induction ls; intuition; simpl in *.
rewrite eqb_refl.
trivial.
case_eq (eqb d a0); intuition;
rewrite H0 in H.
discriminate.
eauto.
Qed.

Lemma arrayLookup_app_none : forall ls d d' v',
arrayLookup ls d = None ->
arrayLookup (ls ++ (d, v') :: nil) d' = arrayLookup ((d, v') :: ls) d'.