(* passage au produit de la propriété pl_tl_length *)
Definition pl_tl_length_prod_def_0 (pl0 pl1 : prec_list) : Prop :=
forall (l : prec_list) (a : ad) (n m : nat),
pl_essence pl0 pl1 <= m ->
pl_tl_length pl0 n ->
pl_tl_length l (S n) \/ l = prec_empty ->
(pl_tl_length pl1 (S n) -> pl_tl_length (pl_produit_0 a pl0 pl1 m l) (S n)) /\
(pl1 = prec_empty ->
(pl_tl_length l (S n) -> pl_tl_length (pl_produit_0 a pl0 pl1 m l) (S n)) /\
(l = prec_empty -> pl_produit_0 a pl0 pl1 m l = prec_empty)).
Definition pl_tl_length_prod_def_1 (pl0 pl1 : prec_list) : Prop :=
forall n m : nat,
pl_tl_length pl0 n ->
pl_tl_length pl1 n ->
pl_essence pl0 pl1 <= m -> pl_tl_length (pl_produit_1 pl0 m pl1) n.

(* Specifications: *)
Lemma pl_tl_length_prod_0 :
forall p : prec_list, pl_tl_length_prod_def_0 p prec_empty.
Lemma pl_tl_length_prod_1 :
forall p : prec_list, pl_tl_length_prod_def_1 p prec_empty.
Lemma pl_tl_length_prod_2 :
forall p : prec_list, pl_tl_length_prod_def_1 prec_empty p.
Lemma pl_tl_length_prod_3 :
forall (a : ad) (la ls p : prec_list),
pl_tl_length_prod_def_0 p ls ->
pl_tl_length_prod_def_1 p la ->
pl_tl_length_prod_def_0 p (prec_cons a la ls).
Lemma pl_tl_length_prod_4 :
forall (a : ad) (la ls p : prec_list),
pl_tl_length_prod_def_0 la p ->
pl_tl_length_prod_def_1 ls p ->
pl_tl_length_prod_def_1 (prec_cons a la ls) p.
Lemma pl_tl_length_prod_5 :
forall p p' : prec_list,
pl_tl_length_prod_def_0 p p' /\ pl_tl_length_prod_def_1 p p'.
Lemma pl_tl_length_prod :
forall (pl0 pl1 : prec_list) (n : nat),
pl_tl_length pl0 n ->
pl_tl_length pl1 n -> pl_tl_length (pl_produit pl0 pl1) n.