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).