(* invariant sur union_mpl_0, sens direct, 1 *)

Definition union_s_prd1 (s : state) : Prop :=
  forall (d : preDTA) (a : ad) (pl : prec_list) (c : ad) (tl : term_list),
  mpl_compat (M1 prec_list a pl) s ->
  state_reconnait d s (app c tl) ->
  state_reconnait d (union_mpl_0 a pl s) (app c tl).

Lemma union_s1d_2 : forall m : state, union_s_prd1 m -> forall m0 : state, union_s_prd1 m0 -> union_s_prd1 (M2 prec_list m m0).