Definition pl_path_incl_length_def (plp : pl_path) 
  (pl : prec_list) : Prop :=
  forall n : nat,
  pl_path_incl plp pl -> pl_tl_length pl n -> pl_path_length plp = n.

Lemma pl_path_incl_length_2 : forall (plp : pl_path) (a : ad) (la ls : prec_list), pl_path_incl plp ls -> pl_path_incl_length_def plp ls -> plp <> pl_path_nil -> pl_path_incl_length_def plp (prec_cons a la ls).