Definition partition'  (f: key -> bool) (s: tree) : (tree*tree) :=
  match partition_aux f s (1%positive, nil) (1%positive,  nil)
   with ((n1,l1),(n2,l2)) => (fst (treeify_g n1 l1), fst (treeify_g n2 l2))
  end.

Lemma fold'_app: forall s d ar, fold' (list key) (@cons key) s (app d ar) = app (fold' (list key) (@cons key) s d) ar.