induction ls1; intuition; simpl in *.
case_eq (eqb a a0); intuition;
rewrite H0 in H.
discriminate.
eauto.
Qed.

Theorem arrayLookupList_pred :
forall (A B : Set) (eqda : EqDec A)P (ls : list (A * list B)) a,
(forall x, In x ls -> P (snd x)) ->
P nil ->
P (arrayLookupList _ ls a).