(** ** Treeify *)
Notation ifpred p n := (if p then pred n else n%nat).
Definition treeify_invariant size (f:treeify_t) :=
forall acc,
size <= length acc ->
let (t,acc') := f acc in
cardinal t = size /\ acc = elements t ++ acc'.

(** ** Filter *)
Lemma filter_aux_elements s f acc :
filter_aux f s acc = List.filter f (elements s) ++ acc.