Section PATHLEN.
Variable uf: t.
Definition F_pathlen (a: elt) (rec: forall b, order uf.(m) b a -> nat) : nat :=
match getlink uf.(m) a with
| inleft (exist a' P) => S (rec a' P)
| inright _ => O
end.
Definition pathlen (a: elt) : nat := Fix uf.(mwf) (fun _ => nat) F_pathlen a.
End PATHLEN.
(* Path length and merge *)

Lemma pathlen_gt_merge:
forall uf a b x y,
repr uf x = repr uf y ->
pathlen uf x > pathlen uf y ->
pathlen (merge uf a b) x > pathlen (merge uf a b) y.