intros x H; unfold FPred, FSucc in |- *.
cut (Fbounded b x); [ intros Fb0 | apply FcanonicBound with (1 := H) ].
generalize (Z_eq_bool_correct (Fnum x) (pPred (vNum b)));
case (Z_eq_bool (Fnum x) (pPred (vNum b))); simpl in |- *.
generalize (Z_eq_bool_correct (nNormMin radix precision) (- pPred (vNum b)));
case (Z_eq_bool (nNormMin radix precision) (- pPred (vNum b)));
simpl in |- *.
intros H'; Contradict H'; apply sym_not_equal; apply Zlt_not_eq; auto.
apply Zlt_le_trans with (- 0%nat)%Z.
apply Zlt_Zopp; unfold pPred in |- *; apply Zlt_succ_pred; simpl in |- *;
apply vNumbMoreThanOne with (3 := pGivesBound); auto.
simpl in |- *; apply Zlt_le_weak; apply nNormPos; auto.
generalize
(Z_eq_bool_correct (nNormMin radix precision) (nNormMin radix precision));
case (Z_eq_bool (nNormMin radix precision) (nNormMin radix precision));
simpl in |- *.
generalize (Z_eq_bool_correct (Zsucc (Fexp x)) (- dExp b));
case (Z_eq_bool (Zsucc (Fexp x)) (- dExp b)); simpl in |- *.
intros H' H'0 H'1 H'2; absurd (- dExp b <= Fexp x)%Z; auto with float.
rewrite <- H'; auto with float zarith.
replace (Zpred (Zsucc (Fexp x))) with (Fexp x);
[ idtac | unfold Zsucc, Zpred in |- *; ring ]; auto.
intros H' H'0 H'1 H'2; rewrite <- H'2; auto.
apply floatEq; auto.
intros H'; case H'; auto.
generalize (Z_eq_bool_correct (Fnum x) (- nNormMin radix precision));
case (Z_eq_bool (Fnum x) (- nNormMin radix precision));
simpl in |- *.
generalize (Z_eq_bool_correct (Fexp x) (- dExp b));
case (Z_eq_bool (Fexp x) (- dExp b)); simpl in |- *.
generalize (Z_eq_bool_correct (Zsucc (Fnum x)) (- pPred (vNum b)));
case (Z_eq_bool (Zsucc (Fnum x)) (- pPred (vNum b)));
simpl in |- *.
intros H0 H1 H2; absurd (Zsucc (Fnum x) <= Fnum x)%Z; auto with zarith.
rewrite H0; rewrite H2; (apply Zle_Zopp; auto with float arith).
unfold pPred in |- *; apply Zle_Zpred; apply ZltNormMinVnum; auto with zarith.
generalize (Z_eq_bool_correct (Zsucc (Fnum x)) (nNormMin radix precision));
case (Z_eq_bool (Zsucc (Fnum x)) (nNormMin radix precision));
simpl in |- *.
intros H' H'0 H'1 H'2; Contradict H'2.
rewrite <- H'; auto with zarith.
replace (Zpred (Zsucc (Fnum x))) with (Fnum x);
[ idtac | unfold Zsucc, Zpred in |- *; ring ]; auto.
intros H' H'0 H'1 H'2 H'3; apply floatEq; auto.
generalize (Z_eq_bool_correct (- pPred (vNum b)) (- pPred (vNum b)));
case (Z_eq_bool (- pPred (vNum b)) (- pPred (vNum b)));
auto.
intros H' H'0 H'1 H'2; rewrite <- H'1.
replace (Zsucc (Zpred (Fexp x))) with (Fexp x);
[ idtac | unfold Zsucc, Zpred in |- *; ring ]; auto.
apply floatEq; auto.
intros H'; case H'; auto.
generalize (Z_eq_bool_correct (Zsucc (Fnum x)) (- pPred (vNum b)));
case (Z_eq_bool (Zsucc (Fnum x)) (- pPred (vNum b)));
simpl in |- *.
intros H'; absurd (- pPred (vNum b) <= Fnum x)%Z; auto with float.
rewrite <- H'; auto with zarith.
apply Zle_Zabs_inv1; auto with float.
unfold pPred in |- *; apply Zle_Zpred; auto with float.
generalize (Z_eq_bool_correct (Zsucc (Fnum x)) (nNormMin radix precision));
case (Z_eq_bool (Zsucc (Fnum x)) (nNormMin radix precision));
simpl in |- *.
generalize (Z_eq_bool_correct (Fexp x) (- dExp b));
case (Z_eq_bool (Fexp x) (- dExp b)); simpl in |- *.
intros H' H'0 H'1 H'2 H'3.
replace (Zpred (Zsucc (Fnum x))) with (Fnum x);
[ idtac | unfold Zsucc, Zpred in |- *; ring ]; auto.
apply floatEq; auto.
intros H' H'0 H'1 H'2 H'3; case H.
intros H'4; absurd (nNormMin radix precision <= Zabs (Fnum x))%Z.
replace (Fnum x) with (Zpred (Zsucc (Fnum x)));
[ idtac | unfold Zsucc, Zpred in |- *; ring ]; auto.
rewrite H'0.
apply Zlt_not_le; rewrite Zabs_eq; auto with zarith.
apply Zle_Zpred; apply nNormPos; auto with float zarith.
apply pNormal_absolu_min with (b := b); auto.
intros H'4; Contradict H'; apply FsubnormalFexp with (1 := H'4).
intros H' H'0 H'1 H'2; apply floatEq; simpl in |- *; auto.
unfold Zpred, Zsucc in |- *; ring.
Qed.

Theorem FSucPred :
forall x : float,
Fcanonic radix b x -> FSucc b radix precision (FPred x) = x.