Definition union_s_prd0 (s : state) : Prop :=
  forall (d : preDTA) (c : ad) (pl : prec_list) (tl : term_list),
  mpl_compat (M1 prec_list c pl) s ->
  state_reconnait d (M1 prec_list c pl) (app c tl) ->
  state_reconnait d (union_mpl_0 c pl s) (app c tl).

Lemma union_s_0d : forall m : state, union_s_prd0 m.