intros x y H' H'0 H'1; rewrite (MSB_abs x); rewrite (MSB_abs y).
case (Zle_or_lt (Fexp (Fabs x)) (Fexp (Fabs y))); simpl in |- *; intros Zle1.
rewrite
MSB_shift with (x := Fabs y) (n := Zabs_nat (Fexp (Fabs y) - Fexp (Fabs x))).
apply MSB_monotoneAux; auto.
unfold FtoRradix in |- *; repeat rewrite Fabs_correct; auto with real arith.
rewrite FshiftCorrect; auto with real arith.
repeat rewrite Fabs_correct; auto with real arith.
repeat rewrite Rabs_Rabsolu; repeat rewrite <- Fabs_correct;
auto with real arith.
unfold Fshift in |- *; simpl in |- *.
rewrite inj_abs; [ ring | auto with zarith ].
apply Fabs_Fzero; auto.
rewrite
MSB_shift with (x := Fabs x) (n := Zabs_nat (Fexp (Fabs x) - Fexp (Fabs y))).
apply MSB_monotoneAux; auto.
unfold FtoRradix in |- *; repeat rewrite Fabs_correct; auto with real arith.
rewrite FshiftCorrect; auto with real arith.
repeat rewrite Fabs_correct; auto with real arith.
repeat rewrite Rabs_Rabsolu; repeat rewrite <- Fabs_correct;
auto with real arith.
unfold Fshift in |- *; simpl in |- *.
rewrite inj_abs; [ ring | auto with zarith ].
apply Fabs_Fzero; auto.
Qed.

Theorem MSB_le_multAux :
forall x y : float,
~ is_Fzero x -> ~ is_Fzero y -> (MSB x + MSB y <= MSB (Fmult x y))%Z.