(* Path length and merge *)
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.

Lemma pathlen_some:
forall a a',
M.get a uf.(m) = Some a' ->
pathlen a = S (pathlen a').