exists w; intuition.
case_eq (mod2 w); intro Hmw.
specialize (IHsz (div2 w)); firstorder.
rewrite wordToNat_wordToNat' in *.
let x' := match goal with H : _ + ?x * _ = _ |- _ => x end in
rename x' into x. (* force non-auto-generated name *)
exists x; intuition.
rewrite mult_assoc.
rewrite (mult_comm x 2).
rewrite mult_comm. simpl mult at 1.
rewrite (plus_Sn_m (2 * wordToNat' (natToWord sz (div2 w)))).
rewrite <- mult_assoc.
rewrite <- mult_plus_distr_l.
rewrite H; clear H.
symmetry; apply div2_odd; auto.
specialize (IHsz (div2 w)); firstorder.
let x' := match goal with H : _ + ?x * _ = _ |- _ => x end in
rename x' into x. (* force non-auto-generated name *)
exists x; intuition.
rewrite mult_assoc.
rewrite (mult_comm x 2).
rewrite <- mult_assoc.
rewrite mult_comm.
rewrite <- mult_plus_distr_l.
match goal with H : _ |- _ => rewrite H; clear H end.
symmetry; apply div2_even; auto.
Qed.

Theorem WS_neq : forall b1 b2 sz (w1 w2 : word sz),
(b1 <> b2 \/ w1 <> w2)
-> WS b1 w1 <> WS b2 w2.