(* une relation de compatibilité entre automates *)

Definition pl_compat (pl0 pl1 : prec_list) : Prop :=
  pl0 = prec_empty /\ pl1 = prec_empty \/
  pl0 <> prec_empty /\ pl1 <> prec_empty.Definition mpl_compat (s0 s1 : state) : Prop :=
  forall (c : ad) (p0 p1 : prec_list),
  MapGet prec_list s0 c = Some p0 ->
  MapGet prec_list s1 c = Some p1 -> pl_compat p0 p1.Definition dta_correct (d : preDTA) : Prop :=
  forall (s0 s1 : state) (a0 a1 : ad),
  MapGet state d a0 = Some s0 ->
  MapGet state d a1 = Some s1 -> mpl_compat s0 s1.Definition dta_compat (d0 d1 : preDTA) : Prop :=
  forall (s0 s1 : state) (a0 a1 : ad),
  MapGet state d0 a0 = Some s0 ->
  MapGet state d1 a1 = Some s1 -> mpl_compat s0 s1.Definition DTA_compat (d0 d1 : DTA) : Prop :=
  match d0, d1 with
  | dta p0 a0, dta p1 a1 => dta_compat p0 p1
  end.

Lemma mpl_compat_3 : forall (s0 s1 : state) (pl : prec_list), mpl_compat (M2 prec_list s0 s1) (M1 prec_list N0 pl) -> mpl_compat s0 (M1 prec_list N0 pl).