(** ** 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).
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.