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).

(* Specifications: *)
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).
Lemma union_s0d_3 : union_s_prd0 (M0 prec_list).
Lemma union_s0d_1 :
forall (a : ad) (a0 : prec_list), union_s_prd0 (M1 prec_list a a0).
Lemma union_s_0d : forall m : state, union_s_prd0 m.
Lemma union_s0d :
forall (s : state) (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).