(** ** Semantic preservation for statements and functions *)

Inductive match_cont: Csharpminor.cont -> Cminor.cont -> compilenv -> exit_env -> callstack -> Prop :=
  | match_Kstop: forall cenv xenv,
      match_cont Csharpminor.Kstop Kstop cenv xenv nil
  | match_Kseq: forall s k ts tk cenv xenv cs,
      transl_stmt cenv xenv s = OK ts ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq s k) (Kseq ts tk) cenv xenv cs
  | match_Kseq2: forall s1 s2 k ts1 tk cenv xenv cs,
      transl_stmt cenv xenv s1 = OK ts1 ->
      match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq (Csharpminor.Sseq s1 s2) k)
                 (Kseq ts1 tk) cenv xenv cs
  | match_Kblock: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kblock k) (Kblock tk) cenv (true :: xenv) cs
  | match_Kblock2: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont k (Kblock tk) cenv (false :: xenv) cs
  | match_Kcall: forall optid fn e le k tfn sp te tk cenv xenv lo hi cs sz cenv',
      transl_funbody cenv sz fn = OK tfn ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kcall optid fn e le k)
                 (Kcall optid tfn (Vptr sp Ptrofs.zero) te tk)
                 cenv' nil
                 (Frame cenv tfn e le te sp lo hi :: cs).(** ** Semantic preservation for statements and functions *)

Inductive match_cont: Csharpminor.cont -> Cminor.cont -> compilenv -> exit_env -> callstack -> Prop :=
  | match_Kstop: forall cenv xenv,
      match_cont Csharpminor.Kstop Kstop cenv xenv nil
  | match_Kseq: forall s k ts tk cenv xenv cs,
      transl_stmt cenv xenv s = OK ts ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq s k) (Kseq ts tk) cenv xenv cs
  | match_Kseq2: forall s1 s2 k ts1 tk cenv xenv cs,
      transl_stmt cenv xenv s1 = OK ts1 ->
      match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq (Csharpminor.Sseq s1 s2) k)
                 (Kseq ts1 tk) cenv xenv cs
  | match_Kblock: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kblock k) (Kblock tk) cenv (true :: xenv) cs
  | match_Kblock2: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont k (Kblock tk) cenv (false :: xenv) cs
  | match_Kcall: forall optid fn e le k tfn sp te tk cenv xenv lo hi cs sz cenv',
      transl_funbody cenv sz fn = OK tfn ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kcall optid fn e le k)
                 (Kcall optid tfn (Vptr sp Ptrofs.zero) te tk)
                 cenv' nil
                 (Frame cenv tfn e le te sp lo hi :: cs).(** ** Semantic preservation for statements and functions *)

Inductive match_cont: Csharpminor.cont -> Cminor.cont -> compilenv -> exit_env -> callstack -> Prop :=
  | match_Kstop: forall cenv xenv,
      match_cont Csharpminor.Kstop Kstop cenv xenv nil
  | match_Kseq: forall s k ts tk cenv xenv cs,
      transl_stmt cenv xenv s = OK ts ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq s k) (Kseq ts tk) cenv xenv cs
  | match_Kseq2: forall s1 s2 k ts1 tk cenv xenv cs,
      transl_stmt cenv xenv s1 = OK ts1 ->
      match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq (Csharpminor.Sseq s1 s2) k)
                 (Kseq ts1 tk) cenv xenv cs
  | match_Kblock: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kblock k) (Kblock tk) cenv (true :: xenv) cs
  | match_Kblock2: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont k (Kblock tk) cenv (false :: xenv) cs
  | match_Kcall: forall optid fn e le k tfn sp te tk cenv xenv lo hi cs sz cenv',
      transl_funbody cenv sz fn = OK tfn ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kcall optid fn e le k)
                 (Kcall optid tfn (Vptr sp Ptrofs.zero) te tk)
                 cenv' nil
                 (Frame cenv tfn e le te sp lo hi :: cs).(** ** Semantic preservation for statements and functions *)

Inductive match_cont: Csharpminor.cont -> Cminor.cont -> compilenv -> exit_env -> callstack -> Prop :=
  | match_Kstop: forall cenv xenv,
      match_cont Csharpminor.Kstop Kstop cenv xenv nil
  | match_Kseq: forall s k ts tk cenv xenv cs,
      transl_stmt cenv xenv s = OK ts ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq s k) (Kseq ts tk) cenv xenv cs
  | match_Kseq2: forall s1 s2 k ts1 tk cenv xenv cs,
      transl_stmt cenv xenv s1 = OK ts1 ->
      match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq (Csharpminor.Sseq s1 s2) k)
                 (Kseq ts1 tk) cenv xenv cs
  | match_Kblock: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kblock k) (Kblock tk) cenv (true :: xenv) cs
  | match_Kblock2: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont k (Kblock tk) cenv (false :: xenv) cs
  | match_Kcall: forall optid fn e le k tfn sp te tk cenv xenv lo hi cs sz cenv',
      transl_funbody cenv sz fn = OK tfn ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kcall optid fn e le k)
                 (Kcall optid tfn (Vptr sp Ptrofs.zero) te tk)
                 cenv' nil
                 (Frame cenv tfn e le te sp lo hi :: cs).(** ** Semantic preservation for statements and functions *)

Inductive match_cont: Csharpminor.cont -> Cminor.cont -> compilenv -> exit_env -> callstack -> Prop :=
  | match_Kstop: forall cenv xenv,
      match_cont Csharpminor.Kstop Kstop cenv xenv nil
  | match_Kseq: forall s k ts tk cenv xenv cs,
      transl_stmt cenv xenv s = OK ts ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq s k) (Kseq ts tk) cenv xenv cs
  | match_Kseq2: forall s1 s2 k ts1 tk cenv xenv cs,
      transl_stmt cenv xenv s1 = OK ts1 ->
      match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq (Csharpminor.Sseq s1 s2) k)
                 (Kseq ts1 tk) cenv xenv cs
  | match_Kblock: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kblock k) (Kblock tk) cenv (true :: xenv) cs
  | match_Kblock2: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont k (Kblock tk) cenv (false :: xenv) cs
  | match_Kcall: forall optid fn e le k tfn sp te tk cenv xenv lo hi cs sz cenv',
      transl_funbody cenv sz fn = OK tfn ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kcall optid fn e le k)
                 (Kcall optid tfn (Vptr sp Ptrofs.zero) te tk)
                 cenv' nil
                 (Frame cenv tfn e le te sp lo hi :: cs).(** ** Semantic preservation for statements and functions *)

Inductive match_cont: Csharpminor.cont -> Cminor.cont -> compilenv -> exit_env -> callstack -> Prop :=
  | match_Kstop: forall cenv xenv,
      match_cont Csharpminor.Kstop Kstop cenv xenv nil
  | match_Kseq: forall s k ts tk cenv xenv cs,
      transl_stmt cenv xenv s = OK ts ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq s k) (Kseq ts tk) cenv xenv cs
  | match_Kseq2: forall s1 s2 k ts1 tk cenv xenv cs,
      transl_stmt cenv xenv s1 = OK ts1 ->
      match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq (Csharpminor.Sseq s1 s2) k)
                 (Kseq ts1 tk) cenv xenv cs
  | match_Kblock: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kblock k) (Kblock tk) cenv (true :: xenv) cs
  | match_Kblock2: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont k (Kblock tk) cenv (false :: xenv) cs
  | match_Kcall: forall optid fn e le k tfn sp te tk cenv xenv lo hi cs sz cenv',
      transl_funbody cenv sz fn = OK tfn ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kcall optid fn e le k)
                 (Kcall optid tfn (Vptr sp Ptrofs.zero) te tk)
                 cenv' nil
                 (Frame cenv tfn e le te sp lo hi :: cs).(** ** Semantic preservation for statements and functions *)

Inductive match_cont: Csharpminor.cont -> Cminor.cont -> compilenv -> exit_env -> callstack -> Prop :=
  | match_Kstop: forall cenv xenv,
      match_cont Csharpminor.Kstop Kstop cenv xenv nil
  | match_Kseq: forall s k ts tk cenv xenv cs,
      transl_stmt cenv xenv s = OK ts ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq s k) (Kseq ts tk) cenv xenv cs
  | match_Kseq2: forall s1 s2 k ts1 tk cenv xenv cs,
      transl_stmt cenv xenv s1 = OK ts1 ->
      match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs ->
      match_cont (Csharpminor.Kseq (Csharpminor.Sseq s1 s2) k)
                 (Kseq ts1 tk) cenv xenv cs
  | match_Kblock: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kblock k) (Kblock tk) cenv (true :: xenv) cs
  | match_Kblock2: forall k tk cenv xenv cs,
      match_cont k tk cenv xenv cs ->
      match_cont k (Kblock tk) cenv (false :: xenv) cs
  | match_Kcall: forall optid fn e le k tfn sp te tk cenv xenv lo hi cs sz cenv',
      transl_funbody cenv sz fn = OK tfn ->
      match_cont k tk cenv xenv cs ->
      match_cont (Csharpminor.Kcall optid fn e le k)
                 (Kcall optid tfn (Vptr sp Ptrofs.zero) te tk)
                 cenv' nil
                 (Frame cenv tfn e le te sp lo hi :: cs).Inductive match_states: Csharpminor.state -> Cminor.state -> Prop :=
  | match_state:
      forall fn s k e le m tfn ts tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s = OK ts)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv xenv cs),
      match_states (Csharpminor.State fn s k e le m)
                   (State tfn ts tk (Vptr sp Ptrofs.zero) te tm)
  | match_state_seq:
      forall fn s1 s2 k e le m tfn ts1 tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s1 = OK ts1)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs),
      match_states (Csharpminor.State fn (Csharpminor.Sseq s1 s2) k e le m)
                   (State tfn ts1 tk (Vptr sp Ptrofs.zero) te tm)
  | match_callstate:
      forall fd args k m tfd targs tk tm f cs cenv
      (TR: transl_fundef fd = OK tfd)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (ISCC: Csharpminor.is_call_cont k)
      (ARGSINJ: Val.inject_list f args targs),
      match_states (Csharpminor.Callstate fd args k m)
                   (Callstate tfd targs tk tm)
  | match_returnstate:
      forall v k m tv tk tm f cs cenv
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (RESINJ: Val.inject f v tv),
      match_states (Csharpminor.Returnstate v k m)
                   (Returnstate tv tk tm).Inductive match_states: Csharpminor.state -> Cminor.state -> Prop :=
  | match_state:
      forall fn s k e le m tfn ts tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s = OK ts)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv xenv cs),
      match_states (Csharpminor.State fn s k e le m)
                   (State tfn ts tk (Vptr sp Ptrofs.zero) te tm)
  | match_state_seq:
      forall fn s1 s2 k e le m tfn ts1 tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s1 = OK ts1)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs),
      match_states (Csharpminor.State fn (Csharpminor.Sseq s1 s2) k e le m)
                   (State tfn ts1 tk (Vptr sp Ptrofs.zero) te tm)
  | match_callstate:
      forall fd args k m tfd targs tk tm f cs cenv
      (TR: transl_fundef fd = OK tfd)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (ISCC: Csharpminor.is_call_cont k)
      (ARGSINJ: Val.inject_list f args targs),
      match_states (Csharpminor.Callstate fd args k m)
                   (Callstate tfd targs tk tm)
  | match_returnstate:
      forall v k m tv tk tm f cs cenv
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (RESINJ: Val.inject f v tv),
      match_states (Csharpminor.Returnstate v k m)
                   (Returnstate tv tk tm).Inductive match_states: Csharpminor.state -> Cminor.state -> Prop :=
  | match_state:
      forall fn s k e le m tfn ts tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s = OK ts)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv xenv cs),
      match_states (Csharpminor.State fn s k e le m)
                   (State tfn ts tk (Vptr sp Ptrofs.zero) te tm)
  | match_state_seq:
      forall fn s1 s2 k e le m tfn ts1 tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s1 = OK ts1)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs),
      match_states (Csharpminor.State fn (Csharpminor.Sseq s1 s2) k e le m)
                   (State tfn ts1 tk (Vptr sp Ptrofs.zero) te tm)
  | match_callstate:
      forall fd args k m tfd targs tk tm f cs cenv
      (TR: transl_fundef fd = OK tfd)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (ISCC: Csharpminor.is_call_cont k)
      (ARGSINJ: Val.inject_list f args targs),
      match_states (Csharpminor.Callstate fd args k m)
                   (Callstate tfd targs tk tm)
  | match_returnstate:
      forall v k m tv tk tm f cs cenv
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (RESINJ: Val.inject f v tv),
      match_states (Csharpminor.Returnstate v k m)
                   (Returnstate tv tk tm).Inductive match_states: Csharpminor.state -> Cminor.state -> Prop :=
  | match_state:
      forall fn s k e le m tfn ts tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s = OK ts)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv xenv cs),
      match_states (Csharpminor.State fn s k e le m)
                   (State tfn ts tk (Vptr sp Ptrofs.zero) te tm)
  | match_state_seq:
      forall fn s1 s2 k e le m tfn ts1 tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s1 = OK ts1)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs),
      match_states (Csharpminor.State fn (Csharpminor.Sseq s1 s2) k e le m)
                   (State tfn ts1 tk (Vptr sp Ptrofs.zero) te tm)
  | match_callstate:
      forall fd args k m tfd targs tk tm f cs cenv
      (TR: transl_fundef fd = OK tfd)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (ISCC: Csharpminor.is_call_cont k)
      (ARGSINJ: Val.inject_list f args targs),
      match_states (Csharpminor.Callstate fd args k m)
                   (Callstate tfd targs tk tm)
  | match_returnstate:
      forall v k m tv tk tm f cs cenv
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (RESINJ: Val.inject f v tv),
      match_states (Csharpminor.Returnstate v k m)
                   (Returnstate tv tk tm).Inductive match_states: Csharpminor.state -> Cminor.state -> Prop :=
  | match_state:
      forall fn s k e le m tfn ts tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s = OK ts)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv xenv cs),
      match_states (Csharpminor.State fn s k e le m)
                   (State tfn ts tk (Vptr sp Ptrofs.zero) te tm)
  | match_state_seq:
      forall fn s1 s2 k e le m tfn ts1 tk sp te tm cenv xenv f lo hi cs sz
      (TRF: transl_funbody cenv sz fn = OK tfn)
      (TR: transl_stmt cenv xenv s1 = OK ts1)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm
               (Frame cenv tfn e le te sp lo hi :: cs)
               (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs),
      match_states (Csharpminor.State fn (Csharpminor.Sseq s1 s2) k e le m)
                   (State tfn ts1 tk (Vptr sp Ptrofs.zero) te tm)
  | match_callstate:
      forall fd args k m tfd targs tk tm f cs cenv
      (TR: transl_fundef fd = OK tfd)
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (ISCC: Csharpminor.is_call_cont k)
      (ARGSINJ: Val.inject_list f args targs),
      match_states (Csharpminor.Callstate fd args k m)
                   (Callstate tfd targs tk tm)
  | match_returnstate:
      forall v k m tv tk tm f cs cenv
      (MINJ: Mem.inject f m tm)
      (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm))
      (MK: match_cont k tk cenv nil cs)
      (RESINJ: Val.inject f v tv),
      match_states (Csharpminor.Returnstate v k m)
                   (Returnstate tv tk tm).Remark val_inject_function_pointer:
  forall bound v fd f tv,
  Genv.find_funct ge v = Some fd ->
  match_globalenvs f bound ->
  Val.inject f v tv ->
  tv = v.

Lemma match_is_call_cont: forall tfn te sp tm k tk cenv xenv cs, match_cont k tk cenv xenv cs -> Csharpminor.is_call_cont k -> exists tk', star step tge (State tfn Sskip tk sp te tm) E0 (State tfn Sskip tk' sp te tm) /\ is_call_cont tk' /\ match_cont k tk' cenv nil cs.