Definition union' (s1 s2: tree) : tree :=
match compare_height s1 s1 s2 s2 with
| Lt => fold' _ insert s1 s2
| Gt => fold' _ insert s2 s1
| Eq => linear_union s1 s2
end.

Lemma filter'_spec : forall s x f,
compatb f ->
valid s ->
(interp (filter' f s) x <-> interp s x /\ f x = true).