induction ls; intuition; simpl in *.
discriminate.
destruct i.
inversion H0; clear H0; subst.
intuition.
eapply IHls; eauto.
Qed.

(* Specifications: *)
Theorem nth_firstIndexOf_None :
forall (A : Set)(eqd : eq_dec A)(ls : list A),
NoDup ls ->
forall (a a' : A) i,
In a ls ->
i <> firstIndexOf eqd ls a O ->
nth_option ls i = Some a' ->
a <> a'.