(** An alternate presentation of infinite traces as
  infinite concatenations of nonempty finite traces. *)

CoInductive traceinf': Type :=
  | Econsinf': forall (t: trace) (T: traceinf'), t <> E0 -> traceinf'.(** An alternate presentation of infinite traces as
  infinite concatenations of nonempty finite traces. *)

CoInductive traceinf': Type :=
  | Econsinf': forall (t: trace) (T: traceinf'), t <> E0 -> traceinf'.Program Definition split_traceinf' (t: trace) (T: traceinf') (NE: t <> E0): event * traceinf' :=
  match t with
  | nil => _
  | e :: nil => (e, T)
  | e :: t' => (e, Econsinf' t' T _)
  end.CoFixpoint traceinf_of_traceinf' (T': traceinf') : traceinf :=
  match T' with
  | Econsinf' t T'' NOTEMPTY =>
      let (e, tl) := split_traceinf' t T'' NOTEMPTY in
      Econsinf e (traceinf_of_traceinf' tl)
  end.Remark unroll_traceinf':
  forall T, T = match T with Econsinf' t T' NE => Econsinf' t T' NE end.Remark unroll_traceinf:
  forall T, T = match T with Econsinf t T' => Econsinf t T' end.(** Translation between values and event values. *)

Inductive eventval_match: eventval -> typ -> val -> Prop :=
  | ev_match_int: forall i,
      eventval_match (EVint i) Tint (Vint i)
  | ev_match_long: forall i,
      eventval_match (EVlong i) Tlong (Vlong i)
  | ev_match_float: forall f,
      eventval_match (EVfloat f) Tfloat (Vfloat f)
  | ev_match_single: forall f,
      eventval_match (EVsingle f) Tsingle (Vsingle f)
  | ev_match_ptr: forall id b ofs,
      Senv.public_symbol ge id = true ->
      Senv.find_symbol ge id = Some b ->
      eventval_match (EVptr_global id ofs) Tptr (Vptr b ofs).(** Translation between values and event values. *)

Inductive eventval_match: eventval -> typ -> val -> Prop :=
  | ev_match_int: forall i,
      eventval_match (EVint i) Tint (Vint i)
  | ev_match_long: forall i,
      eventval_match (EVlong i) Tlong (Vlong i)
  | ev_match_float: forall f,
      eventval_match (EVfloat f) Tfloat (Vfloat f)
  | ev_match_single: forall f,
      eventval_match (EVsingle f) Tsingle (Vsingle f)
  | ev_match_ptr: forall id b ofs,
      Senv.public_symbol ge id = true ->
      Senv.find_symbol ge id = Some b ->
      eventval_match (EVptr_global id ofs) Tptr (Vptr b ofs).(** Translation between values and event values. *)

Inductive eventval_match: eventval -> typ -> val -> Prop :=
  | ev_match_int: forall i,
      eventval_match (EVint i) Tint (Vint i)
  | ev_match_long: forall i,
      eventval_match (EVlong i) Tlong (Vlong i)
  | ev_match_float: forall f,
      eventval_match (EVfloat f) Tfloat (Vfloat f)
  | ev_match_single: forall f,
      eventval_match (EVsingle f) Tsingle (Vsingle f)
  | ev_match_ptr: forall id b ofs,
      Senv.public_symbol ge id = true ->
      Senv.find_symbol ge id = Some b ->
      eventval_match (EVptr_global id ofs) Tptr (Vptr b ofs).(** Translation between values and event values. *)

Inductive eventval_match: eventval -> typ -> val -> Prop :=
  | ev_match_int: forall i,
      eventval_match (EVint i) Tint (Vint i)
  | ev_match_long: forall i,
      eventval_match (EVlong i) Tlong (Vlong i)
  | ev_match_float: forall f,
      eventval_match (EVfloat f) Tfloat (Vfloat f)
  | ev_match_single: forall f,
      eventval_match (EVsingle f) Tsingle (Vsingle f)
  | ev_match_ptr: forall id b ofs,
      Senv.public_symbol ge id = true ->
      Senv.find_symbol ge id = Some b ->
      eventval_match (EVptr_global id ofs) Tptr (Vptr b ofs).(** Translation between values and event values. *)

Inductive eventval_match: eventval -> typ -> val -> Prop :=
  | ev_match_int: forall i,
      eventval_match (EVint i) Tint (Vint i)
  | ev_match_long: forall i,
      eventval_match (EVlong i) Tlong (Vlong i)
  | ev_match_float: forall f,
      eventval_match (EVfloat f) Tfloat (Vfloat f)
  | ev_match_single: forall f,
      eventval_match (EVsingle f) Tsingle (Vsingle f)
  | ev_match_ptr: forall id b ofs,
      Senv.public_symbol ge id = true ->
      Senv.find_symbol ge id = Some b ->
      eventval_match (EVptr_global id ofs) Tptr (Vptr b ofs).(** Translation between values and event values. *)

Inductive eventval_match: eventval -> typ -> val -> Prop :=
  | ev_match_int: forall i,
      eventval_match (EVint i) Tint (Vint i)
  | ev_match_long: forall i,
      eventval_match (EVlong i) Tlong (Vlong i)
  | ev_match_float: forall f,
      eventval_match (EVfloat f) Tfloat (Vfloat f)
  | ev_match_single: forall f,
      eventval_match (EVsingle f) Tsingle (Vsingle f)
  | ev_match_ptr: forall id b ofs,
      Senv.public_symbol ge id = true ->
      Senv.find_symbol ge id = Some b ->
      eventval_match (EVptr_global id ofs) Tptr (Vptr b ofs).Inductive eventval_list_match: list eventval -> list typ -> list val -> Prop :=
  | evl_match_nil:
      eventval_list_match nil nil nil
  | evl_match_cons:
      forall ev1 evl ty1 tyl v1 vl,
      eventval_match ev1 ty1 v1 ->
      eventval_list_match evl tyl vl ->
      eventval_list_match (ev1::evl) (ty1::tyl) (v1::vl).Inductive eventval_list_match: list eventval -> list typ -> list val -> Prop :=
  | evl_match_nil:
      eventval_list_match nil nil nil
  | evl_match_cons:
      forall ev1 evl ty1 tyl v1 vl,
      eventval_match ev1 ty1 v1 ->
      eventval_list_match evl tyl vl ->
      eventval_list_match (ev1::evl) (ty1::tyl) (v1::vl).Inductive eventval_list_match: list eventval -> list typ -> list val -> Prop :=
  | evl_match_nil:
      eventval_list_match nil nil nil
  | evl_match_cons:
      forall ev1 evl ty1 tyl v1 vl,
      eventval_match ev1 ty1 v1 ->
      eventval_list_match evl tyl vl ->
      eventval_list_match (ev1::evl) (ty1::tyl) (v1::vl).

(* Specifications: *)
Lemma eventval_match_type: forall ev ty v, eventval_match ev ty v -> Val.has_type v ty.
Lemma eventval_list_match_length: forall evl tyl vl, eventval_list_match evl tyl vl -> List.length vl = List.length tyl.
Lemma eventval_match_lessdef: forall ev ty v1 v2, eventval_match ev ty v1 -> Val.lessdef v1 v2 -> eventval_match ev ty v2.
Lemma eventval_list_match_lessdef: forall evl tyl vl1, eventval_list_match evl tyl vl1 -> forall vl2, Val.lessdef_list vl1 vl2 -> eventval_list_match evl tyl vl2.
Lemma eventval_match_determ_1: forall ev ty v1 v2, eventval_match ev ty v1 -> eventval_match ev ty v2 -> v1 = v2.
Lemma eventval_match_determ_2: forall ev1 ev2 ty v, eventval_match ev1 ty v -> eventval_match ev2 ty v -> ev1 = ev2.
Lemma eventval_list_match_determ_2: forall evl1 tyl vl, eventval_list_match evl1 tyl vl -> forall evl2, eventval_list_match evl2 tyl vl -> evl1 = evl2.
Lemma eventval_match_type: forall ev ty v, eventval_match ev ty v -> Val.has_type v ty.
Lemma eventval_list_match_length: forall evl tyl vl, eventval_list_match evl tyl vl -> List.length vl = List.length tyl.
Lemma eventval_match_lessdef: forall ev ty v1 v2, eventval_match ev ty v1 -> Val.lessdef v1 v2 -> eventval_match ev ty v2.
Lemma eventval_list_match_lessdef: forall evl tyl vl1, eventval_list_match evl tyl vl1 -> forall vl2, Val.lessdef_list vl1 vl2 -> eventval_list_match evl tyl vl2.
Lemma eventval_match_determ_1: forall ev ty v1 v2, eventval_match ev ty v1 -> eventval_match ev ty v2 -> v1 = v2.
Lemma eventval_match_determ_2: forall ev1 ev2 ty v, eventval_match ev1 ty v -> eventval_match ev2 ty v -> ev1 = ev2.
Lemma eventval_list_match_determ_2: forall evl1 tyl vl, eventval_list_match evl1 tyl vl -> forall evl2, eventval_list_match evl2 tyl vl -> evl1 = evl2.