Section pred.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Local Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Hypothesis precisionNotZero : precision <> 0.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
Definition FPred (x : float) :=
match Z_eq_bool (Fnum x) (- pPred (vNum b)) with
| true => Float (- nNormMin radix precision) (Z.succ (Fexp x))
| false =>
match Z_eq_bool (Fnum x) (nNormMin radix precision) with
| true =>
match Z_eq_bool (Fexp x) (- dExp b) with
| true => Float (Z.pred (Fnum x)) (Fexp x)
| false => Float (pPred (vNum b)) (Z.pred (Fexp x))
end
| false => Float (Z.pred (Fnum x)) (Fexp x)
end
end.
intros x H'; unfold FPred in |- *.
generalize (Z_eq_bool_correct (Fnum x) (- pPred (vNum b)));
case (Z_eq_bool (Fnum x) (- pPred (vNum b))); auto.
intros H'0; Contradict H'0; auto.
Qed.
intros x H' H'0; unfold FPred in |- *.
generalize (Z_eq_bool_correct (Fnum x) (- pPred (vNum b)));
case (Z_eq_bool (Fnum x) (- pPred (vNum b))); auto.
intros H'1; absurd (0%nat < Fnum x)%Z; auto with zarith arith.
apply Zle_not_lt; rewrite H'1; replace (Z_of_nat 0) with (- (0))%Z;
[ apply Zle_Zopp | simpl in |- *; auto ].
unfold pPred in |- *; apply Zle_Zpred; red in |- *; simpl in |- *; auto.
rewrite H'.
apply nNormPos; auto with zarith.
intros H'1;
generalize (Z_eq_bool_correct (Fnum x) (nNormMin radix precision));
case (Z_eq_bool (Fnum x) (nNormMin radix precision)).
intros H'2; generalize (Z_eq_bool_correct (Fexp x) (- dExp b));
case (Z_eq_bool (Fexp x) (- dExp b)); auto.
intros H'3; Contradict H'0; auto.
intros H'2; Contradict H'2; auto.
Qed.
unfold FPred in |- *; simpl in |- *.
generalize (Z_eq_bool_correct (nNormMin radix precision) (- pPred (vNum b)));
case (Z_eq_bool (nNormMin radix precision) (- pPred (vNum b)));
auto.
intros H'0; absurd (0 < pPred (vNum b))%Z; auto with zarith arith.
rewrite <- (Z.opp_involutive (pPred (vNum b))); rewrite <- H'0.
apply Zle_not_lt; replace 0%Z with (- (0))%Z;
[ apply Zle_Zopp | simpl in |- *; auto ].
apply Zlt_le_weak; apply nNormPos; auto with float zarith.
unfold pPred in |- *; apply Zlt_succ_pred; simpl in |- *;
auto with float zarith.
simpl in |- *; apply vNumbMoreThanOne with (3 := pGivesBound); auto.
intros H';
generalize
(Z_eq_bool_correct (nNormMin radix precision) (nNormMin radix precision));
case (Z_eq_bool (nNormMin radix precision) (nNormMin radix precision)).
intros H'0; generalize (Z_eq_bool_correct (- dExp b) (- dExp b));
case (Z_eq_bool (- dExp b) (- dExp b)); auto.
intros H'1; Contradict H'1; auto.
intros H'1; Contradict H'1; auto.
Qed.

Theorem FPredSimpl4 :
forall x : float,
Fnum x <> (- pPred (vNum b))%Z ->
Fnum x <> nNormMin radix precision ->
FPred x = Float (Z.pred (Fnum x)) (Fexp x).