Ltac shatterer := simpl; intuition;
  match goal with
    | [ w : _ |- _ ] => rewrite (shatter_word w); simpl
  end; f_equal; auto.

Theorem eq_rect_split2 : forall n1 n2 n2' (w : word (n1 + n2')) Heq, eq_rect n2' (fun n => word n) (split2 n1 n2' w) n2 Heq = split2 n1 n2 (eq_rect (n1+n2') (fun n => word n) w (n1+n2) (eq_rect_split2_helper _ Heq)).