Fixpoint countLeadingZerosWord ni no: word ni -> word no :=
match ni return word ni -> word no with
| 0 => fun _ => $ 0
| S m => fun e =>
if (weq (split2 m 1 (nat_cast (fun n => word n) (eq_sym (Nat.add_1_r m)) e)) WO~0)
then $ 1 ^+ @countLeadingZerosWord m no (split1 m 1 (nat_cast (fun n => word n) (eq_sym (Nat.add_1_r m)) e))
else $ 0
end.

Lemma combine_wplus sz (w1 w2: word sz):
#w1 + #w2 < pow2 sz ->
forall sz' (w': word sz'),
combine (w1 ^+ w2) w' = combine w1 w' ^+ combine w2 ($ 0).