induction ls; intuition; simpl in *.
destruct a.
destruct p.
simpl.
case_eq (eqb a0 a); intuition.
eapply IHls.
Qed.

Theorem arrayLookup_app_Some :
forall (A B : Set)(eqd : EqDec A)(ls1 ls2 : list (A * B))(a : A)(b : B),
arrayLookup eqd ls1 a = Some b ->
arrayLookup eqd (ls1 ++ ls2) a = Some b.