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_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)).