Definition mpl_compat_8_def (s : state) : Prop :=
  forall (a c : ad) (pl l : prec_list),
  MapGet prec_list s c = Some l ->
  a <> c -> MapGet prec_list (union_mpl_0 a pl s) c = Some l.

Lemma union_s0d_0 : forall (d : preDTA) (c : ad) (pl : prec_list) (tl : term_list), mpl_compat (M1 prec_list c pl) (M0 prec_list) -> state_reconnait d (M1 prec_list c pl) (app c tl) -> state_reconnait d (union_mpl_0 c pl (M0 prec_list)) (app c tl).