Definition union_s_rpl_def (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 (union_mpl_0 a pl s) (app c tl) ->
  state_reconnait d (M1 prec_list a pl) (app c tl) \/
  state_reconnait d s (app c tl).

(* Specifications: *)
Lemma union_s_rpl_0 : union_s_rpl_def (M0 prec_list).
Lemma union_s_rpl_1 : forall (a : ad) (a0 : prec_list), union_s_rpl_def (M1 prec_list a a0).
Lemma union_s_rpl_2 : forall m : state, union_s_rpl_def m -> forall m0 : state, union_s_rpl_def m0 -> union_s_rpl_def (M2 prec_list m m0).
Lemma union_s_rpl_3 : forall m : state, union_s_rpl_def m.
Lemma union_s_rpl : forall (s : state) (d : preDTA) (a : ad) (pl : prec_list) (c : ad) (tl : term_list), mpl_compat (M1 prec_list a pl) s -> state_reconnait d (union_mpl_0 a pl s) (app c tl) -> state_reconnait d (M1 prec_list a pl) (app c tl) \/ state_reconnait d s (app c tl).