Definition pl_produit_ref_ok_0 (la pl : prec_list) : Prop :=
  forall (a b : ad) (l : prec_list) (n : nat),
  prec_occur (pl_produit_0 a la pl n l) b ->
  (exists a0 : ad,
     (exists a1 : ad,
        b = iad_conv a0 a1 /\ prec_occur la a0 /\ prec_occur pl a1)) \/
  (exists a1 : ad, b = iad_conv a a1 /\ prec_occur pl a1) \/ prec_occur l b.Definition pl_produit_ref_ok_1 (p0 p1 : prec_list) : Prop :=
  forall (b : ad) (n : nat),
  prec_occur (pl_produit_1 p0 n p1) b ->
  exists a0 : ad,
    (exists a1 : ad,
       b = iad_conv a0 a1 /\ prec_occur p0 a0 /\ prec_occur p1 a1).

Lemma inter_DTA_main_state_correct_invar : forall d0 d1 : DTA, DTA_main_state_correct d0 -> DTA_main_state_correct d1 -> DTA_main_state_correct (inter d0 d1).