Fixpoint countLeadingZerosWord_nat ni: word ni -> nat :=
match ni return word ni -> nat 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_nat m (split1 m 1 (nat_cast (fun n => word n) (eq_sym (Nat.add_1_r m)) e))
else 0
end.

(* These counterexamples will hopefully be found by users who use commands such as "Search ((_ ^+ _) ^% _)" *)
Lemma wmod_plus_distr_does_not_hold: ~ forall sz (a b m: word sz),
m <> $0 ->
(a ^+ b) ^% m = ((a ^% m) ^+ (b ^% m)) ^% m.