Notation Zeven_bool_succ := Z.even_succ (only parsing).
Notation Zeven_bool_pred := Z.even_pred (only parsing).
Notation Zodd_bool_succ := Z.odd_succ (only parsing).
Notation Zodd_bool_pred := Z.odd_pred (only parsing).
(******************************************************************)
(** * Definition of [Z.quot2], [Z.div2] and properties wrt [Zeven] and [Zodd] *)
(** Properties of [Z.div2] *)

Lemma Zdiv2_odd_eqn n : n = 2*(Z.div2 n) + if Z.odd n then 1 else 0.
Proof (Z.div2_odd n).