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

Theorem eq_rect_split1_eq1 : forall n1 n1' n2 (w : word (n1 + n2)) Heq Heq1, eq_rect n1 (fun n => word n) (split1 n1 n2 w) n1' Heq = split1 n1' n2 (eq_rect (n1+n2) (fun n => word n) w (n1'+n2) Heq1).