Fixpoint break_cont (k: cont) : cont :=
match k with
| Kseq s :: k' => break_cont k'
| Kloop1 _ _ :: k' => k'
| Kloop2 _ _ :: k' => k'
| Kswitch :: k' => k'
| _ =>  nil (* stuck *)
end.
Inductive corestate :=
| State: forall (ve: env) (te: temp_env) (k: cont), corestate
| ExtCall: forall (ef: external_function) (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 * list val) :=
match c with
| State _ _ k => None
| ExtCall ef args lid ve te k => Some (ef, args)
end.
Definition cl_after_external (vret: option val) (c: corestate) : option corestate :=
match vret, c with
| Some v, ExtCall ef args (Some id) ve te k => Some (State ve (PTree.set id v te) k)
| None, ExtCall ef args (Some id) ve te k => Some (State ve (PTree.set id Vundef te) k)
| Some v, ExtCall ef args None ve te k => Some (State ve te k)
| None, ExtCall ef 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
| LSnil => None
| LScons _ 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) m = Some v ->
Clight.assign_loc ge (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 cc vf vargs f m1 ve' le',
Cop.classify_fun (typeof a) = Cop.fun_case_f tyargs tyres cc ->
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 cc ->
list_norepet (var_names f.(fn_params) ++ var_names f.(fn_temps)) ->
forall (NRV: list_norepet (var_names f.(fn_vars))),
Clight.alloc_variables ge 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 cc vf vargs ef,
Cop.classify_fun (typeof a) = Cop.fun_case_f tyargs tyres cc ->
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 cc) ->
cl_step ge (State ve te (Kseq (Scall optid a al) :: k)) m (ExtCall ef 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) m = 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 ge ve) = Some m' ->
match optexp with None => v' = Vundef
| Some a => exists v, Clight.eval_expr ge ve te m a v
/\ Cop.sem_cast v (typeof a) f.(fn_return) m = Some v'
end ->
match optid with None => True /\ te''=te'
| Some id => True /\ 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 v n,
Clight.eval_expr ge ve te m a v ->
Cop.sem_switch_arg v (typeof a) = Some 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 cc_default 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.
Fixpoint params_of_types (i: positive) (l : list type) : list (ident * type) :=
match l with
| nil => nil
| t :: l => (i, t) :: params_of_types (i+1)%positive l
end.
Fixpoint typelist2list (tl: typelist) : list type :=
match tl with
| Tcons t r => t::typelist2list r
| Tnil => nil
end.
Definition params_of_fundef (f: fundef) : list type :=
match f with
| Internal {| fn_params := fn_params |} => map snd fn_params
| External _ t _ _ => typelist2list t
end.
Inductive val_casted_list: list val -> typelist -> Prop :=
| vcl_nil:
val_casted_list nil Tnil
| vcl_cons: forall v1 vl ty1 tyl,
val_casted v1 ty1 -> val_casted_list vl tyl ->
val_casted_list (v1 :: vl) (Tcons  ty1 tyl).
Definition cl_initial_core (ge: genv) (v: val) (args: list val) (q: corestate) : Prop :=
match v with
Vptr b i =>
if Ptrofs.eq_dec i Ptrofs.zero then
match Genv.find_funct_ptr ge b with
Some f =>
match type_of_fundef f with Tfunction targs _ c =>
c = cc_default /\
val_casted_list args targs /\
Val.has_type_list args (Ctypes.typlist_of_typelist targs) /\
q = State empty_env (temp_bindings 1%positive (v::args))
(Kseq (Scall None
(Etempvar 1%positive (type_of_fundef f))
(map (fun x => Etempvar (fst x) (snd x))
(params_of_types 2%positive
(params_of_fundef f)))) ::
Kseq (Sloop Sskip Sskip) :: nil)
| _ => False end
| _ => False end
else False
| _ => False
end.

(* Specifications: *)
Lemma cl_corestep_not_at_external:
forall ge m q m' q', cl_step ge q m q' m' -> cl_at_external q = None.
Lemma cl_corestep_not_halted :
forall ge m q m' q', cl_step ge q m q' m' -> cl_halted q = None.
Lemma cl_after_at_external_excl :
forall retv q q', cl_after_external retv q = Some q' -> cl_at_external q' = None.