(* union de deux états dans un meme preDTA *)

(* définition d'un état *)

Fixpoint union_pl (pl0 : prec_list) : prec_list -> prec_list :=
  fun pl1 : prec_list =>
  match pl0 with
  | prec_empty => pl1
  | prec_cons a pl00 pl01 => prec_cons a pl00 (union_pl pl01 pl1)
  end.Fixpoint union_mpl_0 (c : ad) (pl : prec_list) (s : state) {struct s} :
 state :=
  match s with
  | M0 => M1 prec_list c pl
  | M1 c0 pl0 =>
      if Neqb c c0
      then M1 prec_list c (union_pl pl pl0)
      else MapMerge prec_list (M1 prec_list c pl) (M1 prec_list c0 pl0)
  | M2 s0 s1 =>
      match c with
      | N0 => M2 prec_list (union_mpl_0 N0 pl s0) s1
      | Npos p =>
          match p with
          | xH => M2 prec_list s0 (union_mpl_0 N0 pl s1)
          | xO p' => M2 prec_list (union_mpl_0 (Npos p') pl s0) s1
          | xI p' => M2 prec_list s0 (union_mpl_0 (Npos p') pl s1)
          end
      end
  end.Fixpoint union_mpl (s0 : state) : state -> state :=
  fun s1 : state =>
  match s0, s1 with
  | M0, M0 => M0 prec_list
  | M0, M2 s10 s11 => M2 prec_list s10 s11
  | _, M1 c1 pl1 => union_mpl_0 c1 pl1 s0
  | M1 c0 pl0, _ => union_mpl_0 c0 pl0 s1
  | M2 s00 s01, M0 => M2 prec_list s00 s01
  | M2 s00 s01, M2 s10 s11 =>
      M2 prec_list (union_mpl s00 s10) (union_mpl s01 s11)
  end.

Lemma union_pl_1 : forall pl : prec_list, union_pl prec_empty pl = pl.