(**********************************************************************)
(** About parity *)

Notation two_or_two_plus_one := Z_modulo_2 (only parsing).(**********************************************************************)
(** The biggest power of 2 that is stricly less than [a]

    Easy to compute: replace all "1" of the binary representation by
    "0", except the first "1" (or the first one :-) *)

Fixpoint floor_pos (a:positive) : positive :=
  match a with
    | xH => 1%positive
    | xO a' => xO (floor_pos a')
    | xI b' => xO (floor_pos b')
  end.Definition floor (a:positive) := Zpos (floor_pos a).

Theorem Z_lt_abs_induction : forall P:Z -> Prop, (forall n:Z, (forall m:Z, Z.abs m < Z.abs n -> P m) -> P n) -> forall n:Z, P n.