induction ls; intuition; simpl in *.
unfold arrayLookupList.
simpl.
case_eq (eqb a a0); intuition.
specialize (X (a0, b)).
intuition.
eapply IHls; intuition.
Qed.

Theorem arrayLookup_Some_impl_In :
forall (A B : Set)(eqd : EqDec A)(ls : list (A * B)) a b,
arrayLookup _ ls a = Some b ->
In (a, b) ls.