float 1%nat (MSB radix f1))); auto.
apply oneExp_le; auto with zarith.
unfold FtoRradix in |- *; rewrite <- Fabs_correct1 with (x := f1);
auto with float arith.
apply MSB_le_abs; auto.
Qed.

Theorem MSBtoZero :
forall f1 f2 : float,
ToZeroP b radix f1 f2 ->
~ is_Fzero f1 -> ~ is_Fzero f2 -> MSB radix f1 = MSB radix f2.