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_1 : forall (plp : pl_path) (a : ad) (la ls : prec_list), pl_path_incl plp la -> pl_path_incl_length_def plp la -> pl_path_incl_length_def (pl_path_cons a plp) (prec_cons a la ls).