Correctness of instruction selection for operators
Require Import Builtins.
Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import ExtValues.
Require Import Memory.
Require Import Globalenvs.
Require Import Cminor.
Require Import Op.
Require Import CminorSel.
Require Import Builtins1.
Require Import SelectOp.
Require Import Events.
Require Import OpHelpers.
Require Import OpHelpersproof.
Require Import DecBoolOps.
Require Import Lia.
Local Open Scope cminorsel_scope.
Local Open Scope string_scope.
Useful lemmas and tactics
The following are trivial lemmas and custom tactics that help
perform backward (inversion) and forward reasoning over the evaluation
of operator applications.
Ltac EvalOp :=
eapply eval_Eop;
eauto with evalexpr.
Ltac InvEval1 :=
match goal with
| [
H: (
eval_expr _ _ _ _ _ (
Eop _ Enil)
_) |-
_ ] =>
inv H;
InvEval1
| [
H: (
eval_expr _ _ _ _ _ (
Eop _ (
_ :::
Enil))
_) |-
_ ] =>
inv H;
InvEval1
| [
H: (
eval_expr _ _ _ _ _ (
Eop _ (
_ :::
_ :::
Enil))
_) |-
_ ] =>
inv H;
InvEval1
| [
H: (
eval_exprlist _ _ _ _ _ Enil _) |-
_ ] =>
inv H;
InvEval1
| [
H: (
eval_exprlist _ _ _ _ _ (
_ :::
_)
_) |-
_ ] =>
inv H;
InvEval1
|
_ =>
idtac
end.
Ltac InvEval2 :=
match goal with
| [
H: (
eval_operation _ _ _ nil _ =
Some _) |-
_ ] =>
simpl in H;
inv H
| [
H: (
eval_operation _ _ _ (
_ ::
nil)
_ =
Some _) |-
_ ] =>
simpl in H;
FuncInv
| [
H: (
eval_operation _ _ _ (
_ ::
_ ::
nil)
_ =
Some _) |-
_ ] =>
simpl in H;
FuncInv
| [
H: (
eval_operation _ _ _ (
_ ::
_ ::
_ ::
nil)
_ =
Some _) |-
_ ] =>
simpl in H;
FuncInv
|
_ =>
idtac
end.
Ltac InvEval :=
InvEval1;
InvEval2;
InvEval2.
Ltac TrivialExists :=
match goal with
| [ |-
exists v,
_ /\
Val.lessdef ?
a v ] =>
exists a;
split; [
EvalOp |
auto]
end.
Correctness of the smart constructors
Section CMCONSTR.
Variable prog:
program.
Variable hf:
helper_functions.
Hypothesis HELPERS:
helper_functions_declared prog hf.
Let ge :=
Genv.globalenv prog.
Variable sp:
val.
Variable e:
env.
Variable m:
mem.
Ltac UseHelper :=
decompose [
Logic.and]
arith_helpers_correct;
eauto.
Ltac DeclHelper :=
red in HELPERS;
decompose [
Logic.and]
HELPERS;
eauto.
Lemma eval_helper:
forall le id name sg args vargs vres,
eval_exprlist ge sp e m le args vargs ->
helper_declared prog id name sg ->
external_implements name sg vargs vres ->
eval_expr ge sp e m le (
Eexternal id sg args)
vres.
Proof.
Corollary eval_helper_1:
forall le id name sg arg1 varg1 vres,
eval_expr ge sp e m le arg1 varg1 ->
helper_declared prog id name sg ->
external_implements name sg (
varg1::
nil)
vres ->
eval_expr ge sp e m le (
Eexternal id sg (
arg1 :::
Enil))
vres.
Proof.
intros.
eapply eval_helper;
eauto.
constructor;
auto.
constructor.
Qed.
Corollary eval_helper_2:
forall le id name sg arg1 arg2 varg1 varg2 vres,
eval_expr ge sp e m le arg1 varg1 ->
eval_expr ge sp e m le arg2 varg2 ->
helper_declared prog id name sg ->
external_implements name sg (
varg1::
varg2::
nil)
vres ->
eval_expr ge sp e m le (
Eexternal id sg (
arg1 :::
arg2 :::
Enil))
vres.
Proof.
intros.
eapply eval_helper;
eauto.
constructor;
auto.
constructor;
auto.
constructor.
Qed.
We now show that the code generated by "smart constructor" functions
such as
Selection.notint behaves as expected. Continuing the
notint example, we show that if the expression
e
evaluates to some integer value
Vint n, then
Selection.notint e
evaluates to a value
Vint (Int.not n) which is indeed the integer
negation of the value of
e.
All proofs follow a common pattern:
-
Reasoning by case over the result of the classification functions
(such as add_match for integer addition), gathering additional
information on the shape of the argument expressions in the non-default
cases.
-
Inversion of the evaluations of the arguments, exploiting the additional
information thus gathered.
-
Equational reasoning over the arithmetic operations performed,
using the lemmas from the Int and Float modules.
-
Construction of an evaluation derivation for the expression returned
by the smart constructor.
(
cstr:
expr ->
expr) (
sem:
val ->
val) :
Prop :=
forall le a x,
eval_expr ge sp e m le a x ->
exists v,
eval_expr ge sp e m le (
cstr a)
v /\
Val.lessdef (
sem x)
v.
Definition binary_constructor_sound (
cstr:
expr ->
expr ->
expr) (
sem:
val ->
val ->
val) :
Prop :=
forall le a x b y,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
exists v,
eval_expr ge sp e m le (
cstr a b)
v /\
Val.lessdef (
sem x y)
v.
Theorem eval_addrsymbol:
forall le id ofs,
exists v,
eval_expr ge sp e m le (
addrsymbol id ofs)
v /\
Val.lessdef (
Genv.symbol_address ge id ofs)
v.
Proof.
intros.
unfold addrsymbol.
econstructor;
split.
EvalOp.
simpl;
eauto.
auto.
Qed.
Theorem eval_addrstack:
forall le ofs,
exists v,
eval_expr ge sp e m le (
addrstack ofs)
v /\
Val.lessdef (
Val.offset_ptr sp ofs)
v.
Proof.
intros.
unfold addrstack.
econstructor;
split.
EvalOp.
simpl;
eauto.
auto.
Qed.
Theorem eval_addimm_shlimm:
forall sh k2,
unary_constructor_sound (
addimm_shlimm sh k2) (
fun x =>
ExtValues.addx sh x (
Vint k2)).
Proof.
Theorem eval_addimm:
forall n,
unary_constructor_sound (
addimm n) (
fun x =>
Val.add x (
Vint n)).
Proof.
Lemma eval_addx:
forall n,
binary_constructor_sound (
add_shlimm n) (
ExtValues.addx n).
Proof.
Theorem eval_add:
binary_constructor_sound add Val.add.
Proof.
Theorem eval_sub:
binary_constructor_sound sub Val.sub.
Proof.
Theorem eval_negint:
unary_constructor_sound negint (
fun v =>
Val.sub Vzero v).
Proof.
red;
intros until x.
unfold negint.
case (
negint_match a);
intros;
InvEval.
TrivialExists.
TrivialExists.
Qed.
Theorem eval_shlimm:
forall n,
unary_constructor_sound (
fun a =>
shlimm a n)
(
fun x =>
Val.shl x (
Vint n)).
Proof.
Theorem eval_shruimm:
forall n,
unary_constructor_sound (
fun a =>
shruimm a n)
(
fun x =>
Val.shru x (
Vint n)).
Proof.
Theorem eval_shrimm:
forall n,
unary_constructor_sound (
fun a =>
shrimm a n)
(
fun x =>
Val.shr x (
Vint n)).
Proof.
Lemma eval_mulimm_base:
forall n,
unary_constructor_sound (
mulimm_base n) (
fun x =>
Val.mul x (
Vint n)).
Proof.
Theorem eval_mulimm:
forall n,
unary_constructor_sound (
mulimm n) (
fun x =>
Val.mul x (
Vint n)).
Proof.
Theorem eval_mul:
binary_constructor_sound mul Val.mul.
Proof.
Theorem eval_mulhs:
binary_constructor_sound mulhs Val.mulhs.
Proof.
Theorem eval_mulhu:
binary_constructor_sound mulhu Val.mulhu.
Proof.
Theorem eval_andimm:
forall n,
unary_constructor_sound (
andimm n) (
fun x =>
Val.and x (
Vint n)).
Proof.
Theorem eval_and:
binary_constructor_sound and Val.and.
Proof.
Theorem eval_orimm:
forall n,
unary_constructor_sound (
orimm n) (
fun x =>
Val.or x (
Vint n)).
Proof.
Remark eval_same_expr:
forall a1 a2 le v1 v2,
same_expr_pure a1 a2 =
true ->
eval_expr ge sp e m le a1 v1 ->
eval_expr ge sp e m le a2 v2 ->
a1 =
a2 /\
v1 =
v2.
Proof.
intros until v2.
destruct a1;
simpl;
try (
intros;
discriminate).
destruct a2;
simpl;
try (
intros;
discriminate).
case (
ident_eq i i0);
intros.
subst i0.
inversion H0.
inversion H1.
split.
auto.
congruence.
discriminate.
Qed.
Lemma int_eq_commut:
forall x y :
int,
(
Int.eq x y) = (
Int.eq y x).
Proof.
Theorem eval_or:
binary_constructor_sound or Val.or.
Proof.
unfold or;
red;
intros.
assert (
DEFAULT:
exists v,
eval_expr ge sp e m le (
Eop Oor (
a:::
b:::
Enil))
v /\
Val.lessdef (
Val.or x y)
v)
by TrivialExists.
assert (
ROR:
forall v n1 n2,
Int.add n1 n2 =
Int.iwordsize ->
Val.lessdef (
Val.or (
Val.shl v (
Vint n1)) (
Val.shru v (
Vint n2)))
(
Val.ror v (
Vint n2))).
{
intros.
destruct v;
simpl;
auto.
destruct (
Int.ltu n1 Int.iwordsize)
eqn:
N1;
auto.
destruct (
Int.ltu n2 Int.iwordsize)
eqn:
N2;
auto.
simpl.
rewrite <-
Int.or_ror;
auto. }
destruct (
or_match a b);
InvEval.
-
rewrite Val.or_commut.
apply eval_orimm;
auto.
-
apply eval_orimm;
auto.
-
predSpec Int.eq Int.eq_spec (
Int.add n1 n2)
Int.iwordsize;
auto.
destruct (
same_expr_pure t1 t2)
eqn:?;
auto.
InvEval.
exploit eval_same_expr;
eauto.
intros [
EQ1 EQ2];
subst.
exists (
Val.ror v0 (
Vint n2));
split.
EvalOp.
apply ROR;
auto.
-
predSpec Int.eq Int.eq_spec (
Int.add n1 n2)
Int.iwordsize;
auto.
destruct (
same_expr_pure t1 t2)
eqn:?;
auto.
InvEval.
exploit eval_same_expr;
eauto.
intros [
EQ1 EQ2];
subst.
exists (
Val.ror v1 (
Vint n2));
split.
EvalOp.
rewrite Val.or_commut.
apply ROR;
auto.
-
TrivialExists;
simpl;
congruence.
-
rewrite Val.or_commut.
TrivialExists;
simpl;
congruence.
-
set (
zstop := (
int_highest_bit mask)).
set (
zstart := (
Int.unsigned start)).
destruct (
is_bitfield _ _)
eqn:
Risbitfield.
+
destruct (
and_dec _ _)
as [[
Rmask Rnmask] | ].
*
simpl in H6.
injection H6.
clear H6.
intro.
subst y.
subst x.
TrivialExists.
simpl.
f_equal.
unfold insf.
rewrite Risbitfield.
rewrite Rmask.
rewrite Rnmask.
simpl.
unfold bitfield_mask.
subst v0.
subst zstart.
rewrite Int.repr_unsigned.
reflexivity.
*
apply DEFAULT.
+
apply DEFAULT.
-
set (
zstop := (
int_highest_bit mask)).
set (
zstart := (
Int.unsigned start)).
destruct (
is_bitfield _ _)
eqn:
Risbitfield.
+
destruct (
and_dec _ _)
as [[
Rmask Rnmask] | ].
*
simpl in H6.
injection H6.
clear H6.
intro.
subst y.
subst x.
TrivialExists.
rewrite Val.or_commut.
simpl.
f_equal.
unfold insf.
rewrite Risbitfield.
rewrite Rmask.
rewrite Rnmask.
simpl.
unfold bitfield_mask.
subst v1.
subst zstart.
rewrite Int.repr_unsigned.
reflexivity.
*
apply DEFAULT.
+
apply DEFAULT.
-
set (
zstop := (
int_highest_bit mask)).
set (
zstart := 0).
destruct (
is_bitfield _ _)
eqn:
Risbitfield.
+
destruct (
and_dec _ _)
as [[
Rmask Rnmask] | ].
*
subst y.
subst x.
TrivialExists.
simpl.
f_equal.
unfold insf.
rewrite Risbitfield.
rewrite Rmask.
rewrite Rnmask.
simpl.
unfold bitfield_mask.
subst zstart.
rewrite (
Val.or_commut (
Val.and v1 _)).
rewrite (
Val.or_commut (
Val.and v1 _)).
destruct v0;
simpl;
trivial.
unfold Int.ltu,
Int.iwordsize,
Int.zwordsize.
rewrite Int.unsigned_repr.
**
rewrite Int.unsigned_repr.
***
simpl.
rewrite Int.shl_zero.
reflexivity.
***
simpl.
unfold Int.max_unsigned,
Int.modulus.
simpl.
lia.
**
unfold Int.max_unsigned,
Int.modulus.
simpl.
lia.
*
clear Risbitfield.
clear o.
clear zstop.
set (
zstop := (
int_highest_bit nmask)).
destruct (
is_bitfield _ _)
eqn:
Risbitfield.
++
destruct (
and_dec _ _)
as [[
Rmask Rnmask] | ].
**
subst y.
subst x.
TrivialExists.
simpl.
f_equal.
rewrite Val.or_commut.
unfold insf.
rewrite Risbitfield.
rewrite Rmask.
rewrite Rnmask.
simpl.
unfold bitfield_mask.
subst zstart.
rewrite (
Val.or_commut (
Val.and v0 _)).
rewrite (
Val.or_commut (
Val.and v0 _)).
destruct v1;
simpl;
trivial.
unfold Int.ltu,
Int.iwordsize,
Int.zwordsize.
rewrite Int.unsigned_repr.
***
rewrite Int.unsigned_repr.
****
simpl.
rewrite Int.shl_zero.
reflexivity.
****
simpl.
unfold Int.max_unsigned,
Int.modulus.
simpl.
lia.
***
unfold Int.max_unsigned,
Int.modulus.
simpl.
lia.
**
apply DEFAULT.
++
apply DEFAULT.
+
apply DEFAULT.
-
apply DEFAULT.
Qed.
Theorem eval_xorimm:
forall n,
unary_constructor_sound (
xorimm n) (
fun x =>
Val.xor x (
Vint n)).
Proof.
Theorem eval_xor:
binary_constructor_sound xor Val.xor.
Proof.
Theorem eval_notint:
unary_constructor_sound notint Val.notint.
Proof.
assert (
forall v,
Val.lessdef (
Val.notint (
Val.notint v))
v).
destruct v;
simpl;
auto.
rewrite Int.not_involutive;
auto.
unfold notint;
red;
intros until x;
case (
notint_match a);
intros;
InvEval.
-
TrivialExists;
simpl;
congruence.
-
TrivialExists;
simpl;
congruence.
-
TrivialExists;
simpl;
congruence.
-
TrivialExists;
simpl;
congruence.
-
TrivialExists;
simpl;
congruence.
-
TrivialExists;
simpl;
congruence.
-
subst x.
exists (
Val.and v1 v0);
split;
trivial.
econstructor.
constructor.
eassumption.
constructor.
eassumption.
constructor.
simpl.
reflexivity.
-
subst x.
exists (
Val.and v1 (
Vint n));
split;
trivial.
econstructor.
constructor.
eassumption.
constructor.
simpl.
reflexivity.
-
subst x.
exists (
Val.or v1 v0);
split;
trivial.
econstructor.
constructor.
eassumption.
constructor.
eassumption.
constructor.
simpl.
reflexivity.
-
subst x.
exists (
Val.or v1 (
Vint n));
split;
trivial.
econstructor.
constructor.
eassumption.
constructor.
simpl.
reflexivity.
-
subst x.
exists (
Val.xor v1 v0);
split;
trivial.
econstructor.
constructor.
eassumption.
constructor.
eassumption.
constructor.
simpl.
reflexivity.
-
subst x.
exists (
Val.xor v1 (
Vint n));
split;
trivial.
econstructor.
constructor.
eassumption.
constructor.
simpl.
reflexivity.
-
subst x.
TrivialExists.
simpl.
destruct v0;
destruct v1;
simpl;
trivial.
f_equal.
f_equal.
rewrite Int.not_and_or_not.
rewrite Int.not_involutive.
apply Int.or_commut.
-
subst x.
TrivialExists.
simpl.
destruct v1;
simpl;
trivial.
f_equal.
f_equal.
rewrite Int.not_and_or_not.
rewrite Int.not_involutive.
reflexivity.
-
subst x.
TrivialExists.
simpl.
destruct v0;
destruct v1;
simpl;
trivial.
f_equal.
f_equal.
rewrite Int.not_or_and_not.
rewrite Int.not_involutive.
apply Int.and_commut.
-
subst x.
TrivialExists.
simpl.
destruct v1;
simpl;
trivial.
f_equal.
f_equal.
rewrite Int.not_or_and_not.
rewrite Int.not_involutive.
reflexivity.
-
subst x.
exists v1;
split;
trivial.
-
TrivialExists.
-
TrivialExists.
Qed.
Theorem eval_divs_base:
forall le a b x y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.divs x y =
Some z ->
exists v,
eval_expr ge sp e m le (
divs_base a b)
v /\
Val.lessdef z v.
Proof.
Theorem eval_mods_base:
forall le a b x y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.mods x y =
Some z ->
exists v,
eval_expr ge sp e m le (
mods_base a b)
v /\
Val.lessdef z v.
Proof.
Theorem eval_divu_base:
forall le a b x y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.divu x y =
Some z ->
exists v,
eval_expr ge sp e m le (
divu_base a b)
v /\
Val.lessdef z v.
Proof.
Theorem eval_modu_base:
forall le a b x y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.modu x y =
Some z ->
exists v,
eval_expr ge sp e m le (
modu_base a b)
v /\
Val.lessdef z v.
Proof.
Theorem eval_shrximm:
forall le a n x z,
eval_expr ge sp e m le a x ->
Val.shrx x (
Vint n) =
Some z ->
exists v,
eval_expr ge sp e m le (
shrximm a n)
v /\
Val.lessdef z v.
Proof.
Theorem eval_shl:
binary_constructor_sound shl Val.shl.
Proof.
Theorem eval_shr:
binary_constructor_sound shr Val.shr.
Proof.
Theorem eval_shru:
binary_constructor_sound shru Val.shru.
Proof.
Theorem eval_negf:
unary_constructor_sound negf Val.negf.
Proof.
red; intros. TrivialExists.
Qed.
Theorem eval_absf:
unary_constructor_sound absf Val.absf.
Proof.
red; intros. TrivialExists.
Qed.
Theorem eval_addf:
binary_constructor_sound addf Val.addf.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_subf:
binary_constructor_sound subf Val.subf.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_mulf:
binary_constructor_sound mulf Val.mulf.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_negfs:
unary_constructor_sound negfs Val.negfs.
Proof.
red; intros. TrivialExists.
Qed.
Theorem eval_absfs:
unary_constructor_sound absfs Val.absfs.
Proof.
red; intros. TrivialExists.
Qed.
Theorem eval_addfs:
binary_constructor_sound addfs Val.addfs.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_subfs:
binary_constructor_sound subfs Val.subfs.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_mulfs:
binary_constructor_sound mulfs Val.mulfs.
Proof.
red; intros; TrivialExists.
Qed.
Section COMP_IMM.
Variable default:
comparison ->
int ->
condition.
Variable intsem:
comparison ->
int ->
int ->
bool.
Variable sem:
comparison ->
val ->
val ->
val.
Hypothesis sem_int:
forall c x y,
sem c (
Vint x) (
Vint y) =
Val.of_bool (
intsem c x y).
Hypothesis sem_undef:
forall c v,
sem c Vundef v =
Vundef.
Hypothesis sem_eq:
forall x y,
sem Ceq (
Vint x) (
Vint y) =
Val.of_bool (
Int.eq x y).
Hypothesis sem_ne:
forall x y,
sem Cne (
Vint x) (
Vint y) =
Val.of_bool (
negb (
Int.eq x y)).
Hypothesis sem_default:
forall c v n,
sem c v (
Vint n) =
Val.of_optbool (
eval_condition (
default c n) (
v ::
nil)
m).
Lemma eval_compimm:
forall le c a n2 x,
eval_expr ge sp e m le a x ->
exists v,
eval_expr ge sp e m le (
compimm default intsem c a n2)
v
/\
Val.lessdef (
sem c x (
Vint n2))
v.
Proof.
Hypothesis sem_swap:
forall c x y,
sem (
swap_comparison c)
x y =
sem c y x.
Lemma eval_compimm_swap:
forall le c a n2 x,
eval_expr ge sp e m le a x ->
exists v,
eval_expr ge sp e m le (
compimm default intsem (
swap_comparison c)
a n2)
v
/\
Val.lessdef (
sem c (
Vint n2)
x)
v.
Proof.
End COMP_IMM.
Theorem eval_comp:
forall c,
binary_constructor_sound (
comp c) (
Val.cmp c).
Proof.
Theorem eval_compu:
forall c,
binary_constructor_sound (
compu c) (
Val.cmpu (
Mem.valid_pointer m)
c).
Proof.
Theorem eval_compf:
forall c,
binary_constructor_sound (
compf c) (
Val.cmpf c).
Proof.
intros;
red;
intros.
unfold compf.
TrivialExists.
Qed.
Theorem eval_compfs:
forall c,
binary_constructor_sound (
compfs c) (
Val.cmpfs c).
Proof.
intros;
red;
intros.
unfold compfs.
TrivialExists.
Qed.
Theorem eval_cast8signed:
unary_constructor_sound cast8signed (
Val.sign_ext 8).
Proof.
Theorem eval_cast8unsigned:
unary_constructor_sound cast8unsigned (
Val.zero_ext 8).
Proof.
Theorem eval_cast16signed:
unary_constructor_sound cast16signed (
Val.sign_ext 16).
Proof.
Theorem eval_cast16unsigned:
unary_constructor_sound cast16unsigned (
Val.zero_ext 16).
Proof.
Theorem eval_intoffloat:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.intoffloat x =
Some y ->
exists v,
eval_expr ge sp e m le (
intoffloat a)
v /\
Val.lessdef y v.
Proof.
intros;
unfold intoffloat.
TrivialExists.
simpl.
rewrite H0.
reflexivity.
Qed.
Theorem eval_intuoffloat:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.intuoffloat x =
Some y ->
exists v,
eval_expr ge sp e m le (
intuoffloat a)
v /\
Val.lessdef y v.
Proof.
intros;
unfold intuoffloat.
TrivialExists.
simpl.
rewrite H0.
reflexivity.
Qed.
Theorem eval_floatofintu:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.floatofintu x =
Some y ->
exists v,
eval_expr ge sp e m le (
floatofintu a)
v /\
Val.lessdef y v.
Proof.
intros.
unfold Val.floatofintu in *.
unfold floatofintu.
destruct (
floatofintu_match a).
-
InvEval.
TrivialExists.
-
InvEval.
TrivialExists.
constructor.
econstructor.
constructor.
eassumption.
constructor.
simpl.
f_equal.
constructor.
simpl.
destruct x;
simpl;
trivial;
try discriminate.
f_equal.
inv H0.
f_equal.
rewrite Float.of_intu_of_longu.
reflexivity.
Qed.
Theorem eval_floatofint:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.floatofint x =
Some y ->
exists v,
eval_expr ge sp e m le (
floatofint a)
v /\
Val.lessdef y v.
Proof.
intros.
unfold floatofint.
destruct (
floatofint_match a).
-
InvEval.
TrivialExists.
-
InvEval.
TrivialExists.
constructor.
econstructor.
constructor.
eassumption.
constructor.
simpl.
f_equal.
constructor.
simpl.
destruct x;
simpl;
trivial;
try discriminate.
f_equal.
inv H0.
f_equal.
rewrite Float.of_int_of_long.
reflexivity.
Qed.
Theorem eval_intofsingle:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.intofsingle x =
Some y ->
exists v,
eval_expr ge sp e m le (
intofsingle a)
v /\
Val.lessdef y v.
Proof.
intros;
unfold intofsingle.
TrivialExists.
simpl.
rewrite H0.
reflexivity.
Qed.
Theorem eval_singleofint:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.singleofint x =
Some y ->
exists v,
eval_expr ge sp e m le (
singleofint a)
v /\
Val.lessdef y v.
Proof.
intros;
unfold singleofint;
TrivialExists.
simpl.
rewrite H0.
reflexivity.
Qed.
Theorem eval_intuofsingle:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.intuofsingle x =
Some y ->
exists v,
eval_expr ge sp e m le (
intuofsingle a)
v /\
Val.lessdef y v.
Proof.
intros;
unfold intuofsingle.
TrivialExists.
simpl.
rewrite H0.
reflexivity.
Qed.
Theorem eval_singleofintu:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.singleofintu x =
Some y ->
exists v,
eval_expr ge sp e m le (
singleofintu a)
v /\
Val.lessdef y v.
Proof.
intros;
unfold intuofsingle.
TrivialExists.
simpl.
rewrite H0.
reflexivity.
Qed.
Theorem eval_singleoffloat:
unary_constructor_sound singleoffloat Val.singleoffloat.
Proof.
Theorem eval_floatofsingle:
unary_constructor_sound floatofsingle Val.floatofsingle.
Proof.
Theorem eval_addressing:
forall le chunk a v b ofs,
eval_expr ge sp e m le a v ->
v =
Vptr b ofs ->
match addressing chunk a with (
mode,
args) =>
exists vl,
eval_exprlist ge sp e m le args vl /\
eval_addressing ge sp mode vl =
Some v
end.
Proof.
intros until v.
unfold addressing;
case (
addressing_match a);
intros;
InvEval.
-
exists (@
nil val);
split.
eauto with evalexpr.
simpl.
auto.
-
destruct (
orb _ _).
+
exists (
Vptr b ofs0 ::
nil);
split.
constructor.
EvalOp.
simpl.
congruence.
constructor.
simpl.
rewrite Ptrofs.add_zero.
congruence.
+
exists (@
nil val);
split.
constructor.
simpl;
auto.
-
exists (
v1 ::
nil);
split.
eauto with evalexpr.
simpl.
destruct v1;
simpl in H;
try discriminate.
-
exists (
v1 ::
nil);
split.
eauto with evalexpr.
simpl.
destruct v1;
simpl in H;
try discriminate.
destruct Archi.ptr64 eqn:
SF;
inv H.
simpl.
auto.
-
destruct (
Compopts.optim_xsaddr tt).
+
destruct (
Z.eq_dec _ _).
*
exists (
v1 ::
v2 ::
nil);
split.
repeat (
constructor;
auto).
simpl.
rewrite Int.repr_unsigned.
destruct v2;
simpl in *;
congruence.
*
exists (
v1 ::
v0 ::
nil);
split.
repeat (
constructor;
auto).
econstructor.
repeat (
constructor;
auto).
eassumption.
simpl.
congruence.
simpl.
congruence.
+
exists (
v1 ::
v0 ::
nil);
split.
repeat (
constructor;
auto).
econstructor.
repeat (
constructor;
auto).
eassumption.
simpl.
congruence.
simpl.
congruence.
-
unfold addxl in *.
destruct (
Compopts.optim_xsaddr tt).
+
unfold int_of_shift1_4 in *.
destruct (
Z.eq_dec _ _).
*
exists (
v0 ::
v1 ::
nil);
split.
repeat (
constructor;
auto).
simpl.
congruence.
*
eexists;
split.
repeat (
constructor;
auto).
eassumption.
econstructor.
repeat (
constructor;
auto).
eassumption.
simpl.
reflexivity.
simpl.
congruence.
+
eexists;
split.
repeat (
constructor;
auto).
eassumption.
econstructor.
repeat (
constructor;
auto).
eassumption.
simpl.
reflexivity.
simpl.
unfold int_of_shift1_4 in *.
congruence.
-
exists (
v1 ::
v0 ::
nil);
split.
repeat (
constructor;
auto).
simpl.
congruence.
-
exists (
v ::
nil);
split.
eauto with evalexpr.
subst.
simpl.
rewrite Ptrofs.add_zero;
auto.
Qed.
Theorem eval_builtin_arg:
forall a v,
eval_expr ge sp e m nil a v ->
CminorSel.eval_builtin_arg ge sp e m (
builtin_arg a)
v.
Proof.
intros until v.
unfold builtin_arg;
case (
builtin_arg_match a);
intros.
-
InvEval.
constructor.
-
InvEval.
constructor.
-
InvEval.
constructor.
-
InvEval.
simpl in H5.
inv H5.
constructor.
-
InvEval.
subst v.
constructor;
auto.
-
inv H.
InvEval.
simpl in H6;
inv H6.
constructor;
auto.
-
destruct Archi.ptr64 eqn:
SF.
+
constructor;
auto.
+
InvEval.
replace v with (
if Archi.ptr64 then Val.addl v1 (
Vint n)
else Val.add v1 (
Vint n)).
repeat constructor;
auto.
rewrite SF;
auto.
-
destruct Archi.ptr64 eqn:
SF.
+
InvEval.
replace v with (
if Archi.ptr64 then Val.addl v1 (
Vlong n)
else Val.add v1 (
Vlong n)).
repeat constructor;
auto.
+
constructor;
auto.
-
constructor;
auto.
Qed.
Lemma eval_neg_condition0:
forall cond0:
condition0,
forall v1:
val,
forall m:
mem,
(
eval_condition0 (
negate_condition0 cond0)
v1 m) =
option_map negb (
eval_condition0 cond0 v1 m).
Proof.
Lemma select_neg:
forall a b c,
Val.select (
option_map negb a)
b c =
Val.select a c b.
Proof.
destruct a; simpl; trivial.
destruct b; simpl; trivial.
Qed.
Lemma eval_select0:
forall le ty cond0 ac vc a1 v1 a2 v2,
eval_expr ge sp e m le ac vc ->
eval_expr ge sp e m le a1 v1 ->
eval_expr ge sp e m le a2 v2 ->
exists v,
eval_expr ge sp e m le (
select0 ty cond0 a1 a2 ac)
v
/\
Val.lessdef (
Val.select (
eval_condition0 cond0 vc m)
v1 v2 ty)
v.
Proof.
Lemma bool_cond0_ne:
forall ob :
option bool,
forall m,
(
eval_condition0 (
Ccomp0 Cne) (
Val.of_optbool ob)
m) =
ob.
Proof.
destruct ob; simpl; trivial.
intro.
destruct b; reflexivity.
Qed.
Lemma eval_condition_ccomp_swap :
forall c x y m,
eval_condition (
Ccomp (
swap_comparison c)) (
x ::
y ::
nil)
m=
eval_condition (
Ccomp c) (
y ::
x ::
nil)
m.
Proof.
Lemma eval_condition_ccompu_swap :
forall c x y m,
eval_condition (
Ccompu (
swap_comparison c)) (
x ::
y ::
nil)
m=
eval_condition (
Ccompu c) (
y ::
x ::
nil)
m.
Proof.
Lemma eval_condition_ccompl_swap :
forall c x y m,
eval_condition (
Ccompl (
swap_comparison c)) (
x ::
y ::
nil)
m=
eval_condition (
Ccompl c) (
y ::
x ::
nil)
m.
Proof.
Lemma eval_condition_ccomplu_swap :
forall c x y m,
eval_condition (
Ccomplu (
swap_comparison c)) (
x ::
y ::
nil)
m=
eval_condition (
Ccomplu c) (
y ::
x ::
nil)
m.
Proof.
Lemma int_ltu_zero :
forall i,
Int.ltu i Int.zero =
false.
Proof.
Lemma cmpu_bool_Clt :
forall pred v0 b,
Val.cmpu_bool pred Clt v0 (
Vint Int.zero) =
Some b ->
b =
false.
Proof.
intros until b.
intro CMP.
destruct v0;
cbn in CMP;
try discriminate.
inv CMP.
apply int_ltu_zero.
Qed.
Lemma cmpu_bool_Cge :
forall pred v0 b,
Val.cmpu_bool pred Cge v0 (
Vint Int.zero) =
Some b ->
b =
true.
Proof.
intros until b.
intro CMP.
destruct v0;
cbn in CMP;
try discriminate.
inv CMP.
rewrite int_ltu_zero.
reflexivity.
Qed.
Lemma int64_ltu_zero :
forall i,
Int64.ltu i Int64.zero =
false.
Proof.
Lemma cmplu_bool_Clt :
forall pred v0 b,
Val.cmplu_bool pred Clt v0 (
Vlong Int64.zero) =
Some b ->
b =
false.
Proof.
intros until b.
intro CMP.
destruct v0;
cbn in CMP;
try discriminate.
{
inv CMP.
apply int64_ltu_zero.
}
repeat rewrite if_same in CMP.
discriminate.
Qed.
Lemma cmplu_bool_Cge :
forall pred v0 b,
Val.cmplu_bool pred Cge v0 (
Vlong Int64.zero) =
Some b ->
b =
true.
Proof.
intros until b.
intro CMP.
destruct v0;
cbn in CMP;
try discriminate.
{
inv CMP.
rewrite int64_ltu_zero.
reflexivity.
}
repeat rewrite if_same in CMP.
discriminate.
Qed.
Theorem eval_select:
forall le ty cond al vl a1 v1 a2 v2 a b,
select ty cond al a1 a2 =
Some a ->
eval_exprlist ge sp e m le al vl ->
eval_expr ge sp e m le a1 v1 ->
eval_expr ge sp e m le a2 v2 ->
eval_condition cond vl m =
Some b ->
exists v,
eval_expr ge sp e m le a v
/\
Val.lessdef (
Val.select (
Some b)
v1 v2 ty)
v.
Proof.
Theorem eval_divf_base:
forall le a b x y,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
exists v,
eval_expr ge sp e m le (
divf_base a b)
v /\
Val.lessdef (
Val.divf x y)
v.
Proof.
Lemma eval_divfs_base1:
forall le a b x y,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
exists v,
eval_expr ge sp e m le (
divfs_base1 b)
v /\
Val.lessdef (
ExtValues.invfs y)
v.
Proof.
intros;
unfold divfs_base1.
econstructor;
split.
repeat (
try econstructor;
try eassumption).
trivial.
Qed.
Lemma eval_divfs_baseX:
forall le a b x y,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
exists v,
eval_expr ge sp e m le (
divfs_baseX a b)
v /\
Val.lessdef (
Val.divfs x y)
v.
Proof.
Theorem eval_divfs_base:
forall le a b x y,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
exists v,
eval_expr ge sp e m le (
divfs_base a b)
v /\
Val.lessdef (
Val.divfs x y)
v.
Proof.
Platform-specific known builtins
Lemma eval_fma:
forall al a vl v le,
gen_fma al =
Some a ->
eval_exprlist ge sp e m le al vl ->
platform_builtin_sem BI_fma vl =
Some v ->
exists v',
eval_expr ge sp e m le a v' /\
Val.lessdef v v'.
Proof.
unfold gen_fma.
intros until le.
intro Heval.
destruct (
gen_fma_match _)
in *;
try discriminate.
all:
inversion Heval;
subst a;
clear Heval;
intro;
InvEval.
-
subst v1.
TrivialExists.
destruct v0;
simpl;
trivial;
destruct v2;
simpl;
trivial;
destruct v3;
simpl;
trivial.
-
intro Heval.
simpl in Heval.
inv Heval.
TrivialExists.
destruct v0;
simpl;
trivial;
destruct v1;
simpl;
trivial;
destruct v2;
simpl;
trivial.
Qed.
Lemma eval_fmaf:
forall al a vl v le,
gen_fmaf al =
Some a ->
eval_exprlist ge sp e m le al vl ->
platform_builtin_sem BI_fmaf vl =
Some v ->
exists v',
eval_expr ge sp e m le a v' /\
Val.lessdef v v'.
Proof.
unfold gen_fmaf.
intros until le.
intro Heval.
destruct (
gen_fmaf_match _)
in *;
try discriminate.
all:
inversion Heval;
subst a;
clear Heval;
intro;
InvEval.
-
subst v1.
TrivialExists.
destruct v0;
simpl;
trivial;
destruct v2;
simpl;
trivial;
destruct v3;
simpl;
trivial.
-
intro Heval.
simpl in Heval.
inv Heval.
TrivialExists.
destruct v0;
simpl;
trivial;
destruct v1;
simpl;
trivial;
destruct v2;
simpl;
trivial.
Qed.
Lemma eval_abs:
forall al a vl v le,
gen_abs al =
Some a ->
eval_exprlist ge sp e m le al vl ->
platform_builtin_sem BI_abs vl =
Some v ->
exists v',
eval_expr ge sp e m le a v' /\
Val.lessdef v v'.
Proof.
unfold gen_abs.
intros until le.
intros SELECT Heval BUILTIN.
inv Heval.
discriminate.
inv H0. 2:
discriminate.
cbn in BUILTIN.
inv BUILTIN.
inv SELECT.
econstructor;
split.
{
repeat (
econstructor +
eassumption).
}
destruct v1;
try constructor.
cbn.
unfold int_abs,
int_absdiff,
Z_abs_diff.
change (
Int.signed Int.zero)
with 0%
Z.
rewrite Z.sub_0_r.
constructor.
Qed.
Lemma eval_absl:
forall al a vl v le,
gen_absl al =
Some a ->
eval_exprlist ge sp e m le al vl ->
platform_builtin_sem BI_absl vl =
Some v ->
exists v',
eval_expr ge sp e m le a v' /\
Val.lessdef v v'.
Proof.
unfold gen_abs.
intros until le.
intros SELECT Heval BUILTIN.
inv Heval.
discriminate.
inv H0. 2:
discriminate.
cbn in BUILTIN.
inv BUILTIN.
inv SELECT.
econstructor;
split.
{
repeat (
econstructor +
eassumption).
}
destruct v1;
try constructor.
cbn.
unfold long_abs,
long_absdiff,
Z_abs_diff.
change (
Int64.signed Int64.zero)
with 0%
Z.
rewrite Z.sub_0_r.
constructor.
Qed.
Require FPDivision32.
Theorem eval_platform_builtin:
forall bf al a vl v le,
platform_builtin bf al =
Some a ->
eval_exprlist ge sp e m le al vl ->
platform_builtin_sem bf vl =
Some v ->
exists v',
eval_expr ge sp e m le a v' /\
Val.lessdef v v'.
Proof.
End CMCONSTR.