(** * Transition semantics *)

(** The general form of a transition semantics. *)

Record semantics : Type := Semantics_gen {
  state: Type;
  genvtype: Type;
  step : genvtype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: genvtype;
  symbolenv: Senv.t
}.(** * Transition semantics *)

(** The general form of a transition semantics. *)

Record semantics : Type := Semantics_gen {
  state: Type;
  genvtype: Type;
  step : genvtype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: genvtype;
  symbolenv: Senv.t
}.(** * Transition semantics *)

(** The general form of a transition semantics. *)

Record semantics : Type := Semantics_gen {
  state: Type;
  genvtype: Type;
  step : genvtype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: genvtype;
  symbolenv: Senv.t
}.(** * Transition semantics *)

(** The general form of a transition semantics. *)

Record semantics : Type := Semantics_gen {
  state: Type;
  genvtype: Type;
  step : genvtype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: genvtype;
  symbolenv: Senv.t
}.(** * Transition semantics *)

(** The general form of a transition semantics. *)

Record semantics : Type := Semantics_gen {
  state: Type;
  genvtype: Type;
  step : genvtype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: genvtype;
  symbolenv: Senv.t
}.(** * Transition semantics *)

(** The general form of a transition semantics. *)

Record semantics : Type := Semantics_gen {
  state: Type;
  genvtype: Type;
  step : genvtype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: genvtype;
  symbolenv: Senv.t
}.(** * Transition semantics *)

(** The general form of a transition semantics. *)

Record semantics : Type := Semantics_gen {
  state: Type;
  genvtype: Type;
  step : genvtype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: genvtype;
  symbolenv: Senv.t
}.(** * Transition semantics *)

(** The general form of a transition semantics. *)

Record semantics : Type := Semantics_gen {
  state: Type;
  genvtype: Type;
  step : genvtype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: genvtype;
  symbolenv: Senv.t
}.(** * Transition semantics *)

(** The general form of a transition semantics. *)

Record semantics : Type := Semantics_gen {
  state: Type;
  genvtype: Type;
  step : genvtype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: genvtype;
  symbolenv: Senv.t
}.(** The form used in earlier CompCert versions, for backward compatibility. *)

Definition Semantics {state funtype vartype: Type}
                     (step: Genv.t funtype vartype -> state -> trace -> state -> Prop)
                     (initial_state: state -> Prop)
                     (final_state: state -> int -> Prop)
                     (globalenv: Genv.t funtype vartype) :=
  {| state := state;
     genvtype := Genv.t funtype vartype;
     step := step;
     initial_state := initial_state;
     final_state := final_state;
     globalenv := globalenv;
     symbolenv := Genv.to_senv globalenv |}.(** Handy notations. *)

Notation " 'Step' L " := (step L (globalenv L)) (at level 1) : smallstep_scope.Notation " 'Star' L " := (star (step L) (globalenv L)) (at level 1) : smallstep_scope.Notation " 'Plus' L " := (plus (step L) (globalenv L)) (at level 1) : smallstep_scope.Notation " 'Forever_silent' L " := (forever_silent (step L) (globalenv L)) (at level 1) : smallstep_scope.Notation " 'Forever_reactive' L " := (forever_reactive (step L) (globalenv L)) (at level 1) : smallstep_scope.Notation " 'Nostep' L " := (nostep (step L) (globalenv L)) (at level 1) : smallstep_scope.(** * Forward simulations between two transition semantics. *)

(** The general form of a forward simulation. *)

Record fsim_properties (L1 L2: semantics) (index: Type)
                       (order: index -> index -> Prop)
                       (match_states: index -> state L1 -> state L2 -> Prop) : Prop := {
    fsim_order_wf: well_founded order;
    fsim_match_initial_states:
      forall s1, initial_state L1 s1 ->
      exists i, exists s2, initial_state L2 s2 /\ match_states i s1 s2;
    fsim_match_final_states:
      forall i s1 s2 r,
      match_states i s1 s2 -> final_state L1 s1 r -> final_state L2 s2 r;
    fsim_simulation:
      forall s1 t s1', Step L1 s1 t s1' ->
      forall i s2, match_states i s1 s2 ->
      exists i', exists s2',
         (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ order i' i))
      /\ match_states i' s1' s2';
    fsim_public_preserved:
      forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
  }.(** * Forward simulations between two transition semantics. *)

(** The general form of a forward simulation. *)

Record fsim_properties (L1 L2: semantics) (index: Type)
                       (order: index -> index -> Prop)
                       (match_states: index -> state L1 -> state L2 -> Prop) : Prop := {
    fsim_order_wf: well_founded order;
    fsim_match_initial_states:
      forall s1, initial_state L1 s1 ->
      exists i, exists s2, initial_state L2 s2 /\ match_states i s1 s2;
    fsim_match_final_states:
      forall i s1 s2 r,
      match_states i s1 s2 -> final_state L1 s1 r -> final_state L2 s2 r;
    fsim_simulation:
      forall s1 t s1', Step L1 s1 t s1' ->
      forall i s2, match_states i s1 s2 ->
      exists i', exists s2',
         (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ order i' i))
      /\ match_states i' s1' s2';
    fsim_public_preserved:
      forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
  }.(** * Forward simulations between two transition semantics. *)

(** The general form of a forward simulation. *)

Record fsim_properties (L1 L2: semantics) (index: Type)
                       (order: index -> index -> Prop)
                       (match_states: index -> state L1 -> state L2 -> Prop) : Prop := {
    fsim_order_wf: well_founded order;
    fsim_match_initial_states:
      forall s1, initial_state L1 s1 ->
      exists i, exists s2, initial_state L2 s2 /\ match_states i s1 s2;
    fsim_match_final_states:
      forall i s1 s2 r,
      match_states i s1 s2 -> final_state L1 s1 r -> final_state L2 s2 r;
    fsim_simulation:
      forall s1 t s1', Step L1 s1 t s1' ->
      forall i s2, match_states i s1 s2 ->
      exists i', exists s2',
         (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ order i' i))
      /\ match_states i' s1' s2';
    fsim_public_preserved:
      forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
  }.(** * Forward simulations between two transition semantics. *)

(** The general form of a forward simulation. *)

Record fsim_properties (L1 L2: semantics) (index: Type)
                       (order: index -> index -> Prop)
                       (match_states: index -> state L1 -> state L2 -> Prop) : Prop := {
    fsim_order_wf: well_founded order;
    fsim_match_initial_states:
      forall s1, initial_state L1 s1 ->
      exists i, exists s2, initial_state L2 s2 /\ match_states i s1 s2;
    fsim_match_final_states:
      forall i s1 s2 r,
      match_states i s1 s2 -> final_state L1 s1 r -> final_state L2 s2 r;
    fsim_simulation:
      forall s1 t s1', Step L1 s1 t s1' ->
      forall i s2, match_states i s1 s2 ->
      exists i', exists s2',
         (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ order i' i))
      /\ match_states i' s1' s2';
    fsim_public_preserved:
      forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
  }.(** * Forward simulations between two transition semantics. *)

(** The general form of a forward simulation. *)

Record fsim_properties (L1 L2: semantics) (index: Type)
                       (order: index -> index -> Prop)
                       (match_states: index -> state L1 -> state L2 -> Prop) : Prop := {
    fsim_order_wf: well_founded order;
    fsim_match_initial_states:
      forall s1, initial_state L1 s1 ->
      exists i, exists s2, initial_state L2 s2 /\ match_states i s1 s2;
    fsim_match_final_states:
      forall i s1 s2 r,
      match_states i s1 s2 -> final_state L1 s1 r -> final_state L2 s2 r;
    fsim_simulation:
      forall s1 t s1', Step L1 s1 t s1' ->
      forall i s2, match_states i s1 s2 ->
      exists i', exists s2',
         (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ order i' i))
      /\ match_states i' s1' s2';
    fsim_public_preserved:
      forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
  }.(** * Forward simulations between two transition semantics. *)

(** The general form of a forward simulation. *)

Record fsim_properties (L1 L2: semantics) (index: Type)
                       (order: index -> index -> Prop)
                       (match_states: index -> state L1 -> state L2 -> Prop) : Prop := {
    fsim_order_wf: well_founded order;
    fsim_match_initial_states:
      forall s1, initial_state L1 s1 ->
      exists i, exists s2, initial_state L2 s2 /\ match_states i s1 s2;
    fsim_match_final_states:
      forall i s1 s2 r,
      match_states i s1 s2 -> final_state L1 s1 r -> final_state L2 s2 r;
    fsim_simulation:
      forall s1 t s1', Step L1 s1 t s1' ->
      forall i s2, match_states i s1 s2 ->
      exists i', exists s2',
         (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ order i' i))
      /\ match_states i' s1' s2';
    fsim_public_preserved:
      forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
  }.Inductive forward_simulation (L1 L2: semantics) : Prop :=
  Forward_simulation (index: Type)
                     (order: index -> index -> Prop)
                     (match_states: index -> state L1 -> state L2 -> Prop)
                     (props: fsim_properties L1 L2 index order match_states).Inductive forward_simulation (L1 L2: semantics) : Prop :=
  Forward_simulation (index: Type)
                     (order: index -> index -> Prop)
                     (match_states: index -> state L1 -> state L2 -> Prop)
                     (props: fsim_properties L1 L2 index order match_states).

(* Specifications: *)
Lemma fsim_simulation': forall L1 L2 index order match_states, fsim_properties L1 L2 index order match_states -> forall i s1 t s1', Step L1 s1 t s1' -> forall s2, match_states i s1 s2 -> (exists i', exists s2', Plus L2 s2 t s2' /\ match_states i' s1' s2') \/ (exists i', order i' i /\ t = E0 /\ match_states i' s1' s2).
Lemma forward_simulation_star_wf: forward_simulation L1 L2.
Lemma forward_simulation_star: forward_simulation L1 L2.
Lemma forward_simulation_plus: forward_simulation L1 L2.
Lemma forward_simulation_step: forward_simulation L1 L2.
Lemma forward_simulation_opt: forward_simulation L1 L2.
Lemma simulation_star: forall s1 t s1', Star L1 s1 t s1' -> forall i s2, match_states i s1 s2 -> exists i', exists s2', Star L2 s2 t s2' /\ match_states i' s1' s2'.
Lemma simulation_plus: forall s1 t s1', Plus L1 s1 t s1' -> forall i s2, match_states i s1 s2 -> (exists i', exists s2', Plus L2 s2 t s2' /\ match_states i' s1' s2') \/ (exists i', clos_trans _ order i' i /\ t = E0 /\ match_states i' s1' s2).
Lemma simulation_forever_silent: forall i s1 s2, Forever_silent L1 s1 -> match_states i s1 s2 -> Forever_silent L2 s2.
Lemma simulation_forever_reactive: forall i s1 s2 T, Forever_reactive L1 s1 T -> match_states i s1 s2 -> Forever_reactive L2 s2 T.
Lemma compose_forward_simulations: forall L1 L2 L3, forward_simulation L1 L2 -> forward_simulation L2 L3 -> forward_simulation L1 L3.