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

Theorem split1_eq_rect_eq1 : forall a a' b H w, split1 a b w = eq_rect _ word (split1 a' b (eq_rect _ word w _ (split1_eq_rect_eq1_helper b H))) _ H.