induction l using list_ind'; intros.
+ simpl in *; inversion H; subst.
exists f; intros.
split; auto.
+ simpl in H; destruct a; inversion H.
+ pose proof IHl p; clear IHl.
destruct a; destruct b; simpl in *.
-  pose proof (H0 (fun p0 => xO (f p0))); clear H0.
apply H1 in H; clear H1.
assumption.
-  pose proof (H0 (fun p0 => xI (f p0))); clear H0.
apply H1 in H; clear H1.
assumption.
- inversion H.
- inversion H.
Qed.

(* Specifications: *)
Lemma posPathInj : forall p, pathToPos (posToPath p) = Some p.
induction p; unfold posToPath, pathToPos in *; simpl in *.