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 union_valid: forall s1 s2, valid s1 -> valid s2 -> valid (union' s1 s2).
Definition diff' (s1 s2: tree) : tree :=
match compare_height s1 s1 s2 s2 with
| Lt => filter' (fun k => negb (member k s2)) s1
| Gt => fold' _ delete s2 s1
| Eq => linear_diff s1 s2
end.