Correctness of the translation from Clight to C#minor.
Require Import Coqlib.
Require Import Errors.
Require Import Maps.
Require Import Integers.
Require Import Floats.
Require Import AST.
Require Import Values.
Require Import Events.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Ctypes.
Require Import Cop.
Require Import Clight.
Require Import Cminor.
Require Import Csharpminor.
Require Import Cshmgen.
Require Import Values_symbolictype.
Require Import Values_symbolic.
Require Import Normalise.
Require Import NormaliseSpec.
Require Import Memory.
Require Import Equivalences.
Require Import ClightEquivalences.
Properties of operations over types
Remark transl_params_types:
forall params,
map typ_of_type (
map snd params) =
typlist_of_typelist (
type_of_params params).
Proof.
induction params; simpl. auto. destruct a as [id ty]; simpl. f_equal; auto.
Qed.
Lemma transl_fundef_sig1:
forall f tf args res cc,
transl_fundef f =
OK tf ->
classify_fun (
type_of_fundef f) =
fun_case_f args res cc ->
funsig tf =
signature_of_type args res cc.
Proof.
Lemma transl_fundef_sig2:
forall f tf args res cc,
transl_fundef f =
OK tf ->
type_of_fundef f =
Tfunction args res cc ->
funsig tf =
signature_of_type args res cc.
Proof.
Properties of the translation functions
Transformation of expressions and statements.
Lemma transl_expr_lvalue:
forall ge e le m a locofs ta,
Clight.eval_lvalue ge e le m a locofs ->
transl_expr a =
OK ta ->
(
exists tb,
transl_lvalue a =
OK tb /\
make_load tb (
typeof a) =
OK ta).
Proof.
intros until ta;
intros EVAL TR.
inv EVAL;
simpl in TR.
-
(
exists (
Eaddrof id);
auto).
-
(
exists (
Eaddrof id);
auto).
-
monadInv TR.
exists x;
auto.
-
rewrite H0 in TR.
monadInv TR.
econstructor;
split.
simpl.
rewrite H0.
rewrite EQ;
rewrite EQ1;
simpl;
eauto.
auto.
-
rewrite H0 in TR.
monadInv TR.
econstructor;
split.
simpl.
rewrite H0.
rewrite EQ;
simpl;
eauto.
auto.
Qed.
Properties of labeled statements
Lemma transl_lbl_stmt_1:
forall tyret nbrk ncnt n sl tsl,
transl_lbl_stmt tyret nbrk ncnt sl =
OK tsl ->
transl_lbl_stmt tyret nbrk ncnt (
Clight.select_switch n sl) =
OK (
select_switch n tsl).
Proof.
Lemma transl_lbl_stmt_2:
forall tyret nbrk ncnt sl tsl,
transl_lbl_stmt tyret nbrk ncnt sl =
OK tsl ->
transl_statement tyret nbrk ncnt (
seq_of_labeled_statement sl) =
OK (
seq_of_lbl_stmt tsl).
Proof.
induction sl; intros.
monadInv H. auto.
monadInv H. simpl. rewrite EQ; simpl. rewrite (IHsl _ EQ1). simpl. auto.
Qed.
Correctness of Csharpminor construction functions
Section CONSTRUCTORS.
Variable ge:
genv.
Lemma make_intconst_correct:
forall n e le m,
eval_expr ge e le m (
make_intconst n) (
Eval (
Eint n)).
Proof.
Lemma make_floatconst_correct:
forall n e le m,
eval_expr ge e le m (
make_floatconst n) (
Eval (
Efloat n)).
Proof.
Lemma make_singleconst_correct:
forall n e le m,
eval_expr ge e le m (
make_singleconst n) (
Eval (
Esingle n)).
Proof.
Lemma make_longconst_correct:
forall n e le m,
eval_expr ge e le m (
make_longconst n) (
Eval (
Elong n)).
Proof.
Lemma make_singleoffloat_correct:
forall a n e le m,
eval_expr ge e le m a (
Eval (
Efloat n)) ->
eval_expr ge e le m (
make_singleoffloat a) (
expr_cast_gen Tfloat Signed Tsingle Signed (
Eval (
Efloat n))).
Proof.
intros. econstructor. eauto. eauto.
Qed.
Lemma make_floatofsingle_correct:
forall a n e le m,
eval_expr ge e le m a (
Eval (
Esingle n)) ->
eval_expr ge e le m (
make_floatofsingle a) (
expr_cast_gen Tsingle Signed Tfloat Signed (
Eval (
Esingle n))).
Proof.
intros. econstructor. eauto. auto.
Qed.
Lemma make_floatofint_correct:
forall a n sg e le m,
eval_expr ge e le m a (
Eval (
Eint n)) ->
eval_expr ge e le m (
make_floatofint a sg)
(
expr_cast_gen Tint sg Tfloat Signed (
Eval (
Eint n))).
Proof.
Hint Resolve make_intconst_correct make_floatconst_correct make_longconst_correct
make_singleconst_correct make_singleoffloat_correct make_floatofsingle_correct
make_floatofint_correct:
cshm.
Hint Constructors eval_expr eval_exprlist:
cshm.
Hint Extern 2 (@
eq trace _ _) =>
traceEq:
cshm.
Lemma nec_cmp_ne_zero:
forall v,
same_eval
(
Eunop OpBoolval Tint v)
(
Val.cmp Cne v (
Eval (
Eint Int.zero))).
Proof.
intros;
constructor;
simpl;
try red;
intros;
simpl;
try tauto.
repeat unfold_eval;
simpl;
NormaliseSpec.simpl_eval'.
destruct (
Int.eq i Int.zero);
simpl;
auto.
Qed.
Lemma cmp_0_or_1:
forall t sg c v1 v2 alloc em z,
ebinop t t Tint (
eSexpr alloc em eb eu)
(
eEval alloc (
Vi Int.zero))
(
fun_of_binop alloc eb (
OpCmp sg c))
v1 v2 =
z ->
z =
Vi Int.zero \/
z =
Vi Int.one.
Proof.
Lemma norm_boolval_cmp:
forall sg c t v1 v2,
same_eval
(
Ebinop (
OpCmp sg c)
t t v1 v2)
(
Eunop OpBoolval Tint (
Ebinop (
OpCmp sg c)
t t v1 v2)).
Proof.
Lemma make_cmp_ne_zero_correct:
forall ge e le m a v v',
eval_expr ge e le m a v ->
(
same_eval v v') ->
exists v'' :
expr_sym,
eval_expr ge e le m (
make_cmp_ne_zero a)
v'' /\
same_eval (
Eunop OpBoolval Tint v')
v''.
Proof.
Lemma make_cast_int_correct:
forall e le m a v v'
sz si,
eval_expr ge e le m a v ->
same_eval v v' ->
exists v'',
eval_expr ge e le m (
make_cast_int a sz si)
v'' /\
same_eval (
expr_cast_int_int sz si v')
v''.
Proof.
Hint Resolve make_cast_int_correct:
cshm.
Lemma eqm_notbool_ceq_cne:
forall sg t v1 v2,
same_eval
(
Eunop OpNotbool Tint
(
Ebinop (
OpCmp sg Ceq)
t t v1 v2))
(
Ebinop (
OpCmp sg Cne)
t t v1 v2).
Proof.
Lemma make_cast_correct:
forall e le m a b v v2 ty1 ty2 v',
make_cast ty1 ty2 a =
OK b ->
eval_expr ge e le m a v ->
same_eval v v2 ->
sem_cast_expr v2 ty1 ty2 =
Some v' ->
exists v'',
eval_expr ge e le m b v'' /\
same_eval v'
v''.
Proof.
intros e le m a b v v2 ty1 ty2 v'
CAST EVAL SE SEM.
unfold make_cast,
sem_cast_expr in *;
destruct (
classify_cast ty1 ty2)
eqn:?;
inv SEM;
inv CAST;
auto;
try (
eexists;
split;
eauto;
symmetry;
auto;
fail).
-
eapply make_cast_int_correct;
eauto.
-
unfold make_singleoffloat,
expr_cast_gen.
eexists;
split; [
repeat (
econstructor;
simpl;
eauto)|].
apply unop_same_eval;
symmetry;
auto.
-
unfold make_floatofsingle,
expr_cast_gen.
eexists;
split; [
repeat (
econstructor;
simpl;
eauto)|].
apply unop_same_eval;
symmetry;
auto.
-
unfold make_floatofint,
expr_cast_gen.
destruct si1;
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]);
apply unop_same_eval;
symmetry;
auto.
-
unfold make_singleofint,
expr_cast_gen.
destruct si1;
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]);
apply unop_same_eval;
symmetry;
auto.
-
unfold make_intoffloat,
expr_cast_gen.
destruct si2;
eapply make_cast_int_correct;
eauto.
econstructor;
simpl;
eauto.
apply unop_same_eval;
auto.
econstructor;
simpl;
eauto.
apply unop_same_eval;
auto.
-
destruct si2;
eapply make_cast_int_correct;
eauto.
econstructor;
simpl;
eauto.
apply unop_same_eval;
auto.
econstructor;
simpl;
eauto.
apply unop_same_eval;
auto.
-
unfold make_longofint,
expr_cast_gen.
destruct si1;
eexists; (
split ;[(
econstructor;
simpl;
eauto)|];
auto);
apply unop_same_eval;
symmetry;
auto.
-
eapply make_cast_int_correct;
eauto.
econstructor;
simpl;
eauto.
apply unop_same_eval;
auto.
-
unfold make_floatoflong,
expr_cast_gen.
simpl.
destruct si1;
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]);
apply unop_same_eval;
symmetry;
auto.
-
unfold make_singleoflong,
expr_cast_gen.
destruct si1;
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]);
apply unop_same_eval;
symmetry;
auto.
-
unfold make_longoffloat,
expr_cast_gen.
destruct si2;
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]);
apply unop_same_eval;
symmetry;
auto.
-
unfold make_longofsingle,
expr_cast_gen.
destruct si2;
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]);
apply unop_same_eval;
symmetry;
auto.
-
unfold make_floatconst,
expr_cast_gen.
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]).
unfold Val.cmpf,
Val.cmpf_bool.
rewrite eqm_notbool_ceq_cne.
apply binop_same_eval;
symmetry;
auto;
reflexivity.
-
unfold make_singleconst.
simpl.
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]).
rewrite eqm_notbool_ceq_cne.
apply binop_same_eval;
symmetry;
auto;
reflexivity.
-
unfold make_longconst.
simpl.
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]).
rewrite eqm_notbool_ceq_cne.
apply binop_same_eval;
symmetry;
auto;
reflexivity.
-
unfold make_intconst.
simpl.
eexists; (
split; [
repeat (
econstructor;
simpl;
eauto)|]).
rewrite eqm_notbool_ceq_cne.
apply binop_same_eval;
symmetry;
auto;
reflexivity.
-
destruct (
ident_eq id1 id2 &&
fieldlist_eq fld1 fld2);
inv H0.
exists v;
split;
auto.
symmetry;
auto.
-
destruct (
ident_eq id1 id2 &&
fieldlist_eq fld1 fld2);
inv H0.
exists v;
split;
auto.
symmetry;
auto.
Qed.
Lemma norm_cne_zero_boolval:
forall v,
same_eval
(
Ebinop (
OpCmp SESigned Cne)
Tint Tint v (
Eval (
Eint Int.zero)))
(
Eunop OpBoolval Tint v).
Proof.
Lemma make_boolean_correct:
forall e le m a v v'
ty,
eval_expr ge e le m a v ->
same_eval v v' ->
exists vb,
eval_expr ge e le m (
make_boolean a ty)
vb
/\
same_eval (
bool_expr v'
ty)
vb.
Proof.
Lemma make_neg_correct:
forall a tya c va va'
v e le m,
same_eval va va' ->
sem_neg_expr va'
tya =
v ->
make_neg a tya =
OK c ->
eval_expr ge e le m a va ->
exists v',
eval_expr ge e le m c v' /\
same_eval v v'.
Proof.
Lemma make_absfloat_correct:
forall a tya c va va'
v e le m,
same_eval va va' ->
sem_absfloat_expr va'
tya =
v ->
make_absfloat a tya =
OK c ->
eval_expr ge e le m a va ->
exists v',
eval_expr ge e le m c v' /\
same_eval v v'.
Proof.
Lemma eqm_notbool_eq_zero:
forall va sg,
same_eval
(
Eunop OpNotbool Tint va)
(
Ebinop (
OpCmp sg Ceq)
Tint Tint va (
Eval (
Eint Int.zero))).
Proof.
Lemma make_notbool_correct:
forall a tya c va va'
v e le m,
same_eval va va' ->
sem_notbool_expr va'
tya =
v ->
make_notbool a tya =
OK c ->
eval_expr ge e le m a va ->
exists v',
eval_expr ge e le m c v' /\
same_eval v v'.
Proof.
Lemma make_notint_correct:
forall a tya c va va'
v e le m,
same_eval va va' ->
sem_notint_expr va'
tya =
v ->
make_notint a tya =
OK c ->
eval_expr ge e le m a va ->
exists v',
eval_expr ge e le m c v' /\
same_eval v v'.
Proof.
Definition binary_constructor_correct
(
make:
expr ->
type ->
expr ->
type ->
res expr)
(
sem:
expr_sym ->
type ->
expr_sym ->
type ->
option expr_sym):
Prop :=
forall a tya b tyb c va vb vaa vbb v e le m,
same_eval va vaa ->
same_eval vb vbb ->
sem vaa tya vbb tyb =
Some v ->
make a tya b tyb =
OK c ->
eval_expr ge e le m a va ->
eval_expr ge e le m b vb ->
exists v',
eval_expr ge e le m c v' /\
same_eval v v'.
Section MAKE_BIN.
Variable sem_int:
signedness ->
expr_sym ->
expr_sym ->
option expr_sym.
Variable sem_long:
signedness ->
expr_sym ->
expr_sym ->
option expr_sym.
Variable sem_float:
expr_sym ->
expr_sym ->
option expr_sym.
Variable sem_single:
expr_sym ->
expr_sym ->
option expr_sym.
Variables iop iopu fop sop lop lopu:
binary_operation.
Hypothesis iop_ok:
forall x y m,
eval_binop iop x y m =
sem_int Signed x y.
Hypothesis iopu_ok:
forall x y m,
eval_binop iopu x y m =
sem_int Unsigned x y.
Hypothesis lop_ok:
forall x y m,
eval_binop lop x y m =
sem_long Signed x y.
Hypothesis lopu_ok:
forall x y m,
eval_binop lopu x y m =
sem_long Unsigned x y.
Hypothesis fop_ok:
forall x y m,
eval_binop fop x y m =
sem_float x y.
Hypothesis sop_ok:
forall x y m,
eval_binop sop x y m =
sem_single x y.
Hypothesis si_norm:
forall sg,
conserves_same_eval (
sem_int sg).
Hypothesis sl_norm:
forall sg,
conserves_same_eval (
sem_long sg).
Hypothesis sf_norm:
conserves_same_eval sem_float.
Hypothesis ss_norm:
conserves_same_eval sem_single.
Lemma make_binarith_correct:
binary_constructor_correct
(
make_binarith iop iopu fop sop lop lopu)
(
sem_binarith_expr sem_int sem_long sem_float sem_single).
Proof.
red;
unfold make_binarith,
sem_binarith_expr;
intros until m;
intros SE1 SE2 SEM MAKE EV1 EV2.
set (
cls :=
classify_binarith tya tyb)
in *.
set (
ty :=
binarith_type cls)
in *.
monadInv MAKE.
destruct (
sem_cast_expr vaa tya ty)
as [
va'|]
eqn:
Ca;
try discriminate.
destruct (
sem_cast_expr vbb tyb ty)
as [
vb'|]
eqn:
Cb;
try discriminate.
exploit make_cast_correct.
eexact EQ.
eauto.
eauto.
eauto.
intros EV1'.
exploit make_cast_correct.
eexact EQ1.
eauto.
eauto.
eauto.
intros EV2'.
destruct EV1'
as [
v'' [
A B]].
destruct EV2'
as [
v''0 [
C D]].
destruct cls;
inv EQ2.
-
generalize (
si_norm s _ _ _ _ B D).
rewrite SEM.
destr_cond_match;
intuition try discriminate.
destruct s;
inv H0;
eexists;
split;
eauto;
repeat (
econstructor;
simpl;
eauto).
rewrite iop_ok;
eauto.
rewrite iopu_ok;
auto.
-
generalize (
sl_norm s _ _ _ _ B D).
rewrite SEM.
destr_cond_match;
intuition try discriminate.
destruct s;
inv H0;
eexists;
split;
eauto;
econstructor;
eauto with cshm.
rewrite lop_ok;
auto.
rewrite lopu_ok;
auto.
-
generalize (
sf_norm _ _ _ _ B D).
rewrite SEM.
destr_cond_match;
intuition try discriminate.
eexists;
split;
eauto;
econstructor;
simpl;
eauto.
erewrite fop_ok;
eauto with cshm.
-
generalize (
ss_norm _ _ _ _ B D).
rewrite SEM.
destr_cond_match;
intuition try discriminate.
eexists;
split;
eauto;
econstructor;
simpl;
eauto.
erewrite sop_ok;
eauto with cshm.
Qed.
Lemma make_binarith_int_correct:
binary_constructor_correct
(
make_binarith_int iop iopu lop lopu)
(
sem_binarith_expr sem_int sem_long (
fun x y =>
None) (
fun x y =>
None)).
Proof.
red;
unfold make_binarith_int,
sem_binarith_expr;
intros until m;
intros SE1 SE2 SEM MAKE EV1 EV2.
set (
cls :=
classify_binarith tya tyb)
in *.
set (
ty :=
binarith_type cls)
in *.
monadInv MAKE.
destruct (
sem_cast_expr vaa tya ty)
as [
va'|]
eqn:
Ca;
try discriminate.
destruct (
sem_cast_expr vbb tyb ty)
as [
vb'|]
eqn:
Cb;
try discriminate.
exploit make_cast_correct.
eexact EQ.
eauto.
eauto.
eauto.
intros EV1'.
exploit make_cast_correct.
eexact EQ1.
eauto.
eauto.
eauto.
intros EV2'.
destruct EV1'
as [
v'' [
A B]].
destruct EV2'
as [
v''0 [
C D]].
destruct cls;
inv EQ2.
-
generalize (
si_norm s _ _ _ _ B D).
rewrite SEM.
destr_cond_match;
intuition try discriminate.
destruct s;
inv H0;
eexists;
split;
eauto;
repeat (
econstructor;
simpl;
eauto).
rewrite iop_ok;
eauto.
rewrite iopu_ok;
auto.
-
generalize (
sl_norm s _ _ _ _ B D).
rewrite SEM.
destr_cond_match;
intuition try discriminate.
destruct s;
inv H0;
eexists;
split;
eauto;
econstructor;
eauto with cshm.
rewrite lop_ok;
auto.
rewrite lopu_ok;
auto.
Qed.
End MAKE_BIN.
Hint Extern 2 (@
eq (
option val)
_ _) => (
simpl;
reflexivity) :
cshm.
Lemma make_add_correct:
binary_constructor_correct make_add sem_add_expr.
Proof.
red;
unfold make_add,
sem_add_expr;
intros until m;
intros SE1 SE2 SEM MAKE EV1 EV2;
destruct (
classify_add tya tyb);
inv MAKE.
-
inv SEM.
eexists;
split;
eauto.
repeat (
econstructor;
simpl;
eauto).
erewrite <- (
binop_same_eval _ _ _ _ _ _ _ SE1 (
same_eval_refl _));
eauto.
apply binop_same_eval.
reflexivity.
inv SE2.
constructor;
intros;
simpl;
auto.
+
rewrite se_typ;
tauto.
+
NormaliseSpec.simpl_eval'.
rewrite se_eval in Heql0;
auto.
rewrite Heql in Heql0;
inv Heql0.
rewrite Int.mul_commut;
auto.
-
inv SEM.
eexists;
split;
eauto.
repeat (
econstructor;
simpl;
eauto).
erewrite <- (
binop_same_eval _ _ _ _ _ _ _ SE2 (
same_eval_refl _));
eauto.
apply binop_same_eval.
reflexivity.
inv SE1.
constructor;
intros;
simpl;
auto.
+
rewrite se_typ;
tauto.
+
NormaliseSpec.simpl_eval'.
rewrite <-
se_eval in Heql;
auto;
rewrite Heql in Heql0;
inv Heql0;
auto.
rewrite Int.mul_commut;
auto.
-
inv SEM.
eexists;
split;
eauto.
repeat (
econstructor;
simpl;
eauto).
erewrite <- (
binop_same_eval _ _ _ _ _ _ _ SE1 (
same_eval_refl _));
eauto.
apply binop_same_eval.
reflexivity.
inv SE2.
constructor;
intros;
simpl;
auto.
+
rewrite se_typ;
tauto.
+
NormaliseSpec.simpl_eval'.
rewrite se_eval in Heql;
auto;
rewrite Heql in Heql1;
inv Heql1.
rewrite Int.mul_commut;
auto.
-
inv SEM.
eexists;
split;
eauto.
repeat (
econstructor;
simpl;
eauto).
erewrite <- (
binop_same_eval _ _ _ _ _ _ _ SE2 (
same_eval_refl _));
eauto.
apply binop_same_eval.
reflexivity.
inv SE1.
constructor;
intros;
simpl;
auto.
+
rewrite se_typ;
tauto.
+
NormaliseSpec.simpl_eval'.
rewrite se_eval in Heql;
auto;
rewrite Heql in Heql1;
inv Heql1.
rewrite Int.mul_commut;
auto.
-
eapply make_binarith_correct in SEM;
eauto;
red;
intros;
auto;
apply binop_same_eval;
auto.
Qed.
Lemma make_sub_correct:
binary_constructor_correct make_sub sem_sub_expr.
Proof.
red;
unfold make_sub,
sem_sub_expr;
intros until m;
intros SE1 SE2 SEM MAKE EV1 EV2;
destruct (
classify_sub tya tyb);
inv MAKE.
-
inv SEM.
eexists;
split;
eauto.
repeat (
econstructor;
simpl;
eauto).
erewrite <- (
binop_same_eval _ _ _ _ _ _ _ SE1 (
same_eval_refl _));
eauto.
apply binop_same_eval.
reflexivity.
inv SE2.
constructor;
intros;
simpl;
auto.
+
rewrite se_typ;
tauto.
+
NormaliseSpec.simpl_eval'.
rewrite <-
se_eval in Heql;
auto;
rewrite Heql in Heql0;
inv Heql0.
rewrite Int.mul_commut;
auto.
-
inv SEM.
eexists;
split;
eauto.
repeat (
econstructor;
simpl;
eauto).
apply binop_same_eval.
apply binop_same_eval;
symmetry;
auto.
reflexivity.
-
inv SEM.
eexists;
split;
eauto.
repeat (
econstructor;
simpl;
eauto).
apply binop_same_eval;
symmetry;
auto.
inv SE2.
constructor;
intros;
simpl;
auto.
+
rewrite se_typ;
tauto.
+
NormaliseSpec.simpl_eval'.
rewrite se_eval in Heql;
auto;
rewrite Heql in Heql0;
inv Heql0.
rewrite Int.mul_commut;
auto.
-
eapply make_binarith_correct in SEM;
eauto;
red;
intros;
auto;
apply binop_same_eval;
auto.
Qed.
Lemma make_mul_correct:
binary_constructor_correct make_mul sem_mul_expr.
Proof.
Lemma make_div_correct:
binary_constructor_correct make_div sem_div_expr.
Proof.
Lemma make_mod_correct:
binary_constructor_correct make_mod sem_mod_expr.
Proof.
Lemma make_and_correct:
binary_constructor_correct make_and sem_and_expr.
Proof.
Lemma make_or_correct:
binary_constructor_correct make_or sem_or_expr.
Proof.
Lemma make_xor_correct:
binary_constructor_correct make_xor sem_xor_expr.
Proof.
Ltac comput val :=
let x :=
fresh in set val as x in *;
vm_compute in x;
subst x.
Remark small_shift_amount_1:
forall i,
Int64.ltu i Int64.iwordsize =
true ->
Int.ltu (
Int64.loword i)
Int64.iwordsize' =
true
/\
Int64.unsigned i =
Int.unsigned (
Int64.loword i).
Proof.
Remark small_shift_amount_2:
forall i,
Int64.ltu i (
Int64.repr 32) =
true ->
Int.ltu (
Int64.loword i)
Int.iwordsize =
true.
Proof.
Lemma small_shift_amount_3:
forall i,
Int.ltu i Int64.iwordsize' =
true ->
Int64.unsigned (
Int64.repr (
Int.unsigned i)) =
Int.unsigned i.
Proof.
Lemma make_shl_correct:
binary_constructor_correct make_shl sem_shl_expr.
Proof.
red;
unfold make_shl,
sem_shl_expr,
sem_shift_expr;
intros until m;
intros SE1 SE2 SEM MAKE EV1 EV2;
destruct (
classify_shift tya tyb);
inv MAKE;
inv SEM.
-
eexists;
split; [
repeat (
econstructor;
simpl;
eauto)|].
apply binop_same_eval;
symmetry;
auto.
-
eexists;
split; [
repeat (
econstructor;
simpl;
eauto)|].
apply binop_same_eval;
symmetry;
auto.
inv SE2.
constructor;
intros;
simpl;
auto.
+
rewrite se_typ;
tauto.
+
NormaliseSpec.simpl_eval'.
-
eexists;
split; [
repeat (
econstructor;
simpl;
eauto)|].
apply binop_same_eval;
symmetry;
auto.
inv SE2.
constructor;
intros;
simpl;
auto.
+
rewrite se_typ;
tauto.
+
unfold_eval.
rewrite se_eval;
simpl;
auto.
unfold NormaliseSpec.convert_t.
NormaliseSpec.simpl_eval'.
destruct s;
simpl;
auto.
-
eexists;
split; [
repeat (
econstructor;
simpl;
eauto)|].
inv SE1;
inv SE2.
constructor;
intros;
simpl;
auto.
+
rewrite se_typ0;
rewrite se_typ.
repeat (
destr_cond_match);
simpl in *;
intuition try congruence.
+
unfold_eval.
rewrite se_eval0;
auto;
rewrite se_eval;
simpl;
auto.
unfold NormaliseSpec.convert_t.
NormaliseSpec.simpl_eval'.
destruct s;
simpl;
auto;
inv_expr_type (
NormaliseSpec.eSexpr alloc em eb eu Tint vbb);
inv_expr_type (
NormaliseSpec.eSexpr alloc em eb eu Tlong vaa);
unfold Int64.loword;
rewrite Int64.unsigned_repr.
rewrite Int.repr_unsigned.
destruct (
Int.ltu i0 Int64.iwordsize')
eqn:
E;
auto.
generalize (
Int.unsigned_range_2 i0).
intros.
split.
omega.
destruct H2.
etransitivity;
eauto.
vm_compute;
congruence.
rewrite Int.repr_unsigned.
destruct (
Int.ltu i0 Int64.iwordsize')
eqn:
E;
auto.
generalize (
Int.unsigned_range_2 i0).
intros.
split.
omega.
destruct H2.
etransitivity;
eauto.
vm_compute;
congruence.
Qed.
Lemma make_shr_correct:
binary_constructor_correct make_shr sem_shr_expr.
Proof.
Lemma make_cmp_correct:
forall cmp a tya b tyb c va va'
vb'
vb v e le m,
same_eval va va' ->
same_eval vb vb' ->
sem_cmp_expr cmp va'
tya vb'
tyb m =
Some v ->
make_cmp cmp a tya b tyb =
OK c ->
eval_expr ge e le m a va ->
eval_expr ge e le m b vb ->
exists v',
eval_expr ge e le m c v' /\
same_eval v v'.
Proof.
unfold sem_cmp_expr,
make_cmp;
intros until m;
intros SE1 SE2 SEM MAKE EV1 EV2;
destruct (
classify_cmp tya tyb);
inv MAKE;
inv SEM.
-
eexists;
split;
eauto; [
repeat (
econstructor;
simpl;
eauto)|].
apply binop_same_eval;
symmetry;
auto.
-
eexists;
split;
eauto; [
repeat (
econstructor;
simpl;
eauto)|].
inv SE1;
inv SE2.
constructor;
intros;
simpl;
auto;
unfold_eval.
+
rewrite se_typ0.
rewrite se_typ.
repeat (
destr_cond_match;
simpl in *;
intuition try discriminate).
+
rewrite se_eval0;
auto.
rewrite se_eval;
auto.
inv_expr_type (
NormaliseSpec.eSexpr alloc em eb eu Tint va');
inv_expr_type (
NormaliseSpec.eSexpr alloc em eb eu Tlong vb').
-
eexists;
split;
eauto; [
repeat (
econstructor;
simpl;
eauto)|].
inv SE1;
inv SE2.
constructor;
intros;
simpl;
auto;
unfold_eval.
+
rewrite se_typ0.
rewrite se_typ.
repeat (
destr_cond_match;
simpl in *;
intuition try discriminate).
+
rewrite se_eval0;
auto.
rewrite se_eval;
auto.
inv_expr_type (
NormaliseSpec.eSexpr alloc em eb eu Tlong va');
inv_expr_type (
NormaliseSpec.eSexpr alloc em eb eu Tint vb').
-
eapply make_binarith_correct in H1;
eauto;
red;
intros;
auto;
apply binop_same_eval;
auto.
Qed.
Lemma transl_unop_correct:
forall op a tya c va va'
v e le m,
transl_unop op a tya =
OK c ->
eval_expr ge e le m a va ->
same_eval va va' ->
sem_unary_operation_expr op va'
tya =
v ->
exists v',
eval_expr ge e le m c v' /\
same_eval v v'.
Proof.
Lemma transl_binop_correct:
forall op a tya b tyb c va vb va'
vb'
v e le m,
same_eval va va' ->
same_eval vb vb' ->
transl_binop op a tya b tyb =
OK c ->
sem_binary_operation_expr op va'
tya vb'
tyb m =
Some v ->
eval_expr ge e le m a va ->
eval_expr ge e le m b vb ->
exists v',
eval_expr ge e le m c v' /\
same_eval v v'.
Proof.
Lemma make_load_correct:
forall addr ty code bofs v v'
e le m,
make_load addr ty =
OK code ->
eval_expr ge e le m addr v' ->
same_eval v'
bofs ->
deref_loc ty m bofs v ->
exists v'',
eval_expr ge e le m code v'' /\
same_eval v v''.
Proof.
unfold make_load;
intros until m;
intros MKLOAD EVEXP SE DEREF.
inv DEREF.
-
rewrite H in MKLOAD.
inv MKLOAD.
eexists;
split;
eauto;
try reflexivity.
econstructor;
eauto.
revert H0.
unfold Mem.loadv.
erewrite same_eval_eqm;
eauto.
symmetry;
auto.
-
rewrite H in MKLOAD.
inv MKLOAD.
eexists;
split;
eauto.
symmetry;
auto.
-
rewrite H in MKLOAD.
inv MKLOAD.
eexists;
split;
eauto.
symmetry;
auto.
Qed.
Lemma encode_val_tot_length:
forall v v'
chunk,
tcheck_expr v =
tcheck_expr v' ->
length (
Mem.encode_val_tot chunk v) =
length (
Mem.encode_val_tot chunk v').
Proof.
Lemma make_memcpy_correct:
forall f dst src ty k e le m b'
ofs'
m'
v v',
eval_expr ge e le m dst v ->
eval_expr ge e le m src v' ->
Mem.mem_norm m v'
Ptr =
Vptr b'
ofs' ->
assign_loc ty m v (
Eval (
Eptr b'
ofs'))
m' ->
access_mode ty =
By_copy ->
exists m'',
step ge (
State f (
make_memcpy dst src ty)
k e le m)
E0 (
State f Sskip k e le m'') /\
mem_equiv m'
m''.
Proof.
Lemma make_store_correct:
forall addr ty rhs code e le m v v'
m'
f k,
make_store addr ty rhs =
OK code ->
eval_expr ge e le m addr v' ->
eval_expr ge e le m rhs v ->
assign_loc ty m v'
v m' ->
exists m'',
step ge (
State f code k e le m)
E0 (
State f Sskip k e le m'') /\
mem_equiv m'
m''.
Proof.
End CONSTRUCTORS.
Basic preservation invariants
Section CORRECTNESS.
Variable prog:
Clight.program.
Variable tprog:
Csharpminor.program.
Hypothesis TRANSL:
transl_program prog =
OK tprog.
Let ge :=
Genv.globalenv prog.
Let tge :=
Genv.globalenv tprog.
Lemma symbols_preserved:
forall s,
Genv.find_symbol tge s =
Genv.find_symbol ge s.
Proof (
Genv.find_symbol_transf_partial2 transl_fundef transl_globvar _ TRANSL).
Lemma functions_translated:
forall m v f,
Genv.find_funct m ge v =
Some f ->
exists tf,
Genv.find_funct m tge v =
Some tf /\
transl_fundef f =
OK tf.
Proof (
Genv.find_funct_transf_partial2 transl_fundef transl_globvar _ TRANSL).
Lemma function_ptr_translated:
forall b f,
Genv.find_funct_ptr ge b =
Some f ->
exists tf,
Genv.find_funct_ptr tge b =
Some tf /\
transl_fundef f =
OK tf.
Proof (
Genv.find_funct_ptr_transf_partial2 transl_fundef transl_globvar _ TRANSL).
Lemma var_info_translated:
forall b v,
Genv.find_var_info ge b =
Some v ->
exists tv,
Genv.find_var_info tge b =
Some tv /\
transf_globvar transl_globvar v =
OK tv.
Proof (
Genv.find_var_info_transf_partial2 transl_fundef transl_globvar _ TRANSL).
Lemma var_info_rev_translated:
forall b tv,
Genv.find_var_info tge b =
Some tv ->
exists v,
Genv.find_var_info ge b =
Some v /\
transf_globvar transl_globvar v =
OK tv.
Proof (
Genv.find_var_info_rev_transf_partial2 transl_fundef transl_globvar _ TRANSL).
Lemma block_is_volatile_preserved:
forall b,
block_is_volatile tge b =
block_is_volatile ge b.
Proof.
Matching between environments
In this section, we define a matching relation between
a Clight local environment and a Csharpminor local environment.
Record match_env (
e:
Clight.env) (
te:
Csharpminor.env) :
Prop :=
mk_match_env {
me_local:
forall id b ty,
e!
id =
Some (
b,
ty) ->
te!
id =
Some(
b,
sizeof ty);
me_local_inv:
forall id b sz,
te!
id =
Some (
b,
sz) ->
exists ty,
e!
id =
Some(
b,
ty)
}.
Lemma match_env_globals:
forall e te id,
match_env e te ->
e!
id =
None ->
te!
id =
None.
Proof.
intros.
destruct (
te!
id)
as [[
b sz] | ]
eqn:?;
auto.
exploit me_local_inv;
eauto.
intros [
ty EQ].
congruence.
Qed.
Lemma match_env_same_blocks:
forall e te,
match_env e te ->
blocks_of_env te =
Clight.blocks_of_env e.
Proof.
Lemma match_env_free_blocks:
forall e te m m',
match_env e te ->
Mem.free_list m (
Clight.blocks_of_env e) =
Some m' ->
Mem.free_list m (
blocks_of_env te) =
Some m'.
Proof.
Lemma match_env_empty:
match_env Clight.empty_env Csharpminor.empty_env.
Proof.
The following lemmas establish the match_env invariant at
the beginning of a function invocation, after allocation of
local variables and initialization of the parameters.
Lemma match_env_alloc_variables:
forall e1 m1 vars e2 m2,
Clight.alloc_variables e1 m1 vars e2 m2 ->
forall te1,
match_env e1 te1 ->
exists te2,
Csharpminor.alloc_variables te1 m1 (
map transl_var vars)
te2 m2
/\
match_env e2 te2.
Proof.
induction 1;
intros;
simpl.
exists te1;
split.
constructor.
auto.
exploit (
IHalloc_variables (
PTree.set id (
b1,
sizeof ty)
te1)).
constructor.
intros until ty0.
repeat rewrite PTree.gsspec.
destruct (
peq id0 id);
intros.
congruence.
eapply me_local;
eauto.
intros until sz.
repeat rewrite PTree.gsspec.
destruct (
peq id0 id);
intros.
exists ty;
congruence.
eapply me_local_inv;
eauto.
intros [
te2 [
ALLOC MENV]].
exists te2;
split.
econstructor;
eauto.
rewrite Zmax_r;
simpl;
eauto.
generalize (
sizeof_pos ty);
omega.
rewrite Zmax_r;
simpl;
eauto.
generalize (
sizeof_pos ty);
omega.
auto.
Qed.
Lemma create_undef_temps_match:
forall temps,
create_undef_temps (
map fst temps) =
Clight.create_undef_temps temps.
Proof.
induction temps; simpl. auto.
destruct a as [id ty]. simpl. decEq. auto.
Qed.
Lemma bind_parameter_temps_match:
forall vars vals le1 le2,
Clight.bind_parameter_temps vars vals le1 =
Some le2 ->
bind_parameters (
map fst vars)
vals le1 =
Some le2.
Proof.
induction vars; simpl; intros.
destruct vals; inv H. auto.
destruct a as [id ty]. destruct vals; try discriminate. auto.
Qed.
Proof of semantic preservation
Semantic preservation for expressions
The proof of semantic preservation for the translation of expressions
relies on simulation diagrams of the following form:
e, le, m, a ------------------- te, le, m, ta
| |
| |
| |
v v
e, le, m, v ------------------- te, le, m, v
Left: evaluation of r-value expression
a in Clight.
Right: evaluation of its translation
ta in Csharpminor.
Top (precondition): matching between environments
e,
te,
plus well-typedness of expression
a.
Bottom (postcondition): the result values
v
are identical in both evaluations.
We state these diagrams as the following properties, parameterized
by the Clight evaluation.
.
Variable e:
Clight.env.
Variable le:
temp_env.
Variable m:
mem.
Variable te:
Csharpminor.env.
Hypothesis MENV:
match_env e te.
Lemma expr_int_expr_all:
forall alloc em v t,
tcheck_expr v =
Some Tint ->
NormaliseSpec.eSexpr alloc em eb eu t v =
NormaliseSpec.cast t (
NormaliseSpec.eSexpr alloc em eb eu Tint v).
Proof.
Lemma is_norm_norm_id:
forall m e r v
(
MN:
Mem.mem_norm m e r =
v)
(
NU:
v <>
Vundef),
Mem.mem_norm m (
expr_of_val v)
r =
v.
Proof.
Lemma transl_expr_lvalue_correct:
(
forall a v,
Clight.eval_expr ge e le m a v ->
forall ta (
TR:
transl_expr a =
OK ta) ,
exists v',
Csharpminor.eval_expr tge te le m ta v' /\
same_eval v v')
/\(
forall a v,
Clight.eval_lvalue ge e le m a v ->
forall ta (
TR:
transl_lvalue a =
OK ta),
exists v',
Csharpminor.eval_expr tge te le m ta v' /\
same_eval v v') .
Proof.
Lemma transl_expr_correct:
forall a v,
Clight.eval_expr ge e le m a v ->
forall ta,
transl_expr a =
OK ta ->
exists v',
Csharpminor.eval_expr tge te le m ta v' /\
same_eval v v'.
Proof.
Lemma transl_lvalue_correct:
forall a v,
Clight.eval_lvalue ge e le m a v ->
forall ta,
transl_lvalue a =
OK ta ->
exists v',
Csharpminor.eval_expr tge te le m ta v' /\
same_eval v v'.
Proof.
Lemma transl_arglist_correct:
forall al tyl vl,
Clight.eval_exprlist ge e le m al tyl vl ->
forall tal,
transl_arglist al tyl =
OK tal ->
exists vl',
list_forall2 same_eval vl vl' /\
Csharpminor.eval_exprlist tge te le m tal vl'.
Proof.
induction 1;
intros tal TAL.
-
monadInv TAL.
exists nil;
split;
constructor.
-
monadInv TAL.
destruct (
IHeval_exprlist _ EQ0)
as [
vl' [
A B]].
destruct (
transl_expr_correct _ _ H _ EQ)
as [
v' [
C D]].
destruct (
make_cast_correct _ _ _ _ _ _ _ _ _ _ _ EQ1 C (
same_eval_sym _ _ D)
H0)
as [
v0 [
E F]].
(
exists (
v0::
vl');
split;
auto).
constructor;
auto.
constructor;
auto.
Qed.
Lemma typlist_of_arglist_eq:
forall m'
al tyl vl,
Clight.eval_exprlist ge e le m'
al tyl vl ->
typlist_of_arglist al tyl =
typlist_of_typelist tyl.
Proof.
clear.
induction 1; simpl.
auto.
f_equal; auto.
Qed.
End EXPR.
Semantic preservation for statements
The simulation diagram for the translation of statements and functions
is a "plus" diagram of the form
I
S1 ------- R1
| |
t | + | t
v v
S2 ------- R2
I I
The invariant
I is the
match_states predicate that we now define.
:
stmt ->
cont ->
stmt ->
cont ->
Prop :=
|
match_transl_0:
forall ts tk,
match_transl ts tk ts tk
|
match_transl_1:
forall ts tk,
match_transl (
Sblock ts)
tk ts (
Kblock tk).
Lemma match_transl_step:
forall ts tk ts'
tk'
f te le m,
match_transl (
Sblock ts)
tk ts'
tk' ->
star step tge (
State f ts'
tk'
te le m)
E0 (
State f ts (
Kblock tk)
te le m).
Proof.
Inductive match_cont:
type ->
nat ->
nat ->
Clight.cont ->
Csharpminor.cont ->
Prop :=
|
match_Kstop:
forall tyret nbrk ncnt,
match_cont tyret nbrk ncnt Clight.Kstop Kstop
|
match_Kseq:
forall tyret nbrk ncnt s k ts tk,
transl_statement tyret nbrk ncnt s =
OK ts ->
match_cont tyret nbrk ncnt k tk ->
match_cont tyret nbrk ncnt
(
Clight.Kseq s k)
(
Kseq ts tk)
|
match_Kloop1:
forall tyret s1 s2 k ts1 ts2 nbrk ncnt tk,
transl_statement tyret 1%
nat 0%
nat s1 =
OK ts1 ->
transl_statement tyret 0%
nat (
S ncnt)
s2 =
OK ts2 ->
match_cont tyret nbrk ncnt k tk ->
match_cont tyret 1%
nat 0%
nat
(
Clight.Kloop1 s1 s2 k)
(
Kblock (
Kseq ts2 (
Kseq (
Sloop (
Sseq (
Sblock ts1)
ts2)) (
Kblock tk))))
|
match_Kloop2:
forall tyret s1 s2 k ts1 ts2 nbrk ncnt tk,
transl_statement tyret 1%
nat 0%
nat s1 =
OK ts1 ->
transl_statement tyret 0%
nat (
S ncnt)
s2 =
OK ts2 ->
match_cont tyret nbrk ncnt k tk ->
match_cont tyret 0%
nat (
S ncnt)
(
Clight.Kloop2 s1 s2 k)
(
Kseq (
Sloop (
Sseq (
Sblock ts1)
ts2)) (
Kblock tk))
|
match_Kswitch:
forall tyret nbrk ncnt k tk,
match_cont tyret nbrk ncnt k tk ->
match_cont tyret 0%
nat (
S ncnt)
(
Clight.Kswitch k)
(
Kblock tk)
|
match_Kcall_some:
forall tyret nbrk ncnt nbrk'
ncnt'
f e k id tf te le le'
tk,
transl_function f =
OK tf ->
match_env e te ->
temp_env_equiv le le' ->
match_cont (
Clight.fn_return f)
nbrk'
ncnt'
k tk ->
match_cont tyret nbrk ncnt
(
Clight.Kcall id f e le k)
(
Kcall id tf te le'
tk).
Inductive match_states:
Clight.state ->
Csharpminor.state ->
Prop :=
|
match_state:
forall f nbrk ncnt s k e le le'
m m'
tf ts tk te ts'
tk'
(
TRF:
transl_function f =
OK tf)
(
TR:
transl_statement (
Clight.fn_return f)
nbrk ncnt s =
OK ts)
(
MTR:
match_transl ts tk ts'
tk')
(
MENV:
match_env e te)
(
MK:
match_cont (
Clight.fn_return f)
nbrk ncnt k tk)
(
TE:
temp_env_equiv le le')
(
ME:
mem_equiv m m'),
match_states (
Clight.State f s k e le m)
(
State tf ts'
tk'
te le'
m')
|
match_callstate:
forall fd args args'
k m m'
tfd tk targs tres cconv
(
TR:
transl_fundef fd =
OK tfd)
(
MK:
match_cont Tvoid 0%
nat 0%
nat k tk)
(
ISCC:
Clight.is_call_cont k)
(
TY:
type_of_fundef fd =
Tfunction targs tres cconv)
(
ME:
mem_equiv m m')
(
ARGS_eq:
list_forall2 (
same_eval)
args args'),
match_states (
Clight.Callstate fd args k m)
(
Callstate tfd args'
tk m')
|
match_returnstate:
forall res res'
k m m'
tk
(
MK:
match_cont Tvoid 0%
nat 0%
nat k tk)
(
MRES:
same_eval res res')
(
ME:
mem_equiv m m'),
match_states (
Clight.Returnstate res k m)
(
Returnstate res'
tk m').
Remark match_states_skip:
forall f e le le'
te nbrk ncnt k tf tk m m',
transl_function f =
OK tf ->
match_env e te ->
match_cont (
Clight.fn_return f)
nbrk ncnt k tk ->
mem_equiv m m' ->
temp_env_equiv le le' ->
match_states (
Clight.State f Clight.Sskip k e le m) (
State tf Sskip tk te le'
m').
Proof.
intros. econstructor; eauto. simpl; reflexivity. constructor.
Qed.
Commutation between label resolution and compilation
Section FIND_LABEL.
Variable lbl:
label.
Variable tyret:
type.
Lemma transl_find_label:
forall s nbrk ncnt k ts tk
(
TR:
transl_statement tyret nbrk ncnt s =
OK ts)
(
MC:
match_cont tyret nbrk ncnt k tk),
match Clight.find_label lbl s k with
|
None =>
find_label lbl ts tk =
None
|
Some (
s',
k') =>
exists ts',
exists tk',
exists nbrk',
exists ncnt',
find_label lbl ts tk =
Some (
ts',
tk')
/\
transl_statement tyret nbrk'
ncnt'
s' =
OK ts'
/\
match_cont tyret nbrk'
ncnt'
k'
tk'
end
with transl_find_label_ls:
forall ls nbrk ncnt k tls tk
(
TR:
transl_lbl_stmt tyret nbrk ncnt ls =
OK tls)
(
MC:
match_cont tyret nbrk ncnt k tk),
match Clight.find_label_ls lbl ls k with
|
None =>
find_label_ls lbl tls tk =
None
|
Some (
s',
k') =>
exists ts',
exists tk',
exists nbrk',
exists ncnt',
find_label_ls lbl tls tk =
Some (
ts',
tk')
/\
transl_statement tyret nbrk'
ncnt'
s' =
OK ts'
/\
match_cont tyret nbrk'
ncnt'
k'
tk'
end.
Proof.
intro s;
case s;
intros;
try (
monadInv TR);
simpl.
-
auto.
-
unfold make_store,
make_memcpy in EQ3.
destruct (
access_mode (
typeof e));
inv EQ3;
auto.
-
auto.
-
simpl in TR.
destruct (
classify_fun (
typeof e));
monadInv TR.
auto.
-
auto.
-
exploit (
transl_find_label s0 nbrk ncnt (
Clight.Kseq s1 k));
eauto.
econstructor;
eauto.
destruct (
Clight.find_label lbl s0 (
Clight.Kseq s1 k))
as [[
s'
k'] | ].
intros [
ts' [
tk' [
nbrk' [
ncnt' [
A [
B C]]]]]].
rewrite A.
exists ts';
exists tk';
exists nbrk';
exists ncnt';
auto.
intro.
rewrite H.
eapply transl_find_label;
eauto.
-
exploit (
transl_find_label s0);
eauto.
destruct (
Clight.find_label lbl s0 k)
as [[
s'
k'] | ].
intros [
ts' [
tk' [
nbrk' [
ncnt' [
A [
B C]]]]]].
rewrite A.
exists ts';
exists tk';
exists nbrk';
exists ncnt';
auto.
intro.
rewrite H.
eapply transl_find_label;
eauto.
-
exploit (
transl_find_label s0 1%
nat 0%
nat (
Kloop1 s0 s1 k));
eauto.
econstructor;
eauto.
destruct (
Clight.find_label lbl s0 (
Kloop1 s0 s1 k))
as [[
s'
k'] | ].
intros [
ts' [
tk' [
nbrk' [
ncnt' [
A [
B C]]]]]].
rewrite A.
exists ts';
exists tk';
exists nbrk';
exists ncnt';
auto.
intro.
rewrite H.
eapply transl_find_label;
eauto.
econstructor;
eauto.
-
auto.
-
auto.
-
simpl in TR.
destruct o;
monadInv TR.
auto.
auto.
-
assert (
exists b,
ts =
Sblock (
Sswitch b x x0)).
{
destruct (
classify_switch (
typeof e));
inv EQ2;
econstructor;
eauto. }
destruct H as [
b EQ3];
rewrite EQ3;
simpl.
eapply transl_find_label_ls with (
k :=
Clight.Kswitch k);
eauto.
econstructor;
eauto.
-
destruct (
ident_eq lbl l).
exists x;
exists tk;
exists nbrk;
exists ncnt;
auto.
eapply transl_find_label;
eauto.
-
auto.
-
intro ls;
case ls;
intros;
monadInv TR;
simpl.
auto.
exploit (
transl_find_label s nbrk ncnt (
Clight.Kseq (
seq_of_labeled_statement l)
k));
eauto.
econstructor;
eauto.
apply transl_lbl_stmt_2;
eauto.
destruct (
Clight.find_label lbl s (
Clight.Kseq (
seq_of_labeled_statement l)
k))
as [[
s'
k'] | ].
intros [
ts' [
tk' [
nbrk' [
ncnt' [
A [
B C]]]]]].
rewrite A.
exists ts';
exists tk';
exists nbrk';
exists ncnt';
auto.
intro.
rewrite H.
eapply transl_find_label_ls;
eauto.
Qed.
End FIND_LABEL.
Properties of call continuations
Lemma match_cont_call_cont:
forall tyret'
nbrk'
ncnt'
tyret nbrk ncnt k tk,
match_cont tyret nbrk ncnt k tk ->
match_cont tyret'
nbrk'
ncnt' (
Clight.call_cont k) (
call_cont tk).
Proof.
induction 1; simpl; intros; auto.
constructor.
econstructor; eauto.
Qed.
Lemma match_cont_is_call_cont:
forall tyret nbrk ncnt k tk tyret'
nbrk'
ncnt',
match_cont tyret nbrk ncnt k tk ->
Clight.is_call_cont k ->
match_cont tyret'
nbrk'
ncnt'
k tk /\
is_call_cont tk.
Proof.
intros. inv H; simpl in H0; try contradiction; simpl.
split; auto; constructor.
split; auto; econstructor; eauto.
Qed.
The simulation proof
Lemma match_states_mem_equiv:
forall f s ts k e e'
te tf tk m1 m2 m2',
mem_equiv m2 m2' ->
match_states (
Clight.State f s k e te m1)
(
State tf ts tk e'
te m2) ->
match_states (
Clight.State f s k e te m1)
(
State tf ts tk e'
te m2').
Proof.
intros f s ts k e e' te tf tk m1 m2 m2' ME MS.
inv MS.
econstructor; eauto.
rewrite ME0; auto.
Qed.
Lemma eval_unop_same_eval:
forall op v1 v sv',
eval_unop op v1 =
Some v ->
same_eval v1 sv' ->
exists sv,
eval_unop op sv' =
Some sv /\
same_eval sv v.
Proof.
intros op v1 v sv'
EU SE.
destruct op;
simpl in *;
inv EU;
(
eexists;
split;
eauto);
try (
apply unop_same_eval;
symmetry;
auto).
Qed.
Lemma eval_binop_same_eval:
forall op v1 v2 v sv'
sv2'
m m',
eval_binop op v1 v2 m =
Some v ->
same_eval v1 sv' ->
same_eval v2 sv2' ->
exists sv,
eval_binop op sv'
sv2'
m' =
Some sv /\
same_eval sv v.
Proof.
intros op v1 v2 v sv'
sv2'
m m'
EB SE SE2.
destruct op;
simpl in *;
inv EB;
(
eexists;
split;
eauto);
try (
apply binop_same_eval;
symmetry;
auto).
Qed.
Lemma mem_equiv_eval_expr:
forall ge e te m m'
a sv,
mem_equiv m m' ->
eval_expr ge e te m a sv ->
exists sv',
eval_expr ge e te m'
a sv' /\
same_eval sv sv'.
Proof.
intros ge0 e te m m'
a sv ME EV.
induction EV.
-
eexists;
split;
eauto.
econstructor;
eauto.
reflexivity.
-
eexists;
split;
eauto.
econstructor;
eauto.
reflexivity.
-
eexists;
split;
eauto.
econstructor;
eauto.
reflexivity.
-
destruct IHEV as [
sv' [
A B]].
destruct (
eval_unop_same_eval _ _ _ _ H B)
as [
sv [
C D]].
eexists;
split.
econstructor;
simpl;
eauto.
symmetry;
auto.
-
destruct IHEV1 as [
sv' [
A B]].
destruct IHEV2 as [
sv'' [
A'
B']].
destruct (
eval_binop_same_eval _ _ _ _ _ _ _ m'
H B B')
as [
sv [
C D]].
eexists;
split.
econstructor;
simpl;
eauto.
symmetry;
auto.
-
destruct IHEV as [
sv' [
A B]].
unfold Mem.loadv in H.
revert H;
destr_cond_match;
intros;
try discriminate.
destruct (
mem_equiv_load chunk m m'
_ _ _ ME H)
as [
v' [
C D]].
eexists;
split.
econstructor;
simpl;
eauto.
unfold Mem.loadv in *.
setoid_rewrite mem_equiv_norm with (
m':=
m).
rewrite <- (
same_eval_eqm _ _ _ _ B).
rewrite Heqv0.
eauto.
symmetry;
auto.
auto.
Qed.
Lemma free_same_eval_cmp:
forall ge e le m a v v',
eval_expr ge e le m a v ->
same_eval v v' ->
exists v'' :
expr_sym,
eval_expr ge e le m (
make_cmp_ne_zero a)
v'' /\
same_eval (
Eunop OpBoolval Tint v')
v''.
Proof.
Lemma alloc_variables_mem_equiv:
forall m e1 vars e2 m2,
alloc_variables e1 m vars e2 m2 ->
forall m',
mem_equiv m m' ->
exists m2',
alloc_variables e1 m'
vars e2 m2' /\
mem_equiv m2 m2'.
Proof.
induction 1;
simpl;
intros.
eexists;
split;
eauto.
econstructor.
eapply alloc_mem_equiv in H;
eauto.
destruct H as [
m2' [
A B]].
apply IHalloc_variables in B.
destruct B as [
m2'0 [
B C]].
eexists;
split;
eauto.
econstructor;
eauto.
Qed.
Lemma varsort_merge_eq:
forall l l1,
VarSort.merge (
map transl_var l)
(
map transl_var l1) =
map transl_var (
VarSortType.merge l l1).
Proof.
induction l.
- destruct l1; simpl; auto.
- simpl. destruct l1.
+ simpl; auto.
+ simpl; destr_cond_match.
* simpl. f_equal. rewrite <- IHl. auto.
* simpl. f_equal.
revert l1. induction l1; simpl; intros; auto.
destr_cond_match. simpl. f_equal.
rewrite <- IHl; auto.
simpl. f_equal.
auto.
Qed.
Lemma mlts_eq:
forall l1 a,
(
VarSort.merge_list_to_stack
(
map
(
fun a0 :
option (
list (
ident *
type)) =>
match a0 with
|
Some l0 =>
Some (
map transl_var l0)
|
None =>
None
end)
l1) (
map transl_var a)) =
(
map
(
fun a0 :
option (
list (
ident *
type)) =>
match a0 with
|
Some l0 =>
Some (
map transl_var l0)
|
None =>
None
end) (
VarSortType.merge_list_to_stack l1 a)).
Proof.
induction l1;
simpl;
intros;
auto.
destruct a;
simpl;
auto.
f_equal.
rewrite <-
IHl1.
simpl.
rewrite varsort_merge_eq.
auto.
Qed.
Lemma varsort_iter_merge_eq:
forall l l1,
VarSort.iter_merge (
map (
fun a =>
match a with
None =>
None
|
Some l1 =>
Some (
map transl_var l1)
end)
l1) (
map transl_var l) =
map transl_var (
VarSortType.iter_merge l1 l).
Proof.
induction l;
simpl;
auto.
-
induction l1;
simpl;
auto.
destruct a;
simpl.
rewrite IHl1.
rewrite varsort_merge_eq;
auto.
auto.
-
intros.
rewrite <-
IHl.
simpl.
auto.
rewrite <-
mlts_eq.
simpl;
auto.
Qed.
Lemma var_sort_eq:
forall l,
VarSort.sort (
map transl_var l) =
map transl_var (
VarSortType.sort l).
Proof.
Lemma transl_step:
forall S1 t S2,
Clight.step2 ge S1 t S2 ->
forall T1,
match_states S1 T1 ->
exists T2,
plus step tge T1 t T2 /\
match_states S2 T2.
Proof.
Lemma transl_initial_states:
forall S,
Clight.initial_state prog S ->
exists R,
initial_state tprog R /\
match_states S R.
Proof.
Lemma transl_final_states:
forall S R r,
match_states S R ->
Clight.final_state S r ->
final_state R r.
Proof.
intros.
inv H0.
inv H.
inv MK.
constructor.
setoid_rewrite <- (
same_eval_eqm m'
Int _ _ MRES).
rewrite <-
H1.
symmetry;
apply mem_equiv_norm;
auto.
Qed.
Theorem transl_program_correct:
forward_simulation (
Clight.semantics2 prog) (
Csharpminor.semantics tprog).
Proof.
End CORRECTNESS.