Definition valid (x: tree) := searchtree None None x /\
(exists n, is_redblack x Red n).
(*  NOW FOR THE HARD PART, DELETION! Here we follow the deletion algorithm of Stefan Kahrs (2001), and in particular, http://www.cs.kent.ac.uk/people/staff/smk/redblack/Untyped.hs *)
Ltac hax :=
repeat (auto; try solve [repeat constructor; auto];
match goal with
| H: is_redblack _ Red _ |- _ => inv H
| H: is_redblack _ _ 0 |- _ => inv H
| H: is_redblack _ _ (S _) |- _ => inv H
| H: infrared _ 0 |- _ => inv H
| H: infrared _ (S _) |- _ => inv H
end).

Lemma searchtree_balance':
forall s1 t s2 lo hi,
searchtree lo (Some t) s1 -> searchtree (Some t) hi s2 ->
searchtree lo hi (balance' s1 t s2).