(*****************************************************)
(**                Division by 2                     *)
Definition half (x : N.t) := snd (half_aux x).
#[global]
Instance half_aux_wd : Proper (N.eq ==> N.eq*N.eq) half_aux.
#[global]
Instance half_wd : Proper (N.eq==>N.eq) half.

Theorem half_double : forall n,
n == 2 * half n \/ n == 1 + 2 * half n.