Section suc.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Local Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionNotZero : precision <> 0.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
Definition FSucc (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.succ (Fnum x)) (Fexp x)
| false => Float (- pPred (vNum b)) (Z.pred (Fexp x))
end
| false => Float (Z.succ (Fnum x)) (Fexp x)
end
end.
intros x H'; unfold FSucc 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 FSucc 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 <= pPred (vNum b))%Z; auto with zarith arith.
rewrite <- H'1; rewrite H'.
unfold nNormMin in |- *; simpl in |- *; auto with zarith.
replace 0%Z with (- (0))%Z; auto with zarith.
unfold pPred in |- *; apply Zle_Zpred; 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 FSucc 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'1; absurd (0%nat <= pPred (vNum b))%Z; auto with zarith arith.
rewrite <- H'1.
unfold nNormMin in |- *; simpl in |- *; auto with zarith.
replace 0%Z with (- (0))%Z; auto with zarith.
unfold pPred in |- *; apply Zle_Zpred; auto with zarith.
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.
intros x H' H'0; unfold FSucc in |- *.
generalize (Z_eq_bool_correct (Fnum x) (pPred (vNum b)));
case (Z_eq_bool (Fnum x) (pPred (vNum b))); auto.
intros H'1; Contradict H'; auto.
intros H'1;
generalize (Z_eq_bool_correct (Fnum x) (- nNormMin radix precision));
case (Z_eq_bool (Fnum x) (- nNormMin radix precision));
auto.
intros H'2; Contradict H'0; auto.
Qed.

Theorem FSuccDiff1 :
forall x : float,
Fnum x <> (- nNormMin radix precision)%Z ->
Fminus radix (FSucc x) x = Float 1%nat (Fexp x) :>R.