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

Theorem arrayLookup_app_None :
forall (A B : Set)(eqd : EqDec A)(ls1 ls2 : list (A * B))(a : A),
arrayLookup eqd ls1 a = None ->
arrayLookup eqd (ls1 ++ ls2) a =
arrayLookup eqd ls2 a.