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

Theorem eq_rect_combine : forall n1 n2 n2' (w1 : word n1) (w2 : word n2') Heq, eq_rect (n1 + n2') (fun n => word n) (combine w1 w2) (n1 + n2) Heq = combine w1 (eq_rect n2' (fun n => word n) w2 n2 (plus_reg_l _ _ _ Heq)).