(** * Alternate operational semantics (big-step) *)

(** We now define another semantics for Cminor without [goto] that follows
  the ``big-step'' style of semantics, also known as natural semantics.
  In this style, just like expressions evaluate to values,
  statements evaluate to``outcomes'' indicating how execution should
  proceed afterwards. *)

Inductive outcome: Type :=
  | Out_normal: outcome                (**r continue in sequence *)
  | Out_exit: nat -> outcome           (**r terminate [n+1] enclosing blocks *)
  | Out_return: option val -> outcome  (**r return immediately to caller *)
  | Out_tailcall_return: val -> outcome.(** * Alternate operational semantics (big-step) *)

(** We now define another semantics for Cminor without [goto] that follows
  the ``big-step'' style of semantics, also known as natural semantics.
  In this style, just like expressions evaluate to values,
  statements evaluate to``outcomes'' indicating how execution should
  proceed afterwards. *)

Inductive outcome: Type :=
  | Out_normal: outcome                (**r continue in sequence *)
  | Out_exit: nat -> outcome           (**r terminate [n+1] enclosing blocks *)
  | Out_return: option val -> outcome  (**r return immediately to caller *)
  | Out_tailcall_return: val -> outcome.(** * Alternate operational semantics (big-step) *)

(** We now define another semantics for Cminor without [goto] that follows
  the ``big-step'' style of semantics, also known as natural semantics.
  In this style, just like expressions evaluate to values,
  statements evaluate to``outcomes'' indicating how execution should
  proceed afterwards. *)

Inductive outcome: Type :=
  | Out_normal: outcome                (**r continue in sequence *)
  | Out_exit: nat -> outcome           (**r terminate [n+1] enclosing blocks *)
  | Out_return: option val -> outcome  (**r return immediately to caller *)
  | Out_tailcall_return: val -> outcome.(** * Alternate operational semantics (big-step) *)

(** We now define another semantics for Cminor without [goto] that follows
  the ``big-step'' style of semantics, also known as natural semantics.
  In this style, just like expressions evaluate to values,
  statements evaluate to``outcomes'' indicating how execution should
  proceed afterwards. *)

Inductive outcome: Type :=
  | Out_normal: outcome                (**r continue in sequence *)
  | Out_exit: nat -> outcome           (**r terminate [n+1] enclosing blocks *)
  | Out_return: option val -> outcome  (**r return immediately to caller *)
  | Out_tailcall_return: val -> outcome.(** * Alternate operational semantics (big-step) *)

(** We now define another semantics for Cminor without [goto] that follows
  the ``big-step'' style of semantics, also known as natural semantics.
  In this style, just like expressions evaluate to values,
  statements evaluate to``outcomes'' indicating how execution should
  proceed afterwards. *)

Inductive outcome: Type :=
  | Out_normal: outcome                (**r continue in sequence *)
  | Out_exit: nat -> outcome           (**r terminate [n+1] enclosing blocks *)
  | Out_return: option val -> outcome  (**r return immediately to caller *)
  | Out_tailcall_return: val -> outcome.(**r already returned to caller via a tailcall *)

Definition outcome_block (out: outcome) : outcome :=
  match out with
  | Out_exit O => Out_normal
  | Out_exit (S n) => Out_exit n
  | out => out
  end.Definition outcome_result_value
    (out: outcome) (retsig: option typ) (vres: val) : Prop :=
  match out with
  | Out_normal => vres = Vundef
  | Out_return None => vres = Vundef
  | Out_return (Some v) => retsig <> None /\ vres = v
  | Out_tailcall_return v => vres = v
  | _ => False
  end.Definition outcome_free_mem
    (out: outcome) (m: mem) (sp: block) (sz: Z) (m': mem) :=
  match out with
  | Out_tailcall_return _ => m' = m
  | _ => Mem.free m sp 0 sz = Some m'
  end.Scheme eval_funcall_ind2 := Minimality for eval_funcall Sort Prop
  with exec_stmt_ind2 := Minimality for exec_stmt Sort Prop.Combined Scheme eval_funcall_exec_stmt_ind2
  from eval_funcall_ind2, exec_stmt_ind2.(** Big-step execution of a whole program *)

Inductive bigstep_program_terminates (p: program): trace -> int -> Prop :=
  | bigstep_program_terminates_intro:
      forall b f m0 t m r,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      funsig f = signature_main ->
      eval_funcall ge m0 f nil t m (Vint r) ->
      bigstep_program_terminates p t r.(** Big-step execution of a whole program *)

Inductive bigstep_program_terminates (p: program): trace -> int -> Prop :=
  | bigstep_program_terminates_intro:
      forall b f m0 t m r,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      funsig f = signature_main ->
      eval_funcall ge m0 f nil t m (Vint r) ->
      bigstep_program_terminates p t r.Inductive bigstep_program_diverges (p: program): traceinf -> Prop :=
  | bigstep_program_diverges_intro:
      forall b f m0 t,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      funsig f = signature_main ->
      evalinf_funcall ge m0 f nil t ->
      bigstep_program_diverges p t.Inductive bigstep_program_diverges (p: program): traceinf -> Prop :=
  | bigstep_program_diverges_intro:
      forall b f m0 t,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      funsig f = signature_main ->
      evalinf_funcall ge m0 f nil t ->
      bigstep_program_diverges p t.Definition bigstep_semantics (p: program) :=
  Bigstep_semantics (bigstep_program_terminates p) (bigstep_program_diverges p).Inductive outcome_state_match
        (sp: val) (e: env) (m: mem) (f: function) (k: cont):
        outcome -> state -> Prop :=
  | osm_normal:
      outcome_state_match sp e m f k
                          Out_normal
                          (State f Sskip k sp e m)
  | osm_exit: forall n,
      outcome_state_match sp e m f k
                          (Out_exit n)
                          (State f (Sexit n) k sp e m)
  | osm_return_none: forall k',
      call_cont k' = call_cont k ->
      outcome_state_match sp e m f k
                          (Out_return None)
                          (State f (Sreturn None) k' sp e m)
  | osm_return_some: forall k' a v,
      call_cont k' = call_cont k ->
      eval_expr ge sp e m a v ->
      outcome_state_match sp e m f k
                          (Out_return (Some v))
                          (State f (Sreturn (Some a)) k' sp e m)
  | osm_tail: forall v,
      outcome_state_match sp e m f k
                          (Out_tailcall_return v)
                          (Returnstate v (call_cont k) m).Inductive outcome_state_match
        (sp: val) (e: env) (m: mem) (f: function) (k: cont):
        outcome -> state -> Prop :=
  | osm_normal:
      outcome_state_match sp e m f k
                          Out_normal
                          (State f Sskip k sp e m)
  | osm_exit: forall n,
      outcome_state_match sp e m f k
                          (Out_exit n)
                          (State f (Sexit n) k sp e m)
  | osm_return_none: forall k',
      call_cont k' = call_cont k ->
      outcome_state_match sp e m f k
                          (Out_return None)
                          (State f (Sreturn None) k' sp e m)
  | osm_return_some: forall k' a v,
      call_cont k' = call_cont k ->
      eval_expr ge sp e m a v ->
      outcome_state_match sp e m f k
                          (Out_return (Some v))
                          (State f (Sreturn (Some a)) k' sp e m)
  | osm_tail: forall v,
      outcome_state_match sp e m f k
                          (Out_tailcall_return v)
                          (Returnstate v (call_cont k) m).Inductive outcome_state_match
        (sp: val) (e: env) (m: mem) (f: function) (k: cont):
        outcome -> state -> Prop :=
  | osm_normal:
      outcome_state_match sp e m f k
                          Out_normal
                          (State f Sskip k sp e m)
  | osm_exit: forall n,
      outcome_state_match sp e m f k
                          (Out_exit n)
                          (State f (Sexit n) k sp e m)
  | osm_return_none: forall k',
      call_cont k' = call_cont k ->
      outcome_state_match sp e m f k
                          (Out_return None)
                          (State f (Sreturn None) k' sp e m)
  | osm_return_some: forall k' a v,
      call_cont k' = call_cont k ->
      eval_expr ge sp e m a v ->
      outcome_state_match sp e m f k
                          (Out_return (Some v))
                          (State f (Sreturn (Some a)) k' sp e m)
  | osm_tail: forall v,
      outcome_state_match sp e m f k
                          (Out_tailcall_return v)
                          (Returnstate v (call_cont k) m).Inductive outcome_state_match
        (sp: val) (e: env) (m: mem) (f: function) (k: cont):
        outcome -> state -> Prop :=
  | osm_normal:
      outcome_state_match sp e m f k
                          Out_normal
                          (State f Sskip k sp e m)
  | osm_exit: forall n,
      outcome_state_match sp e m f k
                          (Out_exit n)
                          (State f (Sexit n) k sp e m)
  | osm_return_none: forall k',
      call_cont k' = call_cont k ->
      outcome_state_match sp e m f k
                          (Out_return None)
                          (State f (Sreturn None) k' sp e m)
  | osm_return_some: forall k' a v,
      call_cont k' = call_cont k ->
      eval_expr ge sp e m a v ->
      outcome_state_match sp e m f k
                          (Out_return (Some v))
                          (State f (Sreturn (Some a)) k' sp e m)
  | osm_tail: forall v,
      outcome_state_match sp e m f k
                          (Out_tailcall_return v)
                          (Returnstate v (call_cont k) m).Inductive outcome_state_match
        (sp: val) (e: env) (m: mem) (f: function) (k: cont):
        outcome -> state -> Prop :=
  | osm_normal:
      outcome_state_match sp e m f k
                          Out_normal
                          (State f Sskip k sp e m)
  | osm_exit: forall n,
      outcome_state_match sp e m f k
                          (Out_exit n)
                          (State f (Sexit n) k sp e m)
  | osm_return_none: forall k',
      call_cont k' = call_cont k ->
      outcome_state_match sp e m f k
                          (Out_return None)
                          (State f (Sreturn None) k' sp e m)
  | osm_return_some: forall k' a v,
      call_cont k' = call_cont k ->
      eval_expr ge sp e m a v ->
      outcome_state_match sp e m f k
                          (Out_return (Some v))
                          (State f (Sreturn (Some a)) k' sp e m)
  | osm_tail: forall v,
      outcome_state_match sp e m f k
                          (Out_tailcall_return v)
                          (Returnstate v (call_cont k) m).Inductive outcome_state_match
        (sp: val) (e: env) (m: mem) (f: function) (k: cont):
        outcome -> state -> Prop :=
  | osm_normal:
      outcome_state_match sp e m f k
                          Out_normal
                          (State f Sskip k sp e m)
  | osm_exit: forall n,
      outcome_state_match sp e m f k
                          (Out_exit n)
                          (State f (Sexit n) k sp e m)
  | osm_return_none: forall k',
      call_cont k' = call_cont k ->
      outcome_state_match sp e m f k
                          (Out_return None)
                          (State f (Sreturn None) k' sp e m)
  | osm_return_some: forall k' a v,
      call_cont k' = call_cont k ->
      eval_expr ge sp e m a v ->
      outcome_state_match sp e m f k
                          (Out_return (Some v))
                          (State f (Sreturn (Some a)) k' sp e m)
  | osm_tail: forall v,
      outcome_state_match sp e m f k
                          (Out_tailcall_return v)
                          (Returnstate v (call_cont k) m).Remark is_call_cont_call_cont:
  forall k, is_call_cont (call_cont k).Remark call_cont_is_call_cont:
  forall k, is_call_cont k -> call_cont k = k.

(* Specifications: *)
Lemma eval_funcall_exec_stmt_steps: (forall m fd args t m' res, eval_funcall ge m fd args t m' res -> forall k, is_call_cont k -> star step ge (Callstate fd args k m) t (Returnstate res k m')) /\(forall f sp e m s t e' m' out, exec_stmt ge f sp e m s t e' m' out -> forall k, exists S, star step ge (State f s k sp e m) t S /\ outcome_state_match sp e' m' f k out S).
Lemma eval_funcall_steps: forall m fd args t m' res, eval_funcall ge m fd args t m' res -> forall k, is_call_cont k -> star step ge (Callstate fd args k m) t (Returnstate res k m').
Lemma exec_stmt_steps: forall f sp e m s t e' m' out, exec_stmt ge f sp e m s t e' m' out -> forall k, exists S, star step ge (State f s k sp e m) t S /\ outcome_state_match sp e' m' f k out S.
Lemma evalinf_funcall_forever: forall m fd args T k, evalinf_funcall ge m fd args T -> forever_plus step ge (Callstate fd args k m) T.
Theorem bigstep_semantics_sound: bigstep_sound (bigstep_semantics prog) (semantics prog).