Locations are a refinement of RTL pseudo-registers, used to reflect
the results of register allocation (file Allocation).
Require Import OrderedType.
Require Import Coqlib.
Require Import Maps.
Require Import Ordered.
Require Import AST.
Require Import Values.
Require Export Machregs.
Representation of locations
A location is either a processor register or (an abstract designation of)
a slot in the activation record of the current function.
Processor registers
Processor registers usable for register allocation are defined
in module Machregs.
Slots in activation records
A slot in an activation record is designated abstractly by a kind,
a type and an integer offset. Three kinds are considered:
-
Local: these are the slots used by register allocation for
pseudo-registers that cannot be assigned a hardware register.
-
Incoming: used to store the parameters of the current function
that cannot reside in hardware registers, as determined by the
calling conventions.
-
Outgoing: used to store arguments to called functions that
cannot reside in hardware registers, as determined by the
calling conventions.
:
Type :=
|
Local
|
Incoming
|
Outgoing.
Morally, the Incoming slots of a function are the Outgoing
slots of its caller function.
The type of a slot indicates how it will be accessed later once mapped to
actual memory locations inside a memory-allocated activation record:
as 32-bit integers/pointers (type Tint) or as 64-bit floats (type Tfloat).
The offset of a slot, combined with its type and its kind, identifies
uniquely the slot and will determine later where it resides within the
memory-allocated activation record. Offsets are always positive.
Lemma slot_eq:
forall (
p q:
slot), {
p =
q} + {
p <>
q}.
Proof.
decide equality.
Defined.
Open Scope Z_scope.
Definition typesize (
ty:
typ) :
Z :=
match ty with
|
Tint => 1
|
Tlong => 2
|
Tfloat => 2
|
Tsingle => 1
|
Tany32 => 1
|
Tany64 => 2
end.
Lemma typesize_pos:
forall (
ty:
typ),
typesize ty > 0.
Proof.
destruct ty; compute; auto.
Qed.
Definition typealign (
ty:
typ) :
Z :=
match ty with
|
Tint => 1
|
Tlong => 2
|
Tfloat => 1
|
Tsingle => 1
|
Tany32 => 1
|
Tany64 => 1
end.
Lemma typealign_pos:
forall (
ty:
typ),
typealign ty > 0.
Proof.
destruct ty; compute; auto.
Qed.
Lemma typealign_typesize:
forall (
ty:
typ), (
typealign ty |
typesize ty).
Proof.
Locations
Locations are just the disjoint union of machine registers and
activation record slots.
Inductive loc :
Type :=
|
R (
r:
mreg)
|
S (
sl:
slot) (
pos:
Z) (
ty:
typ).
Module Loc.
Definition type (
l:
loc) :
typ :=
match l with
|
R r =>
mreg_type r
|
S sl pos ty =>
ty
end.
Lemma eq:
forall (
p q:
loc), {
p =
q} + {
p <>
q}.
Proof.
As mentioned previously, two locations can be different (in the sense
of the
<> mathematical disequality), yet denote
overlapping memory chunks within the activation record.
Given two locations, three cases are possible:
-
They are equal (in the sense of the = equality)
-
They are different and non-overlapping.
-
They are different but overlapping.
The second case (different and non-overlapping) is characterized
by the following
Loc.diff predicate.
(
l1 l2:
loc) :
Prop :=
match l1,
l2 with
|
R r1,
R r2 =>
r1 <>
r2
|
S s1 d1 t1,
S s2 d2 t2 =>
s1 <>
s2 \/
d1 +
typesize t1 <=
d2 \/
d2 +
typesize t2 <=
d1
|
_,
_ =>
True
end.
Lemma same_not_diff:
forall l, ~(
diff l l).
Proof.
destruct l;
unfold diff;
auto.
red;
intros.
destruct H;
auto.
generalize (
typesize_pos ty);
lia.
Qed.
Lemma diff_not_eq:
forall l1 l2,
diff l1 l2 ->
l1 <>
l2.
Proof.
Lemma diff_sym:
forall l1 l2,
diff l1 l2 ->
diff l2 l1.
Proof.
destruct l1;
destruct l2;
unfold diff;
auto.
intuition.
Qed.
Definition diff_dec (
l1 l2:
loc) : {
Loc.diff l1 l2 } + { ~
Loc.diff l1 l2 }.
Proof.
intros.
destruct l1;
destruct l2;
simpl.
-
destruct (
mreg_eq r r0).
right;
tauto.
left;
auto.
-
left;
auto.
-
left;
auto.
-
destruct (
slot_eq sl sl0).
destruct (
zle (
pos +
typesize ty)
pos0).
left;
auto.
destruct (
zle (
pos0 +
typesize ty0)
pos).
left;
auto.
right;
red;
intros [
P | [
P |
P]].
congruence.
lia.
lia.
left;
auto.
Defined.
We now redefine some standard notions over lists, using the Loc.diff
predicate instead of standard disequality <>.
Loc.notin l ll holds if the location l is different from all locations
in the list ll.
Fixpoint notin (
l:
loc) (
ll:
list loc) {
struct ll} :
Prop :=
match ll with
|
nil =>
True
|
l1 ::
ls =>
diff l l1 /\
notin l ls
end.
Lemma notin_iff:
forall l ll,
notin l ll <-> (
forall l',
In l'
ll ->
Loc.diff l l').
Proof.
induction ll; simpl.
tauto.
rewrite IHll. intuition. subst a. auto.
Qed.
Lemma notin_not_in:
forall l ll,
notin l ll -> ~(
In l ll).
Proof.
Lemma notin_dec (
l:
loc) (
ll:
list loc) : {
notin l ll} + {~
notin l ll}.
Proof.
induction ll;
simpl.
left;
auto.
destruct (
diff_dec l a).
destruct IHll.
left;
auto.
right;
tauto.
right;
tauto.
Defined.
Loc.disjoint l1 l2 is true if the locations in list l1
are different from all locations in list l2.
Definition disjoint (
l1 l2:
list loc) :
Prop :=
forall x1 x2,
In x1 l1 ->
In x2 l2 ->
diff x1 x2.
Lemma disjoint_cons_left:
forall a l1 l2,
disjoint (
a ::
l1)
l2 ->
disjoint l1 l2.
Proof.
unfold disjoint;
intros.
auto with coqlib.
Qed.
Lemma disjoint_cons_right:
forall a l1 l2,
disjoint l1 (
a ::
l2) ->
disjoint l1 l2.
Proof.
unfold disjoint;
intros.
auto with coqlib.
Qed.
Lemma disjoint_sym:
forall l1 l2,
disjoint l1 l2 ->
disjoint l2 l1.
Proof.
Lemma in_notin_diff:
forall l1 l2 ll,
notin l1 ll ->
In l2 ll ->
diff l1 l2.
Proof.
Lemma notin_disjoint:
forall l1 l2,
(
forall x,
In x l1 ->
notin x l2) ->
disjoint l1 l2.
Proof.
intros;
red;
intros.
exploit H;
eauto.
rewrite notin_iff;
intros.
auto.
Qed.
Lemma disjoint_notin:
forall l1 l2 x,
disjoint l1 l2 ->
In x l1 ->
notin x l2.
Proof.
intros;
rewrite notin_iff;
intros.
red in H.
auto.
Qed.
Loc.norepet ll holds if the locations in list ll are pairwise
different.
Inductive norepet :
list loc ->
Prop :=
|
norepet_nil:
norepet nil
|
norepet_cons:
forall hd tl,
notin hd tl ->
norepet tl ->
norepet (
hd ::
tl).
Lemma norepet_dec (
ll:
list loc) : {
norepet ll} + {~
norepet ll}.
Proof.
induction ll.
left;
constructor.
destruct (
notin_dec a ll).
destruct IHll.
left;
constructor;
auto.
right;
red;
intros P;
inv P;
contradiction.
right;
red;
intros P;
inv P;
contradiction.
Defined.
Loc.no_overlap l1 l2 holds if elements of l1 never overlap partially
with elements of l2.
Definition no_overlap (
l1 l2 :
list loc) :=
forall r,
In r l1 ->
forall s,
In s l2 ->
r =
s \/
Loc.diff r s.
End Loc.
Mappings from locations to values
The Locmap module defines mappings from locations to values,
used as evaluation environments for the semantics of the LTL
and Linear intermediate languages.
Set Implicit Arguments.
Module Locmap.
Definition t :=
loc ->
val.
Definition init (
x:
val) :
t :=
fun (
_:
loc) =>
x.
Definition get (
l:
loc) (
m:
t) :
val :=
m l.
The set operation over location mappings reflects the overlapping
properties of locations: changing the value of a location l
invalidates (sets to Vundef) the locations that partially overlap
with l. In other terms, the result of set l v m
maps location l to value v, locations that overlap with l
to Vundef, and locations that are different (and non-overlapping)
from l to their previous values in m. This is apparent in the
``good variables'' properties Locmap.gss and Locmap.gso.
Additionally, the set operation also anticipates the fact that
abstract stack slots are mapped to concrete memory locations
in the Stacking phase. Hence, values stored in stack slots
are normalized according to the type of the slot.
Definition set (
l:
loc) (
v:
val) (
m:
t) :
t :=
fun (
p:
loc) =>
if Loc.eq l p then
match l with R r =>
v |
S sl ofs ty =>
Val.load_result (
chunk_of_type ty)
v end
else if Loc.diff_dec l p then
m p
else Vundef.
Lemma gss:
forall l v m,
(
set l v m)
l =
match l with R r =>
v |
S sl ofs ty =>
Val.load_result (
chunk_of_type ty)
v end.
Proof.
Lemma gss_reg:
forall r v m, (
set (
R r)
v m) (
R r) =
v.
Proof.
Lemma gss_typed:
forall l v m,
Val.has_type v (
Loc.type l) -> (
set l v m)
l =
v.
Proof.
Lemma gso:
forall l v m p,
Loc.diff l p -> (
set l v m)
p =
m p.
Proof.
Fixpoint undef (
ll:
list loc) (
m:
t) {
struct ll} :
t :=
match ll with
|
nil =>
m
|
l1 ::
ll' =>
undef ll' (
set l1 Vundef m)
end.
Lemma guo:
forall ll l m,
Loc.notin l ll -> (
undef ll m)
l =
m l.
Proof.
induction ll;
simpl;
intros.
auto.
destruct H.
rewrite IHll;
auto.
apply gso.
apply Loc.diff_sym;
auto.
Qed.
Lemma gus:
forall ll l m,
In l ll -> (
undef ll m)
l =
Vundef.
Proof.
assert (
P:
forall ll l m,
m l =
Vundef -> (
undef ll m)
l =
Vundef).
induction ll;
simpl;
intros.
auto.
apply IHll.
unfold set.
destruct (
Loc.eq a l).
destruct a.
auto.
destruct ty;
reflexivity.
destruct (
Loc.diff_dec a l);
auto.
induction ll;
simpl;
intros.
contradiction.
destruct H.
apply P.
subst a.
apply gss_typed.
exact I.
auto.
Qed.
Definition getpair (
p:
rpair loc) (
m:
t) :
val :=
match p with
|
One l =>
m l
|
Twolong l1 l2 =>
Val.longofwords (
m l1) (
m l2)
end.
Definition setpair (
p:
rpair mreg) (
v:
val) (
m:
t) :
t :=
match p with
|
One r =>
set (
R r)
v m
|
Twolong hi lo =>
set (
R lo) (
Val.loword v) (
set (
R hi) (
Val.hiword v)
m)
end.
Lemma getpair_exten:
forall p ls1 ls2,
(
forall l,
In l (
regs_of_rpair p) ->
ls2 l =
ls1 l) ->
getpair p ls2 =
getpair p ls1.
Proof.
intros. destruct p; simpl.
apply H; simpl; auto.
f_equal; apply H; simpl; auto.
Qed.
Lemma gpo:
forall p v m l,
forall_rpair (
fun r =>
Loc.diff l (
R r))
p ->
setpair p v m l =
m l.
Proof.
Fixpoint setres (
res:
builtin_res mreg) (
v:
val) (
m:
t) :
t :=
match res with
|
BR r =>
set (
R r)
v m
|
BR_none =>
m
|
BR_splitlong hi lo =>
setres lo (
Val.loword v) (
setres hi (
Val.hiword v)
m)
end.
End Locmap.
Total ordering over locations
Module IndexedTyp <:
INDEXED_TYPE.
Definition t :=
typ.
Definition index (
x:
t) :=
match x with
|
Tany32 => 1%
positive
|
Tint => 2%
positive
|
Tsingle => 3%
positive
|
Tany64 => 4%
positive
|
Tfloat => 5%
positive
|
Tlong => 6%
positive
end.
Lemma index_inj:
forall x y,
index x =
index y ->
x =
y.
Proof.
destruct x; destruct y; simpl; congruence. Qed.
Definition eq :=
typ_eq.
End IndexedTyp.
Module OrderedTyp :=
OrderedIndexed(
IndexedTyp).
Module IndexedSlot <:
INDEXED_TYPE.
Definition t :=
slot.
Definition index (
x:
t) :=
match x with Local => 1%
positive |
Incoming => 2%
positive |
Outgoing => 3%
positive end.
Lemma index_inj:
forall x y,
index x =
index y ->
x =
y.
Proof.
destruct x; destruct y; simpl; congruence. Qed.
Definition eq :=
slot_eq.
End IndexedSlot.
Module OrderedSlot :=
OrderedIndexed(
IndexedSlot).
Module OrderedLoc <:
OrderedType.
Definition t :=
loc.
Definition eq (
x y:
t) :=
x =
y.
Definition lt (
x y:
t) :=
match x,
y with
|
R r1,
R r2 =>
Plt (
IndexedMreg.index r1) (
IndexedMreg.index r2)
|
R _,
S _ _ _ =>
True
|
S _ _ _,
R _ =>
False
|
S sl1 ofs1 ty1,
S sl2 ofs2 ty2 =>
OrderedSlot.lt sl1 sl2 \/ (
sl1 =
sl2 /\
(
ofs1 <
ofs2 \/ (
ofs1 =
ofs2 /\
OrderedTyp.lt ty1 ty2)))
end.
Lemma eq_refl :
forall x :
t,
eq x x.
Proof (@
eq_refl t).
Lemma eq_sym :
forall x y :
t,
eq x y ->
eq y x.
Proof (@
eq_sym t).
Lemma eq_trans :
forall x y z :
t,
eq x y ->
eq y z ->
eq x z.
Proof (@
eq_trans t).
Lemma lt_trans :
forall x y z :
t,
lt x y ->
lt y z ->
lt x z.
Proof.
unfold lt;
intros.
destruct x;
destruct y;
destruct z;
try tauto.
eapply Plt_trans;
eauto.
destruct H.
destruct H0.
left;
eapply OrderedSlot.lt_trans;
eauto.
destruct H0.
subst sl0.
auto.
destruct H.
subst sl.
destruct H0.
auto.
destruct H.
right.
split.
auto.
intuition try lia.
right;
split.
congruence.
eapply OrderedTyp.lt_trans;
eauto.
Qed.
Lemma lt_not_eq :
forall x y :
t,
lt x y -> ~
eq x y.
Proof.
Definition compare :
forall x y :
t,
Compare lt eq x y.
Proof.
Definition eq_dec :=
Loc.eq.
Connection between the ordering defined here and the Loc.diff predicate.
Definition diff_low_bound (
l:
loc) :
loc :=
match l with
|
R mr =>
l
|
S sl ofs ty =>
S sl (
ofs - 1)
Tany64
end.
Definition diff_high_bound (
l:
loc) :
loc :=
match l with
|
R mr =>
l
|
S sl ofs ty =>
S sl (
ofs +
typesize ty - 1)
Tlong
end.
Lemma outside_interval_diff:
forall l l',
lt l' (
diff_low_bound l) \/
lt (
diff_high_bound l)
l' ->
Loc.diff l l'.
Proof.
intros.
destruct l as [
mr |
sl ofs ty];
destruct l'
as [
mr' |
sl'
ofs'
ty'];
simpl in *;
auto.
-
assert (
IndexedMreg.index mr <>
IndexedMreg.index mr').
{
destruct H.
apply not_eq_sym.
apply Plt_ne;
auto.
apply Plt_ne;
auto. }
congruence.
-
assert (
RANGE:
forall ty, 1 <=
typesize ty <= 2).
{
intros;
unfold typesize.
destruct ty0;
lia. }
destruct H.
+
destruct H.
left.
apply not_eq_sym.
apply OrderedSlot.lt_not_eq;
auto.
destruct H.
right.
destruct H0.
right.
generalize (
RANGE ty');
lia.
destruct H0.
assert (
ty' =
Tint \/
ty' =
Tsingle \/
ty' =
Tany32).
{
unfold OrderedTyp.lt in H1.
destruct ty';
auto;
compute in H1;
congruence. }
right.
destruct H2 as [
E|[
E|
E]];
subst ty';
simpl typesize;
lia.
+
destruct H.
left.
apply OrderedSlot.lt_not_eq;
auto.
destruct H.
right.
destruct H0.
left;
lia.
destruct H0.
exfalso.
destruct ty';
compute in H1;
congruence.
Qed.
Lemma diff_outside_interval:
forall l l',
Loc.diff l l' ->
lt l' (
diff_low_bound l) \/
lt (
diff_high_bound l)
l'.
Proof.
End OrderedLoc.