Definition share_metric (n : nat) (s : canonTree) : nat :=
 match n with
 | 0 => 0
 | S n => match le_dec (height s) n with
          | left _ => countBLeafCT n s + 1
          | right _ => 0
         end
 end.

Lemma share_metric_lub : forall s s' n, ~(s'<=s)%ba-> 0 < share_metric n s -> 0 < share_metric n (lub s s') -> (share_metric n s<share_metric n (lub s s')).