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 fold'_valid: forall (f: key -> tree -> tree)
(f_valid: forall k s, valid s -> valid (f k s))
s1 s2,
valid s1 -> valid s2 -> valid (fold' _ f s1 s2).