induction n1 as [|n1 IHn1]; simpl; intros n2 n3 Heq w.
rewrite (UIP_dec eq_nat_dec Heq (refl_equal _)).
apply combine_split.
rewrite (shatter_word w) in *.
simpl.
eapply trans_eq; [ | apply IHn1 with (Heq := plus_assoc _ _ _) ]; clear IHn1.
repeat f_equal.
repeat match goal with
| [ |- context[match ?pf with refl_equal => _ end] ] => generalize pf
end.
simpl.
generalize dependent w.
rewrite plus_assoc.
intros.
rewrite (UIP_dec eq_nat_dec e (refl_equal _)).
rewrite (UIP_dec eq_nat_dec Heq0 (refl_equal _)).
reflexivity.
Qed.

Theorem eq_rect_split1 : forall n1 n1' n2 (w : word (n1' + n2)) Heq,
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) (eq_rect_split1_helper _ Heq)).