Definition whnf_ord (x y : approx) :=
    lexprod value (fun _ => value) (R_exp beta) (fun _ => subterm) 
      (f1 x) (f1 y).

Lemma beta_app_list_ctxt : forall (l : list term) (e : env) (u v : term), ctxt beta e u v -> ctxt beta e (app_list l u) (app_list l v).