(** Relating execution states *)

Inductive match_states: state -> state -> Prop :=
  | match_regular_states:
      forall f s k e le m tf ts tk te tle tm j lo hi tlo thi
        (TRF: transf_function f = OK tf)
        (TRS: simpl_stmt (cenv_for f) s = OK ts)
        (MENV: match_envs j (cenv_for f) e le m lo hi te tle tlo thi)
        (MCONT: match_cont j (cenv_for f) k tk m lo tlo)
        (MINJ: Mem.inject j m tm)
        (COMPAT: compat_cenv (addr_taken_stmt s) (cenv_for f))
        (BOUND: Ple hi (Mem.nextblock m))
        (TBOUND: Ple thi (Mem.nextblock tm)),
      match_states (State f s k e le m)
                   (State tf ts tk te tle tm)
  | match_call_state:
      forall fd vargs k m tfd tvargs tk tm j targs tres cconv
        (TRFD: transf_fundef fd = OK tfd)
        (MCONT: forall cenv, match_cont j cenv k tk m (Mem.nextblock m) (Mem.nextblock tm))
        (MINJ: Mem.inject j m tm)
        (AINJ: Val.inject_list j vargs tvargs)
        (FUNTY: type_of_fundef fd = Tfunction targs tres cconv)
        (ANORM: val_casted_list vargs targs),
      match_states (Callstate fd vargs k m)
                   (Callstate tfd tvargs tk tm)
  | match_return_state:
      forall v k m tv tk tm j
        (MCONT: forall cenv, match_cont j cenv k tk m (Mem.nextblock m) (Mem.nextblock tm))
        (MINJ: Mem.inject j m tm)
        (RINJ: Val.inject j v tv),
      match_states (Returnstate v k m)
                   (Returnstate tv tk tm).(** Relating execution states *)

Inductive match_states: state -> state -> Prop :=
  | match_regular_states:
      forall f s k e le m tf ts tk te tle tm j lo hi tlo thi
        (TRF: transf_function f = OK tf)
        (TRS: simpl_stmt (cenv_for f) s = OK ts)
        (MENV: match_envs j (cenv_for f) e le m lo hi te tle tlo thi)
        (MCONT: match_cont j (cenv_for f) k tk m lo tlo)
        (MINJ: Mem.inject j m tm)
        (COMPAT: compat_cenv (addr_taken_stmt s) (cenv_for f))
        (BOUND: Ple hi (Mem.nextblock m))
        (TBOUND: Ple thi (Mem.nextblock tm)),
      match_states (State f s k e le m)
                   (State tf ts tk te tle tm)
  | match_call_state:
      forall fd vargs k m tfd tvargs tk tm j targs tres cconv
        (TRFD: transf_fundef fd = OK tfd)
        (MCONT: forall cenv, match_cont j cenv k tk m (Mem.nextblock m) (Mem.nextblock tm))
        (MINJ: Mem.inject j m tm)
        (AINJ: Val.inject_list j vargs tvargs)
        (FUNTY: type_of_fundef fd = Tfunction targs tres cconv)
        (ANORM: val_casted_list vargs targs),
      match_states (Callstate fd vargs k m)
                   (Callstate tfd tvargs tk tm)
  | match_return_state:
      forall v k m tv tk tm j
        (MCONT: forall cenv, match_cont j cenv k tk m (Mem.nextblock m) (Mem.nextblock tm))
        (MINJ: Mem.inject j m tm)
        (RINJ: Val.inject j v tv),
      match_states (Returnstate v k m)
                   (Returnstate tv tk tm).(** Relating execution states *)

Inductive match_states: state -> state -> Prop :=
  | match_regular_states:
      forall f s k e le m tf ts tk te tle tm j lo hi tlo thi
        (TRF: transf_function f = OK tf)
        (TRS: simpl_stmt (cenv_for f) s = OK ts)
        (MENV: match_envs j (cenv_for f) e le m lo hi te tle tlo thi)
        (MCONT: match_cont j (cenv_for f) k tk m lo tlo)
        (MINJ: Mem.inject j m tm)
        (COMPAT: compat_cenv (addr_taken_stmt s) (cenv_for f))
        (BOUND: Ple hi (Mem.nextblock m))
        (TBOUND: Ple thi (Mem.nextblock tm)),
      match_states (State f s k e le m)
                   (State tf ts tk te tle tm)
  | match_call_state:
      forall fd vargs k m tfd tvargs tk tm j targs tres cconv
        (TRFD: transf_fundef fd = OK tfd)
        (MCONT: forall cenv, match_cont j cenv k tk m (Mem.nextblock m) (Mem.nextblock tm))
        (MINJ: Mem.inject j m tm)
        (AINJ: Val.inject_list j vargs tvargs)
        (FUNTY: type_of_fundef fd = Tfunction targs tres cconv)
        (ANORM: val_casted_list vargs targs),
      match_states (Callstate fd vargs k m)
                   (Callstate tfd tvargs tk tm)
  | match_return_state:
      forall v k m tv tk tm j
        (MCONT: forall cenv, match_cont j cenv k tk m (Mem.nextblock m) (Mem.nextblock tm))
        (MINJ: Mem.inject j m tm)
        (RINJ: Val.inject j v tv),
      match_states (Returnstate v k m)
                   (Returnstate tv tk tm).(** Relating execution states *)

Inductive match_states: state -> state -> Prop :=
  | match_regular_states:
      forall f s k e le m tf ts tk te tle tm j lo hi tlo thi
        (TRF: transf_function f = OK tf)
        (TRS: simpl_stmt (cenv_for f) s = OK ts)
        (MENV: match_envs j (cenv_for f) e le m lo hi te tle tlo thi)
        (MCONT: match_cont j (cenv_for f) k tk m lo tlo)
        (MINJ: Mem.inject j m tm)
        (COMPAT: compat_cenv (addr_taken_stmt s) (cenv_for f))
        (BOUND: Ple hi (Mem.nextblock m))
        (TBOUND: Ple thi (Mem.nextblock tm)),
      match_states (State f s k e le m)
                   (State tf ts tk te tle tm)
  | match_call_state:
      forall fd vargs k m tfd tvargs tk tm j targs tres cconv
        (TRFD: transf_fundef fd = OK tfd)
        (MCONT: forall cenv, match_cont j cenv k tk m (Mem.nextblock m) (Mem.nextblock tm))
        (MINJ: Mem.inject j m tm)
        (AINJ: Val.inject_list j vargs tvargs)
        (FUNTY: type_of_fundef fd = Tfunction targs tres cconv)
        (ANORM: val_casted_list vargs targs),
      match_states (Callstate fd vargs k m)
                   (Callstate tfd tvargs tk tm)
  | match_return_state:
      forall v k m tv tk tm j
        (MCONT: forall cenv, match_cont j cenv k tk m (Mem.nextblock m) (Mem.nextblock tm))
        (MINJ: Mem.inject j m tm)
        (RINJ: Val.inject j v tv),
      match_states (Returnstate v k m)
                   (Returnstate tv tk tm).(** The simulation diagrams *)

Remark is_liftable_var_charact:
  forall cenv a,
  match is_liftable_var cenv a with
  | Some id => exists ty, a = Evar id ty /\ VSet.mem id cenv = true
  | None => match a with Evar id ty => VSet.mem id cenv = false | _ => True end
  end.Remark simpl_select_switch:
  forall cenv n ls tls,
  simpl_lblstmt cenv ls = OK tls ->
  simpl_lblstmt cenv (select_switch n ls) = OK (select_switch n tls).Remark simpl_seq_of_labeled_statement:
  forall cenv ls tls,
  simpl_lblstmt cenv ls = OK tls ->
  simpl_stmt cenv (seq_of_labeled_statement ls) = OK (seq_of_labeled_statement tls).Remark compat_cenv_select_switch:
  forall cenv n ls,
  compat_cenv (addr_taken_lblstmt ls) cenv ->
  compat_cenv (addr_taken_lblstmt (select_switch n ls)) cenv.Remark addr_taken_seq_of_labeled_statement:
  forall ls, addr_taken_stmt (seq_of_labeled_statement ls) = addr_taken_lblstmt ls.

Lemma simpl_find_label: forall s k ts tk, simpl_stmt cenv s = OK ts -> match_cont f cenv k tk m bound tbound -> compat_cenv (addr_taken_stmt s) cenv -> match find_label lbl s k with | None => find_label lbl ts tk = None | Some(s', k') => exists ts', exists tk', find_label lbl ts tk = Some(ts', tk') /\ compat_cenv (addr_taken_stmt s') cenv /\ simpl_stmt cenv s' = OK ts' /\ match_cont f cenv k' tk' m bound tbound end with simpl_find_label_ls: forall ls k tls tk, simpl_lblstmt cenv ls = OK tls -> match_cont f cenv k tk m bound tbound -> compat_cenv (addr_taken_lblstmt ls) cenv -> match find_label_ls lbl ls k with | None => find_label_ls lbl tls tk = None | Some(s', k') => exists ts', exists tk', find_label_ls lbl tls tk = Some(ts', tk') /\ compat_cenv (addr_taken_stmt s') cenv /\ simpl_stmt cenv s' = OK ts' /\ match_cont f cenv k' tk' m bound tbound end.