induction ls; intuition; simpl in *.
remember (split ls) as z.
destruct z.
simpl in *.
intuition; subst.
rewrite eqb_refl in H.
discriminate.
destruct (eqb a a0).
discriminate.
eauto.
Qed.

Theorem arrayLookup_pair_None :
forall (A B C : Set)(eqda : EqDec A)(eqdb : EqDec B) (ls : list (A * B * C)) a b,
(forall x, In x ls -> fst (fst x) <> a) ->
arrayLookup _ ls (a, b) = None.