(** * Events and traces *)

(** The observable behaviour of programs is stated in terms of
  input/output events, which represent the actions of the program
  that the external world can observe.  CompCert leaves much flexibility as to
  the exact content of events: the only requirement is that they
  do not expose memory states nor pointer values
  (other than pointers to global variables), because these
  are not preserved literally during compilation.  For concreteness,
  we use the following type for events.  Each event represents either:

- A system call (e.g. an input/output operation), recording the
  name of the system call, its parameters, and its result.

- A volatile load from a global memory location, recording the chunk
  and address being read and the value just read.

- A volatile store to a global memory location, recording the chunk
  and address being written and the value stored there.

- An annotation, recording the text of the annotation and the values
  of the arguments.

  The values attached to these events are of the following form.
  As mentioned above, we do not expose pointer values directly.
  Pointers relative to a global variable are shown with the name
  of the variable instead of the block identifier.
*)

Inductive eventval: Type :=
  | EVint: int -> eventval
  | EVlong: int64 -> eventval
  | EVfloat: float -> eventval
  | EVsingle: float32 -> eventval
  | EVptr_global: ident -> ptrofs -> eventval.(** * Events and traces *)

(** The observable behaviour of programs is stated in terms of
  input/output events, which represent the actions of the program
  that the external world can observe.  CompCert leaves much flexibility as to
  the exact content of events: the only requirement is that they
  do not expose memory states nor pointer values
  (other than pointers to global variables), because these
  are not preserved literally during compilation.  For concreteness,
  we use the following type for events.  Each event represents either:

- A system call (e.g. an input/output operation), recording the
  name of the system call, its parameters, and its result.

- A volatile load from a global memory location, recording the chunk
  and address being read and the value just read.

- A volatile store to a global memory location, recording the chunk
  and address being written and the value stored there.

- An annotation, recording the text of the annotation and the values
  of the arguments.

  The values attached to these events are of the following form.
  As mentioned above, we do not expose pointer values directly.
  Pointers relative to a global variable are shown with the name
  of the variable instead of the block identifier.
*)

Inductive eventval: Type :=
  | EVint: int -> eventval
  | EVlong: int64 -> eventval
  | EVfloat: float -> eventval
  | EVsingle: float32 -> eventval
  | EVptr_global: ident -> ptrofs -> eventval.(** * Events and traces *)

(** The observable behaviour of programs is stated in terms of
  input/output events, which represent the actions of the program
  that the external world can observe.  CompCert leaves much flexibility as to
  the exact content of events: the only requirement is that they
  do not expose memory states nor pointer values
  (other than pointers to global variables), because these
  are not preserved literally during compilation.  For concreteness,
  we use the following type for events.  Each event represents either:

- A system call (e.g. an input/output operation), recording the
  name of the system call, its parameters, and its result.

- A volatile load from a global memory location, recording the chunk
  and address being read and the value just read.

- A volatile store to a global memory location, recording the chunk
  and address being written and the value stored there.

- An annotation, recording the text of the annotation and the values
  of the arguments.

  The values attached to these events are of the following form.
  As mentioned above, we do not expose pointer values directly.
  Pointers relative to a global variable are shown with the name
  of the variable instead of the block identifier.
*)

Inductive eventval: Type :=
  | EVint: int -> eventval
  | EVlong: int64 -> eventval
  | EVfloat: float -> eventval
  | EVsingle: float32 -> eventval
  | EVptr_global: ident -> ptrofs -> eventval.(** * Events and traces *)

(** The observable behaviour of programs is stated in terms of
  input/output events, which represent the actions of the program
  that the external world can observe.  CompCert leaves much flexibility as to
  the exact content of events: the only requirement is that they
  do not expose memory states nor pointer values
  (other than pointers to global variables), because these
  are not preserved literally during compilation.  For concreteness,
  we use the following type for events.  Each event represents either:

- A system call (e.g. an input/output operation), recording the
  name of the system call, its parameters, and its result.

- A volatile load from a global memory location, recording the chunk
  and address being read and the value just read.

- A volatile store to a global memory location, recording the chunk
  and address being written and the value stored there.

- An annotation, recording the text of the annotation and the values
  of the arguments.

  The values attached to these events are of the following form.
  As mentioned above, we do not expose pointer values directly.
  Pointers relative to a global variable are shown with the name
  of the variable instead of the block identifier.
*)

Inductive eventval: Type :=
  | EVint: int -> eventval
  | EVlong: int64 -> eventval
  | EVfloat: float -> eventval
  | EVsingle: float32 -> eventval
  | EVptr_global: ident -> ptrofs -> eventval.(** * Events and traces *)

(** The observable behaviour of programs is stated in terms of
  input/output events, which represent the actions of the program
  that the external world can observe.  CompCert leaves much flexibility as to
  the exact content of events: the only requirement is that they
  do not expose memory states nor pointer values
  (other than pointers to global variables), because these
  are not preserved literally during compilation.  For concreteness,
  we use the following type for events.  Each event represents either:

- A system call (e.g. an input/output operation), recording the
  name of the system call, its parameters, and its result.

- A volatile load from a global memory location, recording the chunk
  and address being read and the value just read.

- A volatile store to a global memory location, recording the chunk
  and address being written and the value stored there.

- An annotation, recording the text of the annotation and the values
  of the arguments.

  The values attached to these events are of the following form.
  As mentioned above, we do not expose pointer values directly.
  Pointers relative to a global variable are shown with the name
  of the variable instead of the block identifier.
*)

Inductive eventval: Type :=
  | EVint: int -> eventval
  | EVlong: int64 -> eventval
  | EVfloat: float -> eventval
  | EVsingle: float32 -> eventval
  | EVptr_global: ident -> ptrofs -> eventval.(** * Events and traces *)

(** The observable behaviour of programs is stated in terms of
  input/output events, which represent the actions of the program
  that the external world can observe.  CompCert leaves much flexibility as to
  the exact content of events: the only requirement is that they
  do not expose memory states nor pointer values
  (other than pointers to global variables), because these
  are not preserved literally during compilation.  For concreteness,
  we use the following type for events.  Each event represents either:

- A system call (e.g. an input/output operation), recording the
  name of the system call, its parameters, and its result.

- A volatile load from a global memory location, recording the chunk
  and address being read and the value just read.

- A volatile store to a global memory location, recording the chunk
  and address being written and the value stored there.

- An annotation, recording the text of the annotation and the values
  of the arguments.

  The values attached to these events are of the following form.
  As mentioned above, we do not expose pointer values directly.
  Pointers relative to a global variable are shown with the name
  of the variable instead of the block identifier.
*)

Inductive eventval: Type :=
  | EVint: int -> eventval
  | EVlong: int64 -> eventval
  | EVfloat: float -> eventval
  | EVsingle: float32 -> eventval
  | EVptr_global: ident -> ptrofs -> eventval.Notation access_map := (Maps.PMap.t (Z -> option permission)).Notation delta_map := (Maps.PTree.t (Z -> option (option permission))).Notation delta_content := (Maps.PTree.t (Z -> option memval)).Inductive event: Type :=
  | Event_syscall: string -> list eventval -> eventval -> event
  | Event_vload: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_vstore: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_annot: string -> list eventval -> event
  | Event_acq: val -> delta_map -> delta_content -> event
  | Event_rel: val -> delta_map -> delta_content -> event
  | Event_spawn: val -> delta_map -> delta_content -> delta_map -> delta_content -> event.Inductive event: Type :=
  | Event_syscall: string -> list eventval -> eventval -> event
  | Event_vload: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_vstore: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_annot: string -> list eventval -> event
  | Event_acq: val -> delta_map -> delta_content -> event
  | Event_rel: val -> delta_map -> delta_content -> event
  | Event_spawn: val -> delta_map -> delta_content -> delta_map -> delta_content -> event.Inductive event: Type :=
  | Event_syscall: string -> list eventval -> eventval -> event
  | Event_vload: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_vstore: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_annot: string -> list eventval -> event
  | Event_acq: val -> delta_map -> delta_content -> event
  | Event_rel: val -> delta_map -> delta_content -> event
  | Event_spawn: val -> delta_map -> delta_content -> delta_map -> delta_content -> event.Inductive event: Type :=
  | Event_syscall: string -> list eventval -> eventval -> event
  | Event_vload: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_vstore: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_annot: string -> list eventval -> event
  | Event_acq: val -> delta_map -> delta_content -> event
  | Event_rel: val -> delta_map -> delta_content -> event
  | Event_spawn: val -> delta_map -> delta_content -> delta_map -> delta_content -> event.Inductive event: Type :=
  | Event_syscall: string -> list eventval -> eventval -> event
  | Event_vload: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_vstore: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_annot: string -> list eventval -> event
  | Event_acq: val -> delta_map -> delta_content -> event
  | Event_rel: val -> delta_map -> delta_content -> event
  | Event_spawn: val -> delta_map -> delta_content -> delta_map -> delta_content -> event.Inductive event: Type :=
  | Event_syscall: string -> list eventval -> eventval -> event
  | Event_vload: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_vstore: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_annot: string -> list eventval -> event
  | Event_acq: val -> delta_map -> delta_content -> event
  | Event_rel: val -> delta_map -> delta_content -> event
  | Event_spawn: val -> delta_map -> delta_content -> delta_map -> delta_content -> event.Inductive event: Type :=
  | Event_syscall: string -> list eventval -> eventval -> event
  | Event_vload: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_vstore: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_annot: string -> list eventval -> event
  | Event_acq: val -> delta_map -> delta_content -> event
  | Event_rel: val -> delta_map -> delta_content -> event
  | Event_spawn: val -> delta_map -> delta_content -> delta_map -> delta_content -> event.Inductive event: Type :=
  | Event_syscall: string -> list eventval -> eventval -> event
  | Event_vload: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_vstore: memory_chunk -> ident -> ptrofs -> eventval -> event
  | Event_annot: string -> list eventval -> event
  | Event_acq: val -> delta_map -> delta_content -> event
  | Event_rel: val -> delta_map -> delta_content -> event
  | Event_spawn: val -> delta_map -> delta_content -> delta_map -> delta_content -> event.(** The dynamic semantics for programs collect traces of events.
  Traces are of two kinds: finite (type [trace]) or infinite (type [traceinf]). *)

Definition trace := list event.Definition E0 : trace := nil.Definition Eapp (t1 t2: trace) : trace := t1 ++ t2.CoInductive traceinf : Type :=
  | Econsinf: event -> traceinf -> traceinf.CoInductive traceinf : Type :=
  | Econsinf: event -> traceinf -> traceinf.Fixpoint Eappinf (t: trace) (T: traceinf) {struct t} : traceinf :=
  match t with
  | nil => T
  | ev :: t' => Econsinf ev (Eappinf t' T)
  end.Infix "***" := Eappinf (at level 60, right associativity).(** The following [traceEq] tactic proves equalities between traces
  or infinite traces. *)

Ltac substTraceHyp :=
  match goal with
  | [ H: (@eq trace ?x ?y) |- _ ] =>
       subst x || clear H
  end.Ltac traceEq :=
  repeat substTraceHyp; autorewrite with trace_rewrite; decomposeTraceEq.(** Bisimilarity between infinite traces. *)

CoInductive traceinf_sim: traceinf -> traceinf -> Prop :=
  | traceinf_sim_cons: forall e T1 T2,
      traceinf_sim T1 T2 ->
      traceinf_sim (Econsinf e T1) (Econsinf e T2).(** Bisimilarity between infinite traces. *)

CoInductive traceinf_sim: traceinf -> traceinf -> Prop :=
  | traceinf_sim_cons: forall e T1 T2,
      traceinf_sim T1 T2 ->
      traceinf_sim (Econsinf e T1) (Econsinf e T2).

Lemma traceinf_sim'_sim: forall T1 T2, traceinf_sim' T1 T2 -> traceinf_sim T1 T2.