Inductive cont': Type :=
| Kseq: statement -> cont'       (**r [Kseq s2 k] = after [s1] in [s1;s2] *)
| Kloop1: statement -> statement -> cont'
| Kloop2: statement -> statement  -> cont'
| Kswitch: cont'       (**r catches [break] statements arising out of [switch] *)
| Kcall: forall (l: option ident),                  (**r where to store result *)
function ->                      (**r called (not calling!) function *)
env ->                           (**r local env of calling function *)
temp_env ->                      (**r temporary env of calling function *)
cont'.
Definition cont := list cont'.
(** Pop continuation until a call or stop *)
Fixpoint call_cont (k: cont) : cont :=
match k with
| Kseq s :: k => call_cont k
| Kloop1 _ _ :: k => call_cont k
| Kloop2 _ _ :: k => call_cont k
| Kswitch :: k => call_cont k
| _ => k
end.
Fixpoint current_function (k: cont) : option function :=
match k with
| Kseq s :: k => current_function k
| Kloop1 _ _ :: k => current_function k
| Kloop2 _ _:: k =>current_function k
| Kswitch :: k => current_function k
| Kcall _ f _ _ :: _ => Some f
| _ => None
end.
Fixpoint continue_cont (k: cont) : cont :=
match k with
| Kseq s :: k' => continue_cont k'
| Kloop1 s1 s2 :: k' => Kseq s2 :: Kloop2 s1 s2 :: k'
| Kswitch :: k' => continue_cont k'
| _ => nil (* stuck *)
end.
Fixpoint precontinue_cont (k: cont) : cont :=
match k with
| Kseq s :: k' => precontinue_cont k'
| Kloop1 _ _ :: _ => k
| Kswitch :: k' => precontinue_cont k'
| _ => nil (* stuck *)
end.
Fixpoint break_cont (k: cont) : cont :=
match k with
| Kseq s :: k' => break_cont k'
| Kloop1 _ _ :: k' => k'
| Kloop2 _ _ :: _ => nil  (* stuck *)
| Kswitch :: k' => k'
| _ =>  nil (* stuck *)
end.
Inductive corestate :=
| State: forall (ve: env) (te: temp_env) (k: cont), corestate
| ExtCall: forall (ef: external_function) (sig: signature) (args: list val)
(lid: option ident) (ve: env) (te: temp_env) (k: cont),
corestate.
Fixpoint strip_skip (k: cont) : cont :=
match k with Kseq Sskip :: k' => strip_skip k' | _ => k end.
Definition cl_at_external (c: corestate) : option (external_function * signature *list val) :=
match c with
| State _ _ k => None
| ExtCall ef sig args lid ve te k => Some (ef, sig, args)
end.
Definition cl_after_external (vret: option val) (c: corestate) : option corestate :=
match vret, c with
| Some v, ExtCall ef sig args (Some id) ve te k => Some (State ve (PTree.set id v te) k)
| None, ExtCall ef sig args None ve te k => Some (State ve te k)
| _, _ => None
end.
(** Find the statement and manufacture the continuation corresponding to a label *)
Fixpoint find_label (lbl: label) (s: statement) (k: cont)
{struct s}: option cont :=
match s with
| Ssequence s1 s2 =>
match find_label lbl s1 (Kseq s2 :: k) with
| Some sk => Some sk
| None => find_label lbl s2 k
end
| Sifthenelse a s1 s2 =>
match find_label lbl s1 k with
| Some sk => Some sk
| None => find_label lbl s2 k
end
| Sloop s1 a3 =>
match find_label lbl s1 (Kseq Scontinue :: Kloop1 s1 a3 :: k) with
| Some sk => Some sk
| None => find_label lbl a3 (Kloop2 s1 a3 :: k)
end
| Sswitch e sl =>
find_label_ls lbl sl (Kswitch :: k)
| Slabel lbl' s' =>
if ident_eq lbl lbl' then Some(Kseq s' :: k) else find_label lbl s' k
| _ => None
end
with find_label_ls (lbl: label) (sl: labeled_statements) (k: cont)
{struct sl}: option cont :=
match sl with
| LSdefault s => find_label lbl s k
| LScase _ s sl' =>
match find_label lbl s (Kseq (seq_of_labeled_statement sl') :: k) with
| Some sk => Some sk
| None => find_label_ls lbl sl' k
end
end.
(** Transition relation *)
Inductive cl_step (ge: Clight.genv): forall (q: corestate) (m: mem) (q': corestate) (m': mem), Prop :=
| step_assign: forall ve te k m a1 a2 loc ofs v2 v m',
type_is_volatile (typeof a1) = false ->
Clight.eval_lvalue ge ve te m a1 loc ofs ->
Clight.eval_expr ge ve te m a2 v2 ->
Cop.sem_cast v2 (typeof a2) (typeof a1) = Some v ->
Clight.assign_loc (typeof a1) m loc ofs v m' ->
cl_step ge (State ve te (Kseq (Sassign a1 a2):: k)) m (State ve te k) m'
| step_set:   forall ve te k m id a v,
Clight.eval_expr ge ve te m a v ->
cl_step ge (State ve te (Kseq (Sset id a) :: k)) m (State ve (PTree.set id v te) k) m
| step_call_internal:   forall ve te k m optid a al tyargs tyres vf vargs f m1 ve' le',
Cop.classify_fun (typeof a) = Cop.fun_case_f tyargs tyres ->
Clight.eval_expr ge ve te m a vf ->
Clight.eval_exprlist ge ve te m al tyargs vargs ->
Genv.find_funct ge vf = Some (Internal f) ->
type_of_function f = Tfunction tyargs tyres ->
list_norepet (var_names f.(fn_params) ++ var_names f.(fn_temps)) ->
forall (NRV: list_norepet (var_names f.(fn_vars))),
Clight.alloc_variables empty_env m (f.(fn_vars)) ve' m1 ->
bind_parameter_temps f.(fn_params) vargs (create_undef_temps f.(fn_temps)) = Some
le' ->
cl_step ge (State ve te (Kseq (Scall optid a al) :: k)) m
(State ve' le' (Kseq f.(fn_body) :: Kseq (Sreturn None) :: Kcall optid f ve te :: k)) m1
| step_call_external:   forall ve te k m optid a al tyargs tyres vf vargs ef,
Cop.classify_fun (typeof a) = Cop.fun_case_f tyargs tyres ->
Clight.eval_expr ge ve te m a vf ->
Clight.eval_exprlist ge ve te m al tyargs vargs ->
Genv.find_funct ge vf = Some (External ef tyargs tyres) ->
cl_step ge (State ve te (Kseq (Scall optid a al) :: k)) m (ExtCall ef (signature_of_type tyargs tyres) vargs optid ve te k) m
| step_seq: forall ve te k m s1 s2 st' m',
cl_step ge (State ve te (Kseq s1 :: Kseq s2 :: k)) m st' m' ->
cl_step ge (State ve te (Kseq (Ssequence s1 s2) :: k)) m st' m'
| step_skip: forall ve te k m st' m',
cl_step ge (State ve te k) m st' m' ->
cl_step ge (State ve te (Kseq Sskip :: k)) m st' m'
| step_continue: forall ve te k m st' m',
cl_step ge (State ve te (continue_cont k)) m st' m' ->
cl_step ge (State ve te (Kseq Scontinue :: k)) m st' m'
| step_break: forall ve te k m st' m',
cl_step ge (State ve te (break_cont k)) m st' m' ->
cl_step ge (State ve te (Kseq Sbreak :: k)) m st' m'
| step_ifthenelse:  forall ve te k m a s1 s2 v1 b,
Clight.eval_expr ge ve te m a v1 ->
Cop.bool_val v1 (typeof a) = Some b ->
cl_step ge (State ve te (Kseq (Sifthenelse a s1 s2) :: k)) m (State ve te  (Kseq (if b then s1 else s2) :: k)) m
| step_for: forall ve te k m s1 s2,
cl_step ge (State ve te (Kseq (Sloop s1 s2) :: k)) m
(State ve te (Kseq s1 :: Kseq Scontinue :: Kloop1 s1 s2 :: k)) m
| step_loop2: forall ve te k m a3 s,
cl_step ge (State ve te (Kloop2 s a3 :: k)) m
(State ve te (Kseq s :: Kseq Scontinue :: Kloop1 s a3 :: k)) m
| step_return: forall f ve te optexp optid k m v' m' ve' te' te'' k',
call_cont k = Kcall optid f ve' te' :: k' ->
Mem.free_list m (Clight.blocks_of_env ve) = Some m' ->
match optexp with None => True
| Some a => exists v, Clight.eval_expr ge ve te m a v /\ Cop.sem_cast v (typeof a) f.(fn_return) = Some v'
end ->
match optid with None => f.(fn_return) = Tvoid /\ te''=te'
| Some id => optexp <> None /\ te'' = PTree.set id v' te'
end ->
cl_step ge (State ve te (Kseq (Sreturn optexp) :: k)) m (State ve' te'' k') m'
| step_switch: forall ve te k m a sl n,
Clight.eval_expr ge ve te m a (Vint n) ->
cl_step ge (State ve te (Kseq (Sswitch a sl) :: k)) m
(State ve te (Kseq (seq_of_labeled_statement (select_switch n sl)) :: Kswitch :: k)) m
| step_label: forall ve te k m lbl s st' m',
cl_step ge (State ve te (Kseq s :: k)) m st' m' ->
cl_step ge (State ve te (Kseq (Slabel lbl s) :: k)) m st' m'
| step_goto: forall f ve te k m lbl k'
(* make sure to take a step here, so that every loop ticks the clock *)
(CUR: current_function k = Some f),
find_label lbl f.(fn_body) (Kseq (Sreturn None) :: (call_cont k)) = Some k' ->
cl_step ge (State ve te (Kseq (Sgoto lbl) :: k)) m (State ve te k') m.
Definition vret2v (vret: list val) : val :=
match vret with v::nil => v | _ => Vundef end.
Definition exit_syscall_number : ident := 1%positive.
Definition cl_halted (c: corestate) : option val := None.
Definition empty_function : function := mkfunction Tvoid nil nil nil Sskip.
Fixpoint temp_bindings (i: positive) (vl: list val) :=
match vl with
| nil => PTree.empty val
| v::vl' => PTree.set i v (temp_bindings (i+1)%positive vl')
end.
Definition Tint32s := Tint I32 Signed noattr.
Definition true_expr : Clight.expr := Clight.Econst_int Int.one Tint32s.
Fixpoint typed_params (i: positive) (n: nat) : list (ident * type) :=
match n with
| O => nil
| S n' => (i, Tint32s) :: typed_params (i+1)%positive n'
end.
Definition cl_initial_core (ge: genv) (v: val) (args: list val) : option corestate :=
let tl := typed_params 2%positive (length args)
in Some (State empty_env (temp_bindings 1%positive (v::args))
(Kseq (Scall None
(Etempvar 1%positive (Tfunction (type_of_params tl) Tvoid))
(map (fun x => Etempvar (fst x) (snd x)) tl)) ::
Kseq (Sloop Sskip Sskip) :: nil)).

Lemma cl_corestep_not_at_external:
forall ge m q m' q', cl_step ge q m q' m' -> cl_at_external q = None.