induction ls2; intuition; simpl in *.
intuition; subst.
unfold setLet.
destruct (in_dec eqd a0 ls1); intuition.
unfold setLet.
destruct (in_dec eqd a ls1); subst.
eauto.
simpl.
destruct (eqd a a0); subst.
intuition.
right.
eauto.
Qed.

(* Specifications: *)
Lemma lookupIndex_lt_length :
forall (A : Set)(eqd : eq_dec A)(ls : list A) a def,
In a ls ->
lookupIndex eqd ls a def < length ls.