(* Some basic facts about the relativization operation: It is injective when a is nonempty; a number of facts about nonemptyness, nonfullness and canonical trees; it commutes with union, intersection, mkCanon; and it is an associative operation. *)

Lemma relativ_inv : forall a x y,
nonEmptyTree a ->
shareTreeOrd (relativ_tree a x) (relativ_tree a y) ->
shareTreeOrd x y.