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).