(* définition inductive de la coaccessibilité *)

Inductive coacc : preDTA -> ad -> ad -> Prop :=
  | coacc_id :
      forall (d : preDTA) (a : ad) (s : state),
      MapGet state d a = Some s -> coacc d a a
  | coacc_nxt :
      forall (d : preDTA) (a0 a1 a2 : ad) (s1 s2 : state) 
        (pl : prec_list) (c : ad),
      MapGet state d a2 = Some s2 ->
      MapGet state d a1 = Some s1 ->
      MapGet prec_list s1 c = Some pl ->
      prec_occur pl a2 -> coacc d a0 a1 -> coacc d a0 a2.(* définition inductive de la coaccessibilité *)

Inductive coacc : preDTA -> ad -> ad -> Prop :=
  | coacc_id :
      forall (d : preDTA) (a : ad) (s : state),
      MapGet state d a = Some s -> coacc d a a
  | coacc_nxt :
      forall (d : preDTA) (a0 a1 a2 : ad) (s1 s2 : state) 
        (pl : prec_list) (c : ad),
      MapGet state d a2 = Some s2 ->
      MapGet state d a1 = Some s1 ->
      MapGet prec_list s1 c = Some pl ->
      prec_occur pl a2 -> coacc d a0 a1 -> coacc d a0 a2.(* définition inductive de la coaccessibilité *)

Inductive coacc : preDTA -> ad -> ad -> Prop :=
  | coacc_id :
      forall (d : preDTA) (a : ad) (s : state),
      MapGet state d a = Some s -> coacc d a a
  | coacc_nxt :
      forall (d : preDTA) (a0 a1 a2 : ad) (s1 s2 : state) 
        (pl : prec_list) (c : ad),
      MapGet state d a2 = Some s2 ->
      MapGet state d a1 = Some s1 ->
      MapGet prec_list s1 c = Some pl ->
      prec_occur pl a2 -> coacc d a0 a1 -> coacc d a0 a2.Definition coacc_transitive_def (d : preDTA) (a0 a1 : ad) : Prop :=
  forall a2 : ad, coacc d a0 a1 -> coacc d a2 a0 -> coacc d a2 a1.

(* Specifications: *)
Lemma coacc_transitive_0 : forall (d : preDTA) (a : ad) (s : state), MapGet state d a = Some s -> coacc_transitive_def d a a.
Lemma coacc_transitive_1 : forall (d : preDTA) (a0 a1 a2 : ad) (s1 s2 : state) (pl : prec_list) (c : ad), MapGet state d a2 = Some s2 -> MapGet state d a1 = Some s1 -> MapGet prec_list s1 c = Some pl -> prec_occur pl a2 -> coacc d a0 a1 -> coacc_transitive_def d a0 a1 -> coacc_transitive_def d a0 a2.
Lemma coacc_transitive : forall (d : preDTA) (a0 a1 a2 : ad), coacc d a0 a1 -> coacc d a1 a2 -> coacc d a0 a2.