(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the GNU General Public License as published by  *)
(*  the Free Software Foundation, either version 2 of the License, or  *)
(*  (at your option) any later version.  This file is also distributed *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)
(** A big-step semantics for the Clight language. *)
Section BIGSTEP.
Variable ge: genv.
(** ** Big-step semantics for terminating statements and functions *)
(** The execution of a statement produces an ``outcome'', indicating how the execution terminated: either normally or prematurely through the execution of a [break], [continue] or [return] statement. *)
Inductive outcome: Type :=
| Out_break: outcome                 (**r terminated by [break] *)
| Out_continue: outcome              (**r terminated by [continue] *)
| Out_normal: outcome                (**r terminated normally *)
| Out_return: option (val * type) -> outcome. (**r terminated by [return] *)
Inductive out_normal_or_continue : outcome -> Prop :=
| Out_normal_or_continue_N: out_normal_or_continue Out_normal
| Out_normal_or_continue_C: out_normal_or_continue Out_continue.
Inductive out_break_or_return : outcome -> outcome -> Prop :=
| Out_break_or_return_B: out_break_or_return Out_break Out_normal
| Out_break_or_return_R: forall ov,
out_break_or_return (Out_return ov) (Out_return ov).
Definition outcome_switch (out: outcome) : outcome :=
match out with
| Out_break => Out_normal
| o => o
end.
Definition outcome_result_value (out: outcome) (t: type) (v: val) (m: mem): Prop :=
match out, t with
| Out_normal, Tvoid => v = Vundef
| Out_return None, Tvoid => v = Vundef
| Out_return (Some (v',t')), ty => ty <> Tvoid /\ sem_cast v' t' t m = Some v
| _, _ => False
end.
(** [exec_stmt ge e m1 s t m2 out] describes the execution of the statement [s].  [out] is the outcome for this execution. [m1] is the initial memory state, [m2] the final memory state. [t] is the trace of input/output events performed during this evaluation. *)
Inductive exec_stmt: env -> temp_env -> mem -> statement -> trace -> temp_env -> mem -> outcome -> Prop :=
| exec_Sskip:   forall e le m,
exec_stmt e le m Sskip
E0 le m Out_normal
| exec_Sassign:   forall e le m a1 a2 loc ofs v2 v m',
eval_lvalue ge e le m a1 loc ofs ->
eval_expr ge e le m a2 v2 ->
sem_cast v2 (typeof a2) (typeof a1) m = Some v ->
assign_loc ge (typeof a1) m loc ofs v m' ->
exec_stmt e le m (Sassign a1 a2)
E0 le m' Out_normal
| exec_Sset:     forall e le m id a v,
eval_expr ge e le m a v ->
exec_stmt e le m (Sset id a)
E0 (PTree.set id v le) m Out_normal
| exec_Scall:   forall e le m optid a al tyargs tyres cconv vf vargs f t m' vres,
classify_fun (typeof a) = fun_case_f tyargs tyres cconv ->
eval_expr ge e le m a vf ->
eval_exprlist ge e le m al tyargs vargs ->
Genv.find_funct ge vf = Some f ->
type_of_fundef f = Tfunction tyargs tyres cconv ->
eval_funcall m f vargs t m' vres ->
exec_stmt e le m (Scall optid a al)
t (set_opttemp optid vres le) m' Out_normal
| exec_Sbuiltin:   forall e le m optid ef al tyargs vargs t m' vres,
eval_exprlist ge e le m al tyargs vargs ->
external_call ef ge vargs m t vres m' ->
exec_stmt e le m (Sbuiltin optid ef tyargs al)
t (set_opttemp optid vres le) m' Out_normal
| exec_Sseq_1:   forall e le m s1 s2 t1 le1 m1 t2 le2 m2 out,
exec_stmt e le m s1 t1 le1 m1 Out_normal ->
exec_stmt e le1 m1 s2 t2 le2 m2 out ->
exec_stmt e le m (Ssequence s1 s2)
(t1 ** t2) le2 m2 out
| exec_Sseq_2:   forall e le m s1 s2 t1 le1 m1 out,
exec_stmt e le m s1 t1 le1 m1 out ->
out <> Out_normal ->
exec_stmt e le m (Ssequence s1 s2)
t1 le1 m1 out
| exec_Sifthenelse: forall e le m a s1 s2 v1 b t le' m' out,
eval_expr ge e le m a v1 ->
bool_val v1 (typeof a) m = Some b ->
exec_stmt e le m (if b then s1 else s2) t le' m' out ->
exec_stmt e le m (Sifthenelse a s1 s2)
t le' m' out
| exec_Sreturn_none:   forall e le m,
exec_stmt e le m (Sreturn None)
E0 le m (Out_return None)
| exec_Sreturn_some: forall e le m a v,
eval_expr ge e le m a v ->
exec_stmt e le m (Sreturn (Some a))
E0 le m (Out_return (Some (v, typeof a)))
| exec_Sbreak:   forall e le m,
exec_stmt e le m Sbreak
E0 le m Out_break
| exec_Scontinue:   forall e le m,
exec_stmt e le m Scontinue
E0 le m Out_continue
| exec_Sloop_stop1: forall e le m s1 s2 t le' m' out' out,
exec_stmt e le m s1 t le' m' out' ->
out_break_or_return out' out ->
exec_stmt e le m (Sloop s1 s2)
t le' m' out
| exec_Sloop_stop2: forall e le m s1 s2 t1 le1 m1 out1 t2 le2 m2 out2 out,
exec_stmt e le m s1 t1 le1 m1 out1 ->
out_normal_or_continue out1 ->
exec_stmt e le1 m1 s2 t2 le2 m2 out2 ->
out_break_or_return out2 out ->
exec_stmt e le m (Sloop s1 s2)
(t1**t2) le2 m2 out
| exec_Sloop_loop: forall e le m s1 s2 t1 le1 m1 out1 t2 le2 m2 t3 le3 m3 out,
exec_stmt e le m s1 t1 le1 m1 out1 ->
out_normal_or_continue out1 ->
exec_stmt e le1 m1 s2 t2 le2 m2 Out_normal ->
exec_stmt e le2 m2 (Sloop s1 s2) t3 le3 m3 out ->
exec_stmt e le m (Sloop s1 s2)
(t1**t2**t3) le3 m3 out
| exec_Sswitch:   forall e le m a t v n sl le1 m1 out,
eval_expr ge e le m a v ->
sem_switch_arg v (typeof a) = Some n ->
exec_stmt e le m (seq_of_labeled_statement (select_switch n sl)) t le1 m1 out ->
exec_stmt e le m (Sswitch a sl)
t le1 m1 (outcome_switch out)
(** [eval_funcall m1 fd args t m2 res] describes the invocation of function [fd] with arguments [args].  [res] is the value returned by the call.  *)
with eval_funcall: mem -> fundef -> list val -> trace -> mem -> val -> Prop :=
| eval_funcall_internal: forall le m f vargs t e m1 m2 m3 out vres m4,
alloc_variables ge empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
list_norepet (var_names f.(fn_params) ++ var_names f.(fn_vars)) ->
bind_parameters ge e m1 f.(fn_params) vargs m2 ->
exec_stmt e (create_undef_temps f.(fn_temps)) m2 f.(fn_body) t le m3 out ->
outcome_result_value out f.(fn_return) vres m3 ->
Mem.free_list m3 (blocks_of_env ge e) = Some m4 ->
eval_funcall m (Internal f) vargs t m4 vres
| eval_funcall_external: forall m ef targs tres cconv vargs t vres m',
external_call ef ge vargs m t vres m' ->
eval_funcall m (External ef targs tres cconv) vargs t m' vres.
Scheme exec_stmt_ind2 := Minimality for exec_stmt Sort Prop
with eval_funcall_ind2 := Minimality for eval_funcall Sort Prop.
Combined Scheme exec_stmt_funcall_ind from exec_stmt_ind2, eval_funcall_ind2.
(** ** Big-step semantics for diverging statements and functions *)
(** Coinductive semantics for divergence. [execinf_stmt ge e m s t] holds if the execution of statement [s] diverges, i.e. loops infinitely.  [t] is the possibly infinite trace of observable events performed during the execution. *)
CoInductive execinf_stmt: env -> temp_env -> mem -> statement -> traceinf -> Prop :=
| execinf_Scall:   forall e le m optid a al vf tyargs tyres cconv vargs f t,
classify_fun (typeof a) = fun_case_f tyargs tyres cconv ->
eval_expr ge e le m a vf ->
eval_exprlist ge e le m al tyargs vargs ->
Genv.find_funct ge vf = Some f ->
type_of_fundef f = Tfunction tyargs tyres cconv ->
evalinf_funcall m f vargs t ->
execinf_stmt e le m (Scall optid a al) t
| execinf_Sseq_1:   forall e le m s1 s2 t,
execinf_stmt e le m s1 t ->
execinf_stmt e le m (Ssequence s1 s2) t
| execinf_Sseq_2:   forall e le m s1 s2 t1 le1 m1 t2,
exec_stmt e le m s1 t1 le1 m1 Out_normal ->
execinf_stmt e le1 m1 s2 t2 ->
execinf_stmt e le m (Ssequence s1 s2) (t1 *** t2)
| execinf_Sifthenelse: forall e le m a s1 s2 v1 b t,
eval_expr ge e le m a v1 ->
bool_val v1 (typeof a) m = Some b ->
execinf_stmt e le m (if b then s1 else s2) t ->
execinf_stmt e le m (Sifthenelse a s1 s2) t
| execinf_Sloop_body1: forall e le m s1 s2 t,
execinf_stmt e le m s1 t ->
execinf_stmt e le m (Sloop s1 s2) t
| execinf_Sloop_body2: forall e le m s1 s2 t1 le1 m1 out1 t2,
exec_stmt e le m s1 t1 le1 m1 out1 ->
out_normal_or_continue out1 ->
execinf_stmt e le1 m1 s2 t2 ->
execinf_stmt e le m (Sloop s1 s2) (t1***t2)
| execinf_Sloop_loop: forall e le m s1 s2 t1 le1 m1 out1 t2 le2 m2 t3,
exec_stmt e le m s1 t1 le1 m1 out1 ->
out_normal_or_continue out1 ->
exec_stmt e le1 m1 s2 t2 le2 m2 Out_normal ->
execinf_stmt e le2 m2 (Sloop s1 s2) t3 ->
execinf_stmt e le m (Sloop s1 s2) (t1***t2***t3)
| execinf_Sswitch:   forall e le m a t v n sl,
eval_expr ge e le m a v ->
sem_switch_arg v (typeof a) = Some n ->
execinf_stmt e le m (seq_of_labeled_statement (select_switch n sl)) t ->
execinf_stmt e le m (Sswitch a sl) t
(** [evalinf_funcall ge m fd args t] holds if the invocation of function [fd] on arguments [args] diverges, with observable trace [t]. *)
with evalinf_funcall: mem -> fundef -> list val -> traceinf -> Prop :=
| evalinf_funcall_internal: forall m f vargs t e m1 m2,
alloc_variables ge empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
list_norepet (var_names f.(fn_params) ++ var_names f.(fn_vars)) ->
bind_parameters ge e m1 f.(fn_params) vargs m2 ->
execinf_stmt e (create_undef_temps f.(fn_temps)) m2 f.(fn_body) t ->
evalinf_funcall m (Internal f) vargs t.
End BIGSTEP.
(** Big-step execution of a whole program.  *)
Inductive bigstep_program_terminates (p: program): trace -> int -> Prop :=
| bigstep_program_terminates_intro: forall b f m0 m1 t r,
let ge := globalenv p in
Genv.init_mem p = Some m0 ->
Genv.find_symbol ge p.(prog_main) = Some b ->
Genv.find_funct_ptr ge b = Some f ->
type_of_fundef f = Tfunction Tnil type_int32s cc_default ->
eval_funcall ge m0 f nil t m1 (Vint r) ->
bigstep_program_terminates p t r.
Inductive bigstep_program_diverges (p: program): traceinf -> Prop :=
| bigstep_program_diverges_intro: forall b f m0 t,
let ge := globalenv p in
Genv.init_mem p = Some m0 ->
Genv.find_symbol ge p.(prog_main) = Some b ->
Genv.find_funct_ptr ge b = Some f ->
type_of_fundef f = Tfunction Tnil type_int32s cc_default ->
evalinf_funcall ge m0 f nil t ->
bigstep_program_diverges p t.
Definition bigstep_semantics (p: program) :=
Bigstep_semantics (bigstep_program_terminates p) (bigstep_program_diverges p).
(** * Implication from big-step semantics to transition semantics *)
Section BIGSTEP_TO_TRANSITIONS.
Variable prog: program.
Let ge : genv := globalenv prog.
Inductive outcome_state_match
(e: env) (le: temp_env) (m: mem) (f: function) (k: cont): outcome -> state -> Prop :=
| osm_normal:
outcome_state_match e le m f k Out_normal (State f Sskip k e le m)
| osm_break:
outcome_state_match e le m f k Out_break (State f Sbreak k e le m)
| osm_continue:
outcome_state_match e le m f k Out_continue (State f Scontinue k e le m)
| osm_return_none: forall k',
call_cont k' = call_cont k ->
outcome_state_match e le m f k
(Out_return None) (State f (Sreturn None) k' e le m)
| osm_return_some: forall a v k',
call_cont k' = call_cont k ->
eval_expr ge e le m a v ->
outcome_state_match e le m f k
(Out_return (Some (v,typeof a))) (State f (Sreturn (Some a)) k' e le m).

(* Specifications: *)
Lemma is_call_cont_call_cont:
forall k, is_call_cont k -> call_cont k = k.
Lemma exec_stmt_eval_funcall_steps:
(forall e le m s t le' m' out,
exec_stmt ge e le m s t le' m' out ->
forall f k, exists S,
star step1 ge (State f s k e le m) t S
/\ outcome_state_match e le' m' f k out S)
/\
(forall m fd args t m' res,
eval_funcall ge m fd args t m' res ->
forall k,
is_call_cont k ->
star step1 ge (Callstate fd args k m) t (Returnstate res k m')).
Lemma exec_stmt_steps:
forall e le m s t le' m' out,
exec_stmt ge e le m s t le' m' out ->
forall f k, exists S,
star step1 ge (State f s k e le m) t S
/\ outcome_state_match e le' m' f k out S.
Lemma is_call_cont_call_cont:
forall k, is_call_cont k -> call_cont k = k.
Lemma exec_stmt_eval_funcall_steps:
(forall e le m s t le' m' out,
exec_stmt ge e le m s t le' m' out ->
forall f k, exists S,
star step1 ge (State f s k e le m) t S
/\ outcome_state_match e le' m' f k out S)
/\
(forall m fd args t m' res,
eval_funcall ge m fd args t m' res ->
forall k,
is_call_cont k ->
star step1 ge (Callstate fd args k m) t (Returnstate res k m')).
Lemma exec_stmt_steps:
forall e le m s t le' m' out,
exec_stmt ge e le m s t le' m' out ->
forall f k, exists S,
star step1 ge (State f s k e le m) t S
/\ outcome_state_match e le' m' f k out S.