(* Axioms about splittability. These follow easily from relativization. *)
exists (Node (Leaf true) (Leaf false)).
simpl; intuition.
Defined.
Definition rightTree : canonTree.
exists (Node (Leaf false) (Leaf true)).
simpl; intuition.
Defined.
Definition split (x:canonTree) := (rel x leftTree, rel x rightTree).

Lemma split_nontrivial : forall x1 x2 x,
split x = (x1, x2) ->
(x1 = bot \/ x2 = bot) ->
x = bot.