Section FOdd.
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 precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
(* We define the parity predicates*)
Definition Even (z : Z) : Prop := exists z1 : _, z = (2 * z1)%Z.
Definition Odd (z : Z) : Prop := exists z1 : _, z = (2 * z1 + 1)%Z.
intros n H'; case H'; intros m H'1; exists (Z.succ m).
rewrite H'1; unfold Z.succ in |- *; ring.
Qed.
intros n H'; case H'; intros m H'1; exists m.
rewrite H'1; unfold Z.succ in |- *; ring.
Qed.
Hint Resolve OddSEven EvenSOdd: zarith.
intros n H'; case H'; intros m H'1; exists m.
apply Z.succ_inj; rewrite H'1; (unfold Z.succ in |- *; ring).
Qed.
intros n H'; case H'; intros m H'1; exists (Z.pred m).
apply Z.succ_inj; rewrite H'1; (unfold Z.succ, Z.pred in |- *; ring).
Qed.
exists 0%Z; simpl in |- *; auto.
Qed.
Hint Resolve EvenO: zarith.
exists 0%Z; simpl in |- *; auto.
Qed.
Hint Resolve Odd1: zarith.
intros z H; case H; intros z1 H1; exists (- Z.succ z1)%Z; rewrite H1.
unfold Z.succ in |- *; ring.
Qed.

Theorem EvenOpp : forall z : Z, Even z -> Even (- z).