induction ls; intuition; simpl in *.
discriminate.
case_eq (eqb d a0); intuition.
rewrite H0 in H.
inversion H; clear H; subst.
rewrite eqb_leibniz in H0. subst.
case_eq (eqb d' a0); intuition.
eapply arrayLookup_app_ne.
intuition.
subst.
rewrite eqb_refl in H.
discriminate.
rewrite H0 in H.
case_eq (eqb d' a0); intuition.
eapply IHls; eauto.
Qed.

Theorem arrayLookup_arrayUpdate_eq :
forall arr p v,
arrayLookup (arrayUpdate arr p (Some v)) p = (Some v).