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.
intros z H; case H; intros z1 H1; exists (- z1)%Z; rewrite H1; ring.
Qed.
Hint Resolve OddOpp EvenOpp: zarith.
intros z; case z; simpl in |- *; auto with zarith.
intros p; case p; simpl in |- *; auto with zarith.
intros p1; left; exists (Zpos p1); rewrite Zplus_comm;
simpl in |- *; auto.
intros p1; right; exists (Zpos p1); simpl in |- *; auto.
change
(forall p : positive,
{Odd (- Zpos p)} + {Even (- Zpos p)})
in |- *.
intros p; case p; auto with zarith.
intros p1; left; apply OddOpp; exists (Zpos p1);
rewrite Zplus_comm; simpl in |- *; auto.
intros p1; right; apply EvenOpp; exists (Zpos p1); simpl in |- *; auto.
Qed.
intros n H1; red in |- *; intros H2; case H1; case H2; intros z1 Hz1 z2 Hz2.
absurd (n = n); auto.
pattern n at 1 in |- *; rewrite Hz1; rewrite Hz2;
repeat rewrite (fun x => Zplus_comm x 1).
case z1; case z2; simpl in |- *;
try (intros; red in |- *; intros; discriminate).
intros p p0; case p; simpl in |- *;
try (intros; red in |- *; intros; discriminate).
Qed.
intros n H1; red in |- *; intros H2; case H1; case H2; intros z1 Hz1 z2 Hz2.
absurd (n = n); auto.
pattern n at 1 in |- *; rewrite Hz1; rewrite Hz2;
repeat rewrite (fun x => Zplus_comm x 1).
case z1; case z2; simpl in |- *;
try (intros; red in |- *; intros; discriminate).
intros p p0; case p0; simpl in |- *;
try (intros; red in |- *; intros; discriminate).
Qed.
Hint Resolve OddNEven EvenNOdd: zarith.
intros n m H H0; case H; case H0; intros z1 Hz1 z2 Hz2.
exists (z2 + z1)%Z; try rewrite Hz1; try rewrite Hz2; ring.
Qed.
intros n m H H0; case H; case H0; intros z1 Hz1 z2 Hz2.
exists (z2 + z1 + 1)%Z; try rewrite Hz1; try rewrite Hz2; ring.
Qed.
intros n m H H0; case H; case H0; intros z1 Hz1 z2 Hz2.
exists (z2 + z1)%Z; try rewrite Hz1; try rewrite Hz2; ring.
Qed.
intros n m H H0; case H; case H0; intros z1 Hz1 z2 Hz2.
exists (z2 + z1)%Z; try rewrite Hz1; try rewrite Hz2; ring.
Qed.
Hint Resolve EvenPlus1 EvenPlus2 OddPlus1 OddPlus2: zarith.
intros n m H H0; replace m with (n + m + - n)%Z; auto with zarith.
Qed.
intros n m H H0; replace m with (n + m + - n)%Z; auto with zarith.
Qed.
intros n m H H0; replace n with (n + m + - m)%Z; auto with zarith.
Qed.
intros n m H H0; replace n with (n + m + - m)%Z; auto with zarith.
Qed.
intros n m H; case H; intros z1 Hz1; exists (z1 * m)%Z; rewrite Hz1; ring.
Qed.
intros n m H; case H; intros z1 Hz1; exists (z1 * n)%Z; rewrite Hz1; ring.
Qed.
Hint Resolve EvenMult1 EvenMult2: zarith.
intros n m H1 H2; case H1; case H2; intros z1 Hz1 z2 Hz2;
exists (2 * z1 * z2 + z1 + z2)%Z; rewrite Hz1; rewrite Hz2;
ring.
Qed.
Hint Resolve OddMult: zarith.
intros n m H H0; case (OddEvenDec m); auto; intros Z1.
Contradict H; auto with zarith.
Qed.

Theorem OddMultInv : forall n m : Z, Odd (n * m) -> Odd n.