(* ou l'on trouvera successivement la definition des transitions avec
les preconditions, puis l'action elle_meme, pour finir avec la
definition d'un etat legal comme etant celui obtenu a partir de l'etat
initial et d'un nombre quelconque de transitions *)

Inductive class_trans (c : Config) : Set :=
  | make_copy :
      forall s1 s2 : Site,
      s1 <> s2 -> s2 <> owner -> access s1 (rt c) -> class_trans c
  | receive_dec :
      forall s1 s2 : Site,
      first Message (bm c s1 s2) = value Message dec -> class_trans c
  | receive_inc :
      forall s1 s3 : Site,
      first Message (bm c s1 owner) = value Message (inc_dec s3) ->
      class_trans c
  | receive_copy1 :
      forall s1 s2 : Site,
      rt c s2 = true ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | receive_copy2 :
      forall s2 : Site,
      rt c s2 = false ->
      first Message (bm c owner s2) = value Message copy -> class_trans c
  | receive_copy3 :
      forall s1 s2 : Site,
      rt c s2 = false ->
      s1 <> owner ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | delete_entry :
      forall s : Site,
      st c s = 0%Z -> rt c s = true -> s <> owner -> class_trans c.(* ou l'on trouvera successivement la definition des transitions avec
les preconditions, puis l'action elle_meme, pour finir avec la
definition d'un etat legal comme etant celui obtenu a partir de l'etat
initial et d'un nombre quelconque de transitions *)

Inductive class_trans (c : Config) : Set :=
  | make_copy :
      forall s1 s2 : Site,
      s1 <> s2 -> s2 <> owner -> access s1 (rt c) -> class_trans c
  | receive_dec :
      forall s1 s2 : Site,
      first Message (bm c s1 s2) = value Message dec -> class_trans c
  | receive_inc :
      forall s1 s3 : Site,
      first Message (bm c s1 owner) = value Message (inc_dec s3) ->
      class_trans c
  | receive_copy1 :
      forall s1 s2 : Site,
      rt c s2 = true ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | receive_copy2 :
      forall s2 : Site,
      rt c s2 = false ->
      first Message (bm c owner s2) = value Message copy -> class_trans c
  | receive_copy3 :
      forall s1 s2 : Site,
      rt c s2 = false ->
      s1 <> owner ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | delete_entry :
      forall s : Site,
      st c s = 0%Z -> rt c s = true -> s <> owner -> class_trans c.(* ou l'on trouvera successivement la definition des transitions avec
les preconditions, puis l'action elle_meme, pour finir avec la
definition d'un etat legal comme etant celui obtenu a partir de l'etat
initial et d'un nombre quelconque de transitions *)

Inductive class_trans (c : Config) : Set :=
  | make_copy :
      forall s1 s2 : Site,
      s1 <> s2 -> s2 <> owner -> access s1 (rt c) -> class_trans c
  | receive_dec :
      forall s1 s2 : Site,
      first Message (bm c s1 s2) = value Message dec -> class_trans c
  | receive_inc :
      forall s1 s3 : Site,
      first Message (bm c s1 owner) = value Message (inc_dec s3) ->
      class_trans c
  | receive_copy1 :
      forall s1 s2 : Site,
      rt c s2 = true ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | receive_copy2 :
      forall s2 : Site,
      rt c s2 = false ->
      first Message (bm c owner s2) = value Message copy -> class_trans c
  | receive_copy3 :
      forall s1 s2 : Site,
      rt c s2 = false ->
      s1 <> owner ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | delete_entry :
      forall s : Site,
      st c s = 0%Z -> rt c s = true -> s <> owner -> class_trans c.(* ou l'on trouvera successivement la definition des transitions avec
les preconditions, puis l'action elle_meme, pour finir avec la
definition d'un etat legal comme etant celui obtenu a partir de l'etat
initial et d'un nombre quelconque de transitions *)

Inductive class_trans (c : Config) : Set :=
  | make_copy :
      forall s1 s2 : Site,
      s1 <> s2 -> s2 <> owner -> access s1 (rt c) -> class_trans c
  | receive_dec :
      forall s1 s2 : Site,
      first Message (bm c s1 s2) = value Message dec -> class_trans c
  | receive_inc :
      forall s1 s3 : Site,
      first Message (bm c s1 owner) = value Message (inc_dec s3) ->
      class_trans c
  | receive_copy1 :
      forall s1 s2 : Site,
      rt c s2 = true ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | receive_copy2 :
      forall s2 : Site,
      rt c s2 = false ->
      first Message (bm c owner s2) = value Message copy -> class_trans c
  | receive_copy3 :
      forall s1 s2 : Site,
      rt c s2 = false ->
      s1 <> owner ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | delete_entry :
      forall s : Site,
      st c s = 0%Z -> rt c s = true -> s <> owner -> class_trans c.(* ou l'on trouvera successivement la definition des transitions avec
les preconditions, puis l'action elle_meme, pour finir avec la
definition d'un etat legal comme etant celui obtenu a partir de l'etat
initial et d'un nombre quelconque de transitions *)

Inductive class_trans (c : Config) : Set :=
  | make_copy :
      forall s1 s2 : Site,
      s1 <> s2 -> s2 <> owner -> access s1 (rt c) -> class_trans c
  | receive_dec :
      forall s1 s2 : Site,
      first Message (bm c s1 s2) = value Message dec -> class_trans c
  | receive_inc :
      forall s1 s3 : Site,
      first Message (bm c s1 owner) = value Message (inc_dec s3) ->
      class_trans c
  | receive_copy1 :
      forall s1 s2 : Site,
      rt c s2 = true ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | receive_copy2 :
      forall s2 : Site,
      rt c s2 = false ->
      first Message (bm c owner s2) = value Message copy -> class_trans c
  | receive_copy3 :
      forall s1 s2 : Site,
      rt c s2 = false ->
      s1 <> owner ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | delete_entry :
      forall s : Site,
      st c s = 0%Z -> rt c s = true -> s <> owner -> class_trans c.(* ou l'on trouvera successivement la definition des transitions avec
les preconditions, puis l'action elle_meme, pour finir avec la
definition d'un etat legal comme etant celui obtenu a partir de l'etat
initial et d'un nombre quelconque de transitions *)

Inductive class_trans (c : Config) : Set :=
  | make_copy :
      forall s1 s2 : Site,
      s1 <> s2 -> s2 <> owner -> access s1 (rt c) -> class_trans c
  | receive_dec :
      forall s1 s2 : Site,
      first Message (bm c s1 s2) = value Message dec -> class_trans c
  | receive_inc :
      forall s1 s3 : Site,
      first Message (bm c s1 owner) = value Message (inc_dec s3) ->
      class_trans c
  | receive_copy1 :
      forall s1 s2 : Site,
      rt c s2 = true ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | receive_copy2 :
      forall s2 : Site,
      rt c s2 = false ->
      first Message (bm c owner s2) = value Message copy -> class_trans c
  | receive_copy3 :
      forall s1 s2 : Site,
      rt c s2 = false ->
      s1 <> owner ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | delete_entry :
      forall s : Site,
      st c s = 0%Z -> rt c s = true -> s <> owner -> class_trans c.(* ou l'on trouvera successivement la definition des transitions avec
les preconditions, puis l'action elle_meme, pour finir avec la
definition d'un etat legal comme etant celui obtenu a partir de l'etat
initial et d'un nombre quelconque de transitions *)

Inductive class_trans (c : Config) : Set :=
  | make_copy :
      forall s1 s2 : Site,
      s1 <> s2 -> s2 <> owner -> access s1 (rt c) -> class_trans c
  | receive_dec :
      forall s1 s2 : Site,
      first Message (bm c s1 s2) = value Message dec -> class_trans c
  | receive_inc :
      forall s1 s3 : Site,
      first Message (bm c s1 owner) = value Message (inc_dec s3) ->
      class_trans c
  | receive_copy1 :
      forall s1 s2 : Site,
      rt c s2 = true ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | receive_copy2 :
      forall s2 : Site,
      rt c s2 = false ->
      first Message (bm c owner s2) = value Message copy -> class_trans c
  | receive_copy3 :
      forall s1 s2 : Site,
      rt c s2 = false ->
      s1 <> owner ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | delete_entry :
      forall s : Site,
      st c s = 0%Z -> rt c s = true -> s <> owner -> class_trans c.(* ou l'on trouvera successivement la definition des transitions avec
les preconditions, puis l'action elle_meme, pour finir avec la
definition d'un etat legal comme etant celui obtenu a partir de l'etat
initial et d'un nombre quelconque de transitions *)

Inductive class_trans (c : Config) : Set :=
  | make_copy :
      forall s1 s2 : Site,
      s1 <> s2 -> s2 <> owner -> access s1 (rt c) -> class_trans c
  | receive_dec :
      forall s1 s2 : Site,
      first Message (bm c s1 s2) = value Message dec -> class_trans c
  | receive_inc :
      forall s1 s3 : Site,
      first Message (bm c s1 owner) = value Message (inc_dec s3) ->
      class_trans c
  | receive_copy1 :
      forall s1 s2 : Site,
      rt c s2 = true ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | receive_copy2 :
      forall s2 : Site,
      rt c s2 = false ->
      first Message (bm c owner s2) = value Message copy -> class_trans c
  | receive_copy3 :
      forall s1 s2 : Site,
      rt c s2 = false ->
      s1 <> owner ->
      first Message (bm c s1 s2) = value Message copy -> class_trans c
  | delete_entry :
      forall s : Site,
      st c s = 0%Z -> rt c s = true -> s <> owner -> class_trans c.Definition transition (c : Config) (t : class_trans c) :=
  match t with
  | make_copy s1 s2 h1 h2 h3 => copy_trans c s1 s2
  | receive_dec s1 s2 h1 => rec_dec_trans c s1 s2
  | receive_inc s1 s3 h1 => rec_inc_trans c s1 s3
  | receive_copy1 s1 s2 h1 h2 => rec_copy1_trans c s1 s2
  | receive_copy2 s2 h1 h2 => rec_copy2_trans c s2
  | receive_copy3 s1 s2 h1 h2 h3 => rec_copy3_trans c s1 s2
  | delete_entry s h1 h2 h3 => delete_trans c s
  end.Inductive legal : Config -> Prop :=
  | begin : legal config_init
  | after :
      forall (c : Config) (t : class_trans c),
      legal c -> legal (transition c t).Inductive legal : Config -> Prop :=
  | begin : legal config_init
  | after :
      forall (c : Config) (t : class_trans c),
      legal c -> legal (transition c t).Inductive legal : Config -> Prop :=
  | begin : legal config_init
  | after :
      forall (c : Config) (t : class_trans c),
      legal c -> legal (transition c t).

Lemma four : forall c : Config, legal c -> (st c owner + sigma_yi c)%Z = sigma_xi c.