induction ls using rev_ind; intuition; simpl in *.
econstructor.
rewrite app_length.
rewrite plus_comm.
simpl.
eapply list_pred_rev.
eapply list_pred_impl'.
eapply IHls.
intuition.
rewrite app_nth1.
eauto.
apply allNatsLt_lt.
trivial.
intuition.
rewrite app_nth2.
rewrite minus_diag.
simpl.
trivial.
intuition.
Qed.

(* Specifications: *)
Lemma compMap_length :
forall (A B : Set)(eqd : EqDec B)(ls : list A) x (c : A -> Comp B) ,
In x (getSupport (compMap _ c ls)) ->
length x = length ls.