induction ls1; intuition; simpl in *.
inversion H; clear H; subst.
unfold arrayLookupList.
simpl.
trivial.
inversion H; clear H; subst.
unfold arrayLookupList.
simpl.
destruct a2.
simpl in *.
intuition.
case_eq (eqb a a0); intuition.
rewrite eqb_leibniz in H2.
subst.
intuition.
subst.
rewrite eqb_refl.
trivial.
case_eq (eqb a a1); intuition.
rewrite eqb_leibniz in H3.
subst.
intuition.
subst.
rewrite eqb_refl in H2.
discriminate.
eapply IHls1; intuition.
Qed.

(* Specifications: *)
Theorem In_arrayLookupList :
forall (A B : Set)(eqda : EqDec A)(a : A)(b : B) ls,
In b (arrayLookupList _ ls a) ->
exists ls', In (a, ls') ls /\
In b ls'.