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 balance'_shape: forall n tl k tr, (is_redblack tl Black n /\ infrared tr n \/ infrared tl n /\ is_redblack tr Black n) -> is_redblack (balance' tl k tr) Black (S n).