(*****************************************************)
(**            Logarithm for the base 2              *)
#[global]
Instance log_prewd :
Proper ((N.eq==>N.eq)==>N.eq==>N.eq)
(fun g x => if x<<2 then 0 else S (g (half x))).
#[global]
Instance log_wd : Proper (N.eq==>N.eq) log.

Lemma log_good_step : forall n h1 h2,
(forall m, m < n -> h1 m == h2 m) ->
(if n << 2 then 0 else S (h1 (half n))) ==
(if n << 2 then 0 else S (h2 (half n))).