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_s0d_2 : forall m : Map prec_list, union_s_prd0 m -> forall m0 : Map prec_list, union_s_prd0 m0 -> union_s_prd0 (M2 prec_list m m0).