Require Import Coqlib.
Require Import Maps.
Require Import Compopts.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Events.
Require Import Lattice.
Require Import Kildall.
Require Import Registers.
Require Import Op.
Require Import RTL.
Require Import ValueDomain.
Require Import ValueAOp.
Require Import Liveness.
The dataflow analysis
Definition areg (
ae:
aenv) (
r:
reg) :
aval :=
AE.get r ae.
Definition aregs (
ae:
aenv) (
rl:
list reg) :
list aval :=
List.map (
areg ae)
rl.
Definition mafter_public_call :
amem :=
mtop.
Definition mafter_private_call (
am_before:
amem) :
amem :=
{|
am_stack :=
am_before.(
am_stack);
am_glob :=
PTree.empty _;
am_nonstack :=
Nonstack;
am_top :=
plub (
ab_summary (
am_stack am_before))
Nonstack |}.
Definition transfer_call (
ae:
aenv) (
am:
amem) (
args:
list reg) (
res:
reg) :=
if pincl am.(
am_nonstack)
Nonstack
&&
forallb (
fun r =>
vpincl (
areg ae r)
Nonstack)
args
then
VA.State (
AE.set res (
Ifptr Nonstack)
ae) (
mafter_private_call am)
else
VA.State (
AE.set res Vtop ae)
mafter_public_call.
Inductive builtin_kind :
Type :=
|
Builtin_vload (
chunk:
memory_chunk) (
aaddr:
aval)
|
Builtin_vstore (
chunk:
memory_chunk) (
aaddr av:
aval)
|
Builtin_memcpy (
sz al:
Z) (
adst asrc:
aval)
|
Builtin_annot_val (
av:
aval)
|
Builtin_default.
Definition classify_builtin (
ef:
external_function) (
args:
list reg) (
ae:
aenv) :=
match ef,
args with
|
EF_vload chunk,
a1::
nil =>
Builtin_vload chunk (
areg ae a1)
|
EF_vload_global chunk id ofs,
nil =>
Builtin_vload chunk (
Ptr (
Gl id ofs))
|
EF_vstore chunk,
a1::
a2::
nil =>
Builtin_vstore chunk (
areg ae a1) (
areg ae a2)
|
EF_vstore_global chunk id ofs,
a1::
nil =>
Builtin_vstore chunk (
Ptr (
Gl id ofs)) (
areg ae a1)
|
EF_memcpy sz al,
a1::
a2::
nil =>
Builtin_memcpy sz al (
areg ae a1) (
areg ae a2)
|
EF_annot_val _ _,
a1::
nil =>
Builtin_annot_val (
areg ae a1)
|
_,
_ =>
Builtin_default
end.
Definition transfer_builtin (
ae:
aenv) (
am:
amem) (
rm:
romem) (
ef:
external_function) (
args:
list reg) (
res:
reg) :=
match classify_builtin ef args ae with
|
Builtin_vload chunk aaddr =>
let a :=
if va_strict tt
then vlub (
loadv chunk rm am aaddr) (
vnormalize chunk (
Ifptr Glob))
else vnormalize chunk Vtop in
VA.State (
AE.set res a ae)
am
|
Builtin_vstore chunk aaddr av =>
let am' :=
storev chunk am aaddr av in
VA.State (
AE.set res itop ae) (
mlub am am')
|
Builtin_memcpy sz al adst asrc =>
let p :=
loadbytes am rm (
aptr_of_aval asrc)
in
let am' :=
storebytes am (
aptr_of_aval adst)
sz p in
VA.State (
AE.set res itop ae)
am'
|
Builtin_annot_val av =>
VA.State (
AE.set res av ae)
am
|
Builtin_default =>
transfer_call ae am args res
end.
Definition transfer (
f:
function) (
rm:
romem) (
pc:
node) (
ae:
aenv) (
am:
amem) :
VA.t :=
match f.(
fn_code)!
pc with
|
None =>
VA.Bot
|
Some(
Inop s) =>
VA.State ae am
|
Some(
Iop op args res s) =>
let a :=
eval_static_operation op (
aregs ae args)
in
VA.State (
AE.set res a ae)
am
|
Some(
Iload chunk addr args dst s) =>
let a :=
loadv chunk rm am (
eval_static_addressing addr (
aregs ae args))
in
VA.State (
AE.set dst a ae)
am
|
Some(
Istore chunk addr args src s) =>
let am' :=
storev chunk am (
eval_static_addressing addr (
aregs ae args)) (
areg ae src)
in
VA.State ae am'
|
Some(
Icall sig ros args res s) =>
transfer_call ae am args res
|
Some(
Itailcall sig ros args) =>
VA.Bot
|
Some(
Ibuiltin ef args res s) =>
transfer_builtin ae am rm ef args res
|
Some(
Iannot ef args s) =>
VA.State ae am
|
Some(
Icond cond args s1 s2) =>
VA.State ae am
|
Some(
Ijumptable arg tbl) =>
VA.State ae am
|
Some(
Ireturn arg) =>
VA.Bot
end.
Definition transfer' (
f:
function) (
lastuses:
PTree.t (
list reg)) (
rm:
romem)
(
pc:
node) (
before:
VA.t) :
VA.t :=
match before with
|
VA.Bot =>
VA.Bot
|
VA.State ae am =>
match transfer f rm pc ae am with
|
VA.Bot =>
VA.Bot
|
VA.State ae'
am' =>
let ae'' :=
match lastuses!
pc with
|
None =>
ae'
|
Some regs =>
eforget regs ae'
end in
VA.State ae''
am'
end
end.
Module DS :=
Dataflow_Solver(
VA)(
NodeSetForward).
Definition mfunction_entry :=
{|
am_stack :=
ablock_init Pbot;
am_glob :=
PTree.empty _;
am_nonstack :=
Nonstack;
am_top :=
Nonstack |}.
Definition analyze (
rm:
romem) (
f:
function):
PMap.t VA.t :=
let lu :=
Liveness.last_uses f in
let entry :=
VA.State (
einit_regs f.(
fn_params))
mfunction_entry in
match DS.fixpoint f.(
fn_code)
successors_instr (
transfer'
f lu rm)
f.(
fn_entrypoint)
entry with
|
None =>
PMap.init (
VA.State AE.top mtop)
|
Some res =>
res
end.
Constructing the approximation of read-only globals
Definition store_init_data (
ab:
ablock) (
p:
Z) (
id:
init_data) :
ablock :=
match id with
|
Init_int8 n =>
ablock_store Mint8unsigned ab p (
I n)
|
Init_int16 n =>
ablock_store Mint16unsigned ab p (
I n)
|
Init_int32 n =>
ablock_store Mint32 ab p (
I n)
|
Init_int64 n =>
ablock_store Mint64 ab p (
L n)
|
Init_float32 n =>
ablock_store Mfloat32 ab p
(
if propagate_float_constants tt then FS n else ftop)
|
Init_float64 n =>
ablock_store Mfloat64 ab p
(
if propagate_float_constants tt then F n else ftop)
|
Init_addrof symb ofs =>
ablock_store Mint32 ab p (
Ptr (
Gl symb ofs))
|
Init_space n =>
ab
end.
Fixpoint store_init_data_list (
ab:
ablock) (
p:
Z) (
idl:
list init_data)
{
struct idl}:
ablock :=
match idl with
|
nil =>
ab
|
id ::
idl' =>
store_init_data_list (
store_init_data ab p id) (
p +
Genv.init_data_size id)
idl'
end.
When CompCert is used in separate compilation mode, the
gvar_init
initializer attached to a readonly global variable may not correspond
to the actual initial value of this global. This occurs in two cases:
-
an extern const variable, which is represented by gvar_init = nil;
-
a const variable without an explicit initializer, which is treated
by the linker as a "common" symbol, and is represented by
gvar_init = Init_space sz :: nil.
In both cases, the variable can be defined and initialized in another
compilation unit which is later linked with the current compilation unit.
(
init:
list init_data) :
bool :=
match init with
|
nil =>
false
|
Init_space _ ::
nil =>
false
|
_ =>
true
end.
Definition alloc_global (
rm:
romem) (
idg:
ident *
globdef fundef unit):
romem :=
match idg with
| (
id,
Gfun f) =>
PTree.remove id rm
| (
id,
Gvar v) =>
if v.(
gvar_readonly) &&
negb v.(
gvar_volatile) &&
definitive_initializer v.(
gvar_init)
then PTree.set id (
store_init_data_list (
ablock_init Pbot) 0
v.(
gvar_init))
rm
else PTree.remove id rm
end.
Definition romem_for_program (
p:
program) :
romem :=
List.fold_left alloc_global p.(
prog_defs) (
PTree.empty _).
Soundness proof
Properties of the dataflow solution.
Lemma analyze_entrypoint:
forall rm f vl m bc,
(
forall v,
In v vl ->
vmatch bc v (
Ifptr Nonstack)) ->
mmatch bc m mfunction_entry ->
exists ae am,
(
analyze rm f)!!(
fn_entrypoint f) =
VA.State ae am
/\
ematch bc (
init_regs vl (
fn_params f))
ae
/\
mmatch bc m am.
Proof.
Lemma analyze_successor:
forall f n ae am instr s rm ae'
am',
(
analyze rm f)!!
n =
VA.State ae am ->
f.(
fn_code)!
n =
Some instr ->
In s (
successors_instr instr) ->
transfer f rm n ae am =
VA.State ae'
am' ->
VA.ge (
analyze rm f)!!
s (
transfer f rm n ae am).
Proof.
Lemma analyze_succ:
forall e m rm f n ae am instr s ae'
am'
bc,
(
analyze rm f)!!
n =
VA.State ae am ->
f.(
fn_code)!
n =
Some instr ->
In s (
successors_instr instr) ->
transfer f rm n ae am =
VA.State ae'
am' ->
ematch bc e ae' ->
mmatch bc m am' ->
exists ae''
am'',
(
analyze rm f)!!
s =
VA.State ae''
am''
/\
ematch bc e ae''
/\
mmatch bc m am''.
Proof.
intros.
exploit analyze_successor;
eauto.
rewrite H2.
destruct (
analyze rm f)#
s as [ |
ae''
am''];
simpl;
try tauto.
intros [
A B].
exists ae'',
am''.
split.
auto.
split.
eapply ematch_ge;
eauto.
eauto.
Qed.
Classification of builtin functions
Lemma classify_builtin_sound:
forall bc e ae ef (
ge:
genv)
args m t res m',
ematch bc e ae ->
genv_match bc ge ->
external_call ef ge e##
args m t res m' ->
match classify_builtin ef args ae with
|
Builtin_vload chunk aaddr =>
exists addr,
volatile_load_sem chunk ge (
addr::
nil)
m t res m' /\
vmatch bc addr aaddr
|
Builtin_vstore chunk aaddr av =>
exists addr v,
volatile_store_sem chunk ge (
addr::
v::
nil)
m t res m'
/\
vmatch bc addr aaddr /\
vmatch bc v av
|
Builtin_memcpy sz al adst asrc =>
exists dst,
exists src,
extcall_memcpy_sem sz al ge (
dst::
src::
nil)
m t res m'
/\
vmatch bc dst adst /\
vmatch bc src asrc
|
Builtin_annot_val av =>
m' =
m /\
vmatch bc res av
|
Builtin_default =>
True
end.
Proof.
intros.
unfold classify_builtin;
destruct ef;
auto.
-
destruct args;
auto.
destruct args;
auto.
exists (
e#
p);
split;
eauto.
-
destruct args;
auto.
destruct args;
auto.
destruct args;
auto.
exists (
e#
p), (
e#
p0);
eauto.
-
destruct args;
auto.
simpl in H1.
rewrite volatile_load_global_charact in H1.
destruct H1 as (
b &
A &
B).
exists (
Vptr b ofs);
split;
auto.
constructor.
constructor.
eapply H0;
eauto.
-
destruct args;
auto.
destruct args;
auto.
simpl in H1.
rewrite volatile_store_global_charact in H1.
destruct H1 as (
b &
A &
B).
exists (
Vptr b ofs), (
e#
p);
split;
auto.
split;
eauto.
constructor.
constructor.
eapply H0;
eauto.
-
destruct args;
auto.
destruct args;
auto.
destruct args;
auto.
exists (
e#
p), (
e#
p0);
eauto.
-
destruct args;
auto.
destruct args;
auto.
simpl in H1.
inv H1.
eauto.
Qed.
Constructing block classifications
Definition bc_nostack (
bc:
block_classification) :
Prop :=
forall b,
bc b <>
BCstack.
Section NOSTACK.
Variable bc:
block_classification.
Hypothesis NOSTACK:
bc_nostack bc.
Lemma pmatch_no_stack:
forall b ofs p,
pmatch bc b ofs p ->
pmatch bc b ofs Nonstack.
Proof.
intros. inv H; constructor; congruence.
Qed.
Lemma vmatch_no_stack:
forall v x,
vmatch bc v x ->
vmatch bc v (
Ifptr Nonstack).
Proof.
Lemma smatch_no_stack:
forall m b p,
smatch bc m b p ->
smatch bc m b Nonstack.
Proof.
Lemma mmatch_no_stack:
forall m am astk,
mmatch bc m am ->
mmatch bc m {|
am_stack :=
astk;
am_glob :=
PTree.empty _;
am_nonstack :=
Nonstack;
am_top :=
Nonstack |}.
Proof.
End NOSTACK.
Construction 1: allocating the stack frame at function entry
Ltac splitall :=
repeat (
match goal with |-
_ /\
_ =>
split end).
Theorem allocate_stack:
forall m sz m'
sp bc ge rm am,
Mem.alloc m 0
sz = (
m',
sp) ->
genv_match bc ge ->
romatch bc m rm ->
mmatch bc m am ->
bc_nostack bc ->
exists bc',
bc_incr bc bc'
/\
bc'
sp =
BCstack
/\
genv_match bc'
ge
/\
romatch bc'
m'
rm
/\
mmatch bc'
m'
mfunction_entry
/\ (
forall b,
Plt b sp ->
bc'
b =
bc b)
/\ (
forall v x,
vmatch bc v x ->
vmatch bc'
v (
Ifptr Nonstack)).
Proof.
Construction 2: turn the stack into an "other" block, at public calls or function returns
Theorem anonymize_stack:
forall m sp bc ge rm am,
genv_match bc ge ->
romatch bc m rm ->
mmatch bc m am ->
bc sp =
BCstack ->
exists bc',
bc_nostack bc'
/\
bc'
sp =
BCother
/\ (
forall b,
b <>
sp ->
bc'
b =
bc b)
/\ (
forall v x,
vmatch bc v x ->
vmatch bc'
v Vtop)
/\
genv_match bc'
ge
/\
romatch bc'
m rm
/\
mmatch bc'
m mtop.
Proof.
intros until am;
intros GENV RO MM SP.
Part 1: constructing bc' *)
set (
f :=
fun b =>
if eq_block b sp then BCother else bc b).
assert (
F_stack:
forall b1 b2,
f b1 =
BCstack ->
f b2 =
BCstack ->
b1 =
b2).
{
unfold f;
intros.
destruct (
eq_block b1 sp);
try discriminate.
destruct (
eq_block b2 sp);
try discriminate.
eapply bc_stack;
eauto.
}
assert (
F_glob:
forall b1 b2 id,
f b1 =
BCglob id ->
f b2 =
BCglob id ->
b1 =
b2).
{
unfold f;
intros.
destruct (
eq_block b1 sp);
try discriminate.
destruct (
eq_block b2 sp);
try discriminate.
eapply bc_glob;
eauto.
}
set (
bc' :=
BC f F_stack F_glob).
unfold f in bc'.
Part 2: matching wrt bc' *)
assert (
PM:
forall b ofs p,
pmatch bc b ofs p ->
pmatch bc'
b ofs Ptop).
{
intros.
assert (
pmatch bc b ofs Ptop)
by (
eapply pmatch_top';
eauto).
inv H0.
constructor;
simpl.
destruct (
eq_block b sp);
congruence.
}
assert (
VM:
forall v x,
vmatch bc v x ->
vmatch bc'
v Vtop).
{
induction 1;
constructor;
eauto.
}
assert (
SM:
forall b p,
smatch bc m b p ->
smatch bc'
m b Ptop).
{
intros.
destruct H as [
S1 S2].
split;
intros.
eapply VM.
eapply S1;
eauto.
eapply PM.
eapply S2;
eauto.
}
Conclusions *)
exists bc';
splitall.
-
red;
simpl;
intros.
destruct (
eq_block b sp).
congruence.
red;
intros.
elim n.
eapply bc_stack;
eauto.
-
simpl;
apply dec_eq_true.
-
intros;
simpl;
apply dec_eq_false;
auto.
-
auto.
-
apply genv_match_exten with bc;
auto.
simpl;
intros.
destruct (
eq_block b sp);
intuition congruence.
simpl;
intros.
destruct (
eq_block b sp);
auto.
-
apply romatch_exten with bc;
auto.
simpl;
intros.
destruct (
eq_block b sp);
intuition.
-
constructor;
simpl;
intros.
+
destruct (
eq_block b sp).
congruence.
elim n.
eapply bc_stack;
eauto.
+
rewrite PTree.gempty in H0;
discriminate.
+
destruct (
eq_block b sp).
subst b.
eapply SM.
eapply mmatch_stack;
eauto.
eapply SM.
eapply mmatch_nonstack;
eauto.
+
destruct (
eq_block b sp).
subst b.
eapply SM.
eapply mmatch_stack;
eauto.
eapply SM.
eapply mmatch_top;
eauto.
+
red;
simpl;
intros.
destruct (
eq_block b sp).
subst b.
eapply mmatch_below;
eauto.
congruence.
eapply mmatch_below;
eauto.
Qed.
Construction 3: turn the stack into an invalid block, at private calls
Theorem hide_stack:
forall m sp bc ge rm am,
genv_match bc ge ->
romatch bc m rm ->
mmatch bc m am ->
bc sp =
BCstack ->
pge Nonstack am.(
am_nonstack) ->
exists bc',
bc_nostack bc'
/\
bc'
sp =
BCinvalid
/\ (
forall b,
b <>
sp ->
bc'
b =
bc b)
/\ (
forall v x,
vge (
Ifptr Nonstack)
x ->
vmatch bc v x ->
vmatch bc'
v Vtop)
/\
genv_match bc'
ge
/\
romatch bc'
m rm
/\
mmatch bc'
m mtop.
Proof.
Construction 4: restore the stack after a public call
Theorem return_from_public_call:
forall (
caller callee:
block_classification)
bound sp ge e ae v m rm,
bc_below caller bound ->
callee sp =
BCother ->
caller sp =
BCstack ->
(
forall b,
Plt b bound ->
b <>
sp ->
caller b =
callee b) ->
genv_match caller ge ->
ematch caller e ae ->
Ple bound (
Mem.nextblock m) ->
vmatch callee v Vtop ->
romatch callee m rm ->
mmatch callee m mtop ->
genv_match callee ge ->
bc_nostack callee ->
exists bc,
vmatch bc v Vtop
/\
ematch bc e ae
/\
romatch bc m rm
/\
mmatch bc m mafter_public_call
/\
genv_match bc ge
/\
bc sp =
BCstack
/\ (
forall b,
Plt b sp ->
bc b =
caller b).
Proof.
intros until rm;
intros BELOW SP1 SP2 SAME GE1 EM BOUND RESM RM MM GE2 NOSTACK.
Constructing bc *)
set (
f :=
fun b =>
if eq_block b sp then BCstack else callee b).
assert (
F_stack:
forall b1 b2,
f b1 =
BCstack ->
f b2 =
BCstack ->
b1 =
b2).
{
assert (
forall b,
f b =
BCstack ->
b =
sp).
{
unfold f;
intros.
destruct (
eq_block b sp);
auto.
eelim NOSTACK;
eauto. }
intros.
transitivity sp;
auto.
symmetry;
auto.
}
assert (
F_glob:
forall b1 b2 id,
f b1 =
BCglob id ->
f b2 =
BCglob id ->
b1 =
b2).
{
assert (
forall b id,
f b =
BCglob id ->
callee b =
BCglob id).
{
unfold f;
intros.
destruct (
eq_block b sp).
congruence.
auto. }
intros.
eapply (
bc_glob callee);
eauto.
}
set (
bc :=
BC f F_stack F_glob).
unfold f in bc.
assert (
INCR:
bc_incr caller bc).
{
red;
simpl;
intros.
destruct (
eq_block b sp).
congruence.
symmetry;
apply SAME;
auto.
}
Invariance properties *)
assert (
PM:
forall b ofs p,
pmatch callee b ofs p ->
pmatch bc b ofs Ptop).
{
intros.
assert (
pmatch callee b ofs Ptop)
by (
eapply pmatch_top';
eauto).
inv H0.
constructor;
simpl.
destruct (
eq_block b sp);
congruence.
}
assert (
VM:
forall v x,
vmatch callee v x ->
vmatch bc v Vtop).
{
intros.
assert (
vmatch callee v0 Vtop)
by (
eapply vmatch_top;
eauto).
inv H0;
constructor;
eauto.
}
assert (
SM:
forall b p,
smatch callee m b p ->
smatch bc m b Ptop).
{
intros.
destruct H;
split;
intros.
eapply VM;
eauto.
eapply PM;
eauto.
}
Conclusions *)
exists bc;
splitall.
-
eapply VM;
eauto.
-
eapply ematch_incr;
eauto.
-
apply romatch_exten with callee;
auto.
intros;
simpl.
destruct (
eq_block b sp);
intuition.
-
constructor;
simpl;
intros.
+
apply ablock_init_sound.
destruct (
eq_block b sp).
subst b.
eapply SM.
eapply mmatch_nonstack;
eauto.
congruence.
elim (
NOSTACK b);
auto.
+
rewrite PTree.gempty in H0;
discriminate.
+
destruct (
eq_block b sp).
congruence.
eapply SM;
auto.
eapply mmatch_nonstack;
eauto.
+
eapply SM.
eapply mmatch_top;
eauto.
destruct (
eq_block b sp);
congruence.
+
red;
simpl;
intros.
destruct (
eq_block b sp).
subst b.
eapply mmatch_below;
eauto.
congruence.
eapply mmatch_below;
eauto.
-
eapply genv_match_exten with caller;
eauto.
simpl;
intros.
destruct (
eq_block b sp).
intuition congruence.
split;
intros.
rewrite SAME in H by eauto with va.
auto.
apply <- (
proj1 GE2)
in H.
apply (
proj1 GE1)
in H.
auto.
simpl;
intros.
destruct (
eq_block b sp).
congruence.
rewrite <-
SAME;
eauto with va.
-
simpl.
apply dec_eq_true.
-
simpl;
intros.
destruct (
eq_block b sp).
congruence.
symmetry.
apply SAME;
auto.
eapply Plt_trans.
eauto.
apply BELOW.
congruence.
Qed.
Construction 5: restore the stack after a private call
Theorem return_from_private_call:
forall (
caller callee:
block_classification)
bound sp ge e ae v m rm am,
bc_below caller bound ->
callee sp =
BCinvalid ->
caller sp =
BCstack ->
(
forall b,
Plt b bound ->
b <>
sp ->
caller b =
callee b) ->
genv_match caller ge ->
ematch caller e ae ->
bmatch caller m sp am.(
am_stack) ->
Ple bound (
Mem.nextblock m) ->
vmatch callee v Vtop ->
romatch callee m rm ->
mmatch callee m mtop ->
genv_match callee ge ->
bc_nostack callee ->
exists bc,
vmatch bc v (
Ifptr Nonstack)
/\
ematch bc e ae
/\
romatch bc m rm
/\
mmatch bc m (
mafter_private_call am)
/\
genv_match bc ge
/\
bc sp =
BCstack
/\ (
forall b,
Plt b sp ->
bc b =
caller b).
Proof.
intros until am;
intros BELOW SP1 SP2 SAME GE1 EM CONTENTS BOUND RESM RM MM GE2 NOSTACK.
Constructing bc *)
set (
f :=
fun b =>
if eq_block b sp then BCstack else callee b).
assert (
F_stack:
forall b1 b2,
f b1 =
BCstack ->
f b2 =
BCstack ->
b1 =
b2).
{
assert (
forall b,
f b =
BCstack ->
b =
sp).
{
unfold f;
intros.
destruct (
eq_block b sp);
auto.
eelim NOSTACK;
eauto. }
intros.
transitivity sp;
auto.
symmetry;
auto.
}
assert (
F_glob:
forall b1 b2 id,
f b1 =
BCglob id ->
f b2 =
BCglob id ->
b1 =
b2).
{
assert (
forall b id,
f b =
BCglob id ->
callee b =
BCglob id).
{
unfold f;
intros.
destruct (
eq_block b sp).
congruence.
auto. }
intros.
eapply (
bc_glob callee);
eauto.
}
set (
bc :=
BC f F_stack F_glob).
unfold f in bc.
assert (
INCR1:
bc_incr caller bc).
{
red;
simpl;
intros.
destruct (
eq_block b sp).
congruence.
symmetry;
apply SAME;
auto.
}
assert (
INCR2:
bc_incr callee bc).
{
red;
simpl;
intros.
destruct (
eq_block b sp).
congruence.
auto.
}
Invariance properties *)
assert (
PM:
forall b ofs p,
pmatch callee b ofs p ->
pmatch bc b ofs Nonstack).
{
intros.
assert (
pmatch callee b ofs Ptop)
by (
eapply pmatch_top';
eauto).
inv H0.
constructor;
simpl;
destruct (
eq_block b sp);
congruence.
}
assert (
VM:
forall v x,
vmatch callee v x ->
vmatch bc v (
Ifptr Nonstack)).
{
intros.
assert (
vmatch callee v0 Vtop)
by (
eapply vmatch_top;
eauto).
inv H0;
constructor;
eauto.
}
assert (
SM:
forall b p,
smatch callee m b p ->
smatch bc m b Nonstack).
{
intros.
destruct H;
split;
intros.
eapply VM;
eauto.
eapply PM;
eauto.
}
assert (
BSTK:
bmatch bc m sp (
am_stack am)).
{
apply bmatch_incr with caller;
eauto.
}
Conclusions *)
exists bc;
splitall.
-
eapply VM;
eauto.
-
eapply ematch_incr;
eauto.
-
apply romatch_exten with callee;
auto.
intros;
simpl.
destruct (
eq_block b sp);
intuition.
-
constructor;
simpl;
intros.
+
destruct (
eq_block b sp).
subst b.
exact BSTK.
elim (
NOSTACK b);
auto.
+
rewrite PTree.gempty in H0;
discriminate.
+
destruct (
eq_block b sp).
congruence.
eapply SM;
auto.
eapply mmatch_nonstack;
eauto.
+
destruct (
eq_block b sp).
subst.
apply smatch_ge with (
ab_summary (
am_stack am)).
apply BSTK.
apply pge_lub_l.
apply smatch_ge with Nonstack.
eapply SM.
eapply mmatch_top;
eauto.
apply pge_lub_r.
+
red;
simpl;
intros.
destruct (
eq_block b sp).
subst b.
apply Plt_le_trans with bound.
apply BELOW.
congruence.
auto.
eapply mmatch_below;
eauto.
-
eapply genv_match_exten;
eauto.
simpl;
intros.
destruct (
eq_block b sp);
intuition congruence.
simpl;
intros.
destruct (
eq_block b sp);
congruence.
-
simpl.
apply dec_eq_true.
-
simpl;
intros.
destruct (
eq_block b sp).
congruence.
symmetry.
apply SAME;
auto.
eapply Plt_trans.
eauto.
apply BELOW.
congruence.
Qed.
Construction 6: external call
Theorem external_call_match:
forall ef (
ge:
genv)
vargs m t vres m'
bc rm am,
external_call ef ge vargs m t vres m' ->
genv_match bc ge ->
(
forall v,
In v vargs ->
vmatch bc v Vtop) ->
romatch bc m rm ->
mmatch bc m am ->
bc_nostack bc ->
exists bc',
bc_incr bc bc'
/\ (
forall b,
Plt b (
Mem.nextblock m) ->
bc'
b =
bc b)
/\
vmatch bc'
vres Vtop
/\
genv_match bc'
ge
/\
romatch bc'
m'
rm
/\
mmatch bc'
m'
mtop
/\
bc_nostack bc'
/\ (
forall b ofs n,
Mem.valid_block m b ->
bc b =
BCinvalid ->
Mem.loadbytes m'
b ofs n =
Mem.loadbytes m b ofs n).
Proof.
intros until am;
intros EC GENV ARGS RO MM NOSTACK.
Part 1: using ec_mem_inject *)
exploit (@
external_call_mem_inject ef _ _ ge vargs m t vres m' (
inj_of_bc bc)
m vargs).
apply inj_of_bc_preserves_globals;
auto.
exact EC.
eapply mmatch_inj;
eauto.
eapply mmatch_below;
eauto.
revert ARGS.
generalize vargs.
induction vargs0;
simpl;
intros;
constructor.
eapply vmatch_inj;
eauto.
auto.
intros (
j' &
vres' &
m'' &
EC' &
IRES &
IMEM &
UNCH1 &
UNCH2 &
IINCR &
ISEP).
assert (
JBELOW:
forall b,
Plt b (
Mem.nextblock m) ->
j'
b =
inj_of_bc bc b).
{
intros.
destruct (
inj_of_bc bc b)
as [[
b'
delta] | ]
eqn:
EQ.
eapply IINCR;
eauto.
destruct (
j'
b)
as [[
b''
delta'] | ]
eqn:
EQ';
auto.
exploit ISEP;
eauto.
tauto.
}
Part 2: constructing bc' from j' *)
set (
f :=
fun b =>
if plt b (
Mem.nextblock m)
then bc b
else match j'
b with None =>
BCinvalid |
Some _ =>
BCother end).
assert (
F_stack:
forall b1 b2,
f b1 =
BCstack ->
f b2 =
BCstack ->
b1 =
b2).
{
assert (
forall b,
f b =
BCstack ->
bc b =
BCstack).
{
unfold f;
intros.
destruct (
plt b (
Mem.nextblock m));
auto.
destruct (
j'
b);
discriminate. }
intros.
apply (
bc_stack bc);
auto.
}
assert (
F_glob:
forall b1 b2 id,
f b1 =
BCglob id ->
f b2 =
BCglob id ->
b1 =
b2).
{
assert (
forall b id,
f b =
BCglob id ->
bc b =
BCglob id).
{
unfold f;
intros.
destruct (
plt b (
Mem.nextblock m));
auto.
destruct (
j'
b);
discriminate. }
intros.
eapply (
bc_glob bc);
eauto.
}
set (
bc' :=
BC f F_stack F_glob).
unfold f in bc'.
assert (
INCR:
bc_incr bc bc').
{
red;
simpl;
intros.
apply pred_dec_true.
eapply mmatch_below;
eauto.
}
assert (
BC'
INV:
forall b,
bc'
b <>
BCinvalid ->
exists b'
delta,
j'
b =
Some(
b',
delta)).
{
simpl;
intros.
destruct (
plt b (
Mem.nextblock m)).
exists b, 0.
rewrite JBELOW by auto.
apply inj_of_bc_valid;
auto.
destruct (
j'
b)
as [[
b'
delta] | ].
exists b',
delta;
auto.
congruence.
}
Part 3: injection wrt j' implies matching with top wrt bc' *)
assert (
PMTOP:
forall b b'
delta ofs,
j'
b =
Some (
b',
delta) ->
pmatch bc'
b ofs Ptop).
{
intros.
constructor.
simpl;
unfold f.
destruct (
plt b (
Mem.nextblock m)).
rewrite JBELOW in H by auto.
eapply inj_of_bc_inv;
eauto.
rewrite H;
congruence.
}
assert (
VMTOP:
forall v v',
Val.inject j'
v v' ->
vmatch bc'
v Vtop).
{
intros.
inv H;
constructor.
eapply PMTOP;
eauto.
}
assert (
SMTOP:
forall b,
bc'
b <>
BCinvalid ->
smatch bc'
m'
b Ptop).
{
intros;
split;
intros.
-
exploit BC'
INV;
eauto.
intros (
b' &
delta &
J').
exploit Mem.load_inject.
eexact IMEM.
eauto.
eauto.
intros (
v' &
A &
B).
eapply VMTOP;
eauto.
-
exploit BC'
INV;
eauto.
intros (
b'' &
delta &
J').
exploit Mem.loadbytes_inject.
eexact IMEM.
eauto.
eauto.
intros (
bytes &
A &
B).
inv B.
inv H3.
inv H7.
eapply PMTOP;
eauto.
}
Conclusions *)
exists bc';
splitall.
-
exact INCR.
-
simpl;
intros.
apply pred_dec_true;
auto.
-
eapply VMTOP;
eauto.
-
apply genv_match_exten with bc;
auto.
simpl;
intros;
split;
intros.
rewrite pred_dec_true by (
eapply mmatch_below;
eauto with va).
auto.
destruct (
plt b (
Mem.nextblock m)).
auto.
destruct (
j'
b);
congruence.
simpl;
intros.
rewrite pred_dec_true by (
eapply mmatch_below;
eauto with va).
auto.
-
red;
simpl;
intros.
destruct (
plt b (
Mem.nextblock m)).
exploit RO;
eauto.
intros (
R &
P &
Q).
split;
auto.
split.
apply bmatch_incr with bc;
auto.
apply bmatch_inv with m;
auto.
intros.
eapply Mem.loadbytes_unchanged_on_1.
eapply external_call_readonly;
eauto.
auto.
intros;
red.
apply Q.
intros;
red;
intros;
elim (
Q ofs).
eapply external_call_max_perm with (
m2 :=
m');
eauto.
destruct (
j'
b);
congruence.
-
constructor;
simpl;
intros.
+
apply ablock_init_sound.
apply SMTOP.
simpl;
congruence.
+
rewrite PTree.gempty in H0;
discriminate.
+
apply SMTOP;
auto.
+
apply SMTOP;
auto.
+
red;
simpl;
intros.
destruct (
plt b (
Mem.nextblock m)).
eapply Plt_le_trans.
eauto.
eapply external_call_nextblock;
eauto.
destruct (
j'
b)
as [[
bx deltax] | ]
eqn:
J'.
eapply Mem.valid_block_inject_1;
eauto.
congruence.
-
red;
simpl;
intros.
destruct (
plt b (
Mem.nextblock m)).
apply NOSTACK;
auto.
destruct (
j'
b);
congruence.
-
intros.
eapply Mem.loadbytes_unchanged_on_1;
auto.
apply UNCH1;
auto.
intros;
red.
unfold inj_of_bc;
rewrite H0;
auto.
Qed.
Semantic invariant
Section SOUNDNESS.
Variable prog:
program.
Let ge :
genv :=
Genv.globalenv prog.
Let rm :=
romem_for_program prog.
Inductive sound_stack:
block_classification ->
list stackframe ->
mem ->
block ->
Prop :=
|
sound_stack_nil:
forall bc m bound,
sound_stack bc nil m bound
|
sound_stack_public_call:
forall (
bc:
block_classification)
res f sp pc e stk m bound bc'
bound'
ae
(
STK:
sound_stack bc'
stk m sp)
(
INCR:
Ple bound'
bound)
(
BELOW:
bc_below bc'
bound')
(
SP:
bc sp =
BCother)
(
SP':
bc'
sp =
BCstack)
(
SAME:
forall b,
Plt b bound' ->
b <>
sp ->
bc b =
bc'
b)
(
GE:
genv_match bc'
ge)
(
AN:
VA.ge (
analyze rm f)!!
pc (
VA.State (
AE.set res Vtop ae)
mafter_public_call))
(
EM:
ematch bc'
e ae),
sound_stack bc (
Stackframe res f (
Vptr sp Int.zero)
pc e ::
stk)
m bound
|
sound_stack_private_call:
forall (
bc:
block_classification)
res f sp pc e stk m bound bc'
bound'
ae am
(
STK:
sound_stack bc'
stk m sp)
(
INCR:
Ple bound'
bound)
(
BELOW:
bc_below bc'
bound')
(
SP:
bc sp =
BCinvalid)
(
SP':
bc'
sp =
BCstack)
(
SAME:
forall b,
Plt b bound' ->
b <>
sp ->
bc b =
bc'
b)
(
GE:
genv_match bc'
ge)
(
AN:
VA.ge (
analyze rm f)!!
pc (
VA.State (
AE.set res (
Ifptr Nonstack)
ae) (
mafter_private_call am)))
(
EM:
ematch bc'
e ae)
(
CONTENTS:
bmatch bc'
m sp am.(
am_stack)),
sound_stack bc (
Stackframe res f (
Vptr sp Int.zero)
pc e ::
stk)
m bound.
Inductive sound_state:
state ->
Prop :=
|
sound_regular_state:
forall s f sp pc e m ae am bc
(
STK:
sound_stack bc s m sp)
(
AN: (
analyze rm f)!!
pc =
VA.State ae am)
(
EM:
ematch bc e ae)
(
RO:
romatch bc m rm)
(
MM:
mmatch bc m am)
(
GE:
genv_match bc ge)
(
SP:
bc sp =
BCstack),
sound_state (
State s f (
Vptr sp Int.zero)
pc e m)
|
sound_call_state:
forall s fd args m bc
(
STK:
sound_stack bc s m (
Mem.nextblock m))
(
ARGS:
forall v,
In v args ->
vmatch bc v Vtop)
(
RO:
romatch bc m rm)
(
MM:
mmatch bc m mtop)
(
GE:
genv_match bc ge)
(
NOSTK:
bc_nostack bc),
sound_state (
Callstate s fd args m)
|
sound_return_state:
forall s v m bc
(
STK:
sound_stack bc s m (
Mem.nextblock m))
(
RES:
vmatch bc v Vtop)
(
RO:
romatch bc m rm)
(
MM:
mmatch bc m mtop)
(
GE:
genv_match bc ge)
(
NOSTK:
bc_nostack bc),
sound_state (
Returnstate s v m).
Properties of the sound_stack invariant on call stacks.
Lemma sound_stack_ext:
forall m'
bc stk m bound,
sound_stack bc stk m bound ->
(
forall b ofs n bytes,
Plt b bound ->
bc b =
BCinvalid ->
n >= 0 ->
Mem.loadbytes m'
b ofs n =
Some bytes ->
Mem.loadbytes m b ofs n =
Some bytes) ->
sound_stack bc stk m'
bound.
Proof.
induction 1;
intros INV.
-
constructor.
-
assert (
Plt sp bound')
by eauto with va.
eapply sound_stack_public_call;
eauto.
apply IHsound_stack;
intros.
apply INV.
xomega.
rewrite SAME;
auto.
xomega.
auto.
auto.
-
assert (
Plt sp bound')
by eauto with va.
eapply sound_stack_private_call;
eauto.
apply IHsound_stack;
intros.
apply INV.
xomega.
rewrite SAME;
auto.
xomega.
auto.
auto.
apply bmatch_ext with m;
auto.
intros.
apply INV.
xomega.
auto.
auto.
auto.
Qed.
Lemma sound_stack_inv:
forall m'
bc stk m bound,
sound_stack bc stk m bound ->
(
forall b ofs n,
Plt b bound ->
bc b =
BCinvalid ->
n >= 0 ->
Mem.loadbytes m'
b ofs n =
Mem.loadbytes m b ofs n) ->
sound_stack bc stk m'
bound.
Proof.
Lemma sound_stack_storev:
forall chunk m addr v m'
bc aaddr stk bound,
Mem.storev chunk m addr v =
Some m' ->
vmatch bc addr aaddr ->
sound_stack bc stk m bound ->
sound_stack bc stk m'
bound.
Proof.
Lemma sound_stack_storebytes:
forall m b ofs bytes m'
bc aaddr stk bound,
Mem.storebytes m b (
Int.unsigned ofs)
bytes =
Some m' ->
vmatch bc (
Vptr b ofs)
aaddr ->
sound_stack bc stk m bound ->
sound_stack bc stk m'
bound.
Proof.
Lemma sound_stack_free:
forall m b lo hi m'
bc stk bound,
Mem.free m b lo hi =
Some m' ->
sound_stack bc stk m bound ->
sound_stack bc stk m'
bound.
Proof.
Lemma sound_stack_new_bound:
forall bc stk m bound bound',
sound_stack bc stk m bound ->
Ple bound bound' ->
sound_stack bc stk m bound'.
Proof.
Lemma sound_stack_exten:
forall bc stk m bound (
bc1:
block_classification),
sound_stack bc stk m bound ->
(
forall b,
Plt b bound ->
bc1 b =
bc b) ->
sound_stack bc1 stk m bound.
Proof.
intros.
inv H.
-
constructor.
-
assert (
Plt sp bound')
by eauto with va.
eapply sound_stack_public_call;
eauto.
rewrite H0;
auto.
xomega.
intros.
rewrite H0;
auto.
xomega.
-
assert (
Plt sp bound')
by eauto with va.
eapply sound_stack_private_call;
eauto.
rewrite H0;
auto.
xomega.
intros.
rewrite H0;
auto.
xomega.
Qed.
Preservation of the semantic invariant by one step of execution
Lemma sound_succ_state:
forall bc pc ae am instr ae'
am'
s f sp pc'
e'
m',
(
analyze rm f)!!
pc =
VA.State ae am ->
f.(
fn_code)!
pc =
Some instr ->
In pc' (
successors_instr instr) ->
transfer f rm pc ae am =
VA.State ae'
am' ->
ematch bc e'
ae' ->
mmatch bc m'
am' ->
romatch bc m'
rm ->
genv_match bc ge ->
bc sp =
BCstack ->
sound_stack bc s m'
sp ->
sound_state (
State s f (
Vptr sp Int.zero)
pc'
e'
m').
Proof.
intros.
exploit analyze_succ;
eauto.
intros (
ae'' &
am'' &
AN &
EM &
MM).
econstructor;
eauto.
Qed.
Lemma areg_sound:
forall bc e ae r,
ematch bc e ae ->
vmatch bc (
e#
r) (
areg ae r).
Proof.
intros. apply H.
Qed.
Lemma aregs_sound:
forall bc e ae rl,
ematch bc e ae ->
list_forall2 (
vmatch bc) (
e##
rl) (
aregs ae rl).
Proof.
induction rl;
simpl;
intros.
constructor.
constructor;
auto.
apply areg_sound;
auto.
Qed.
Hint Resolve areg_sound aregs_sound:
va.
Theorem sound_step:
forall st t st',
RTL.step ge st t st' ->
sound_state st ->
sound_state st'.
Proof.
End SOUNDNESS.
Soundness of the initial memory abstraction
Section INITIAL.
Variable prog:
program.
Let ge :=
Genv.globalenv prog.
Lemma initial_block_classification:
forall m,
Genv.init_mem prog =
Some m ->
exists bc,
genv_match bc ge
/\
bc_below bc (
Mem.nextblock m)
/\
bc_nostack bc
/\ (
forall b id,
bc b =
BCglob id ->
Genv.find_symbol ge id =
Some b)
/\ (
forall b,
Mem.valid_block m b ->
bc b <>
BCinvalid).
Proof.
Section INIT.
Variable bc:
block_classification.
Hypothesis GMATCH:
genv_match bc ge.
Lemma store_init_data_summary:
forall ab p id,
pge Glob (
ab_summary ab) ->
pge Glob (
ab_summary (
store_init_data ab p id)).
Proof.
Lemma store_init_data_list_summary:
forall idl ab p,
pge Glob (
ab_summary ab) ->
pge Glob (
ab_summary (
store_init_data_list ab p idl)).
Proof.
Lemma store_init_data_sound:
forall m b p id m'
ab,
Genv.store_init_data ge m b p id =
Some m' ->
bmatch bc m b ab ->
bmatch bc m'
b (
store_init_data ab p id).
Proof.
Lemma store_init_data_list_sound:
forall idl m b p m'
ab,
Genv.store_init_data_list ge m b p idl =
Some m' ->
bmatch bc m b ab ->
bmatch bc m'
b (
store_init_data_list ab p idl).
Proof.
Lemma store_init_data_other:
forall m b p id m'
ab b',
Genv.store_init_data ge m b p id =
Some m' ->
b' <>
b ->
bmatch bc m b'
ab ->
bmatch bc m'
b'
ab.
Proof.
Lemma store_init_data_list_other:
forall b b'
ab idl m p m',
Genv.store_init_data_list ge m b p idl =
Some m' ->
b' <>
b ->
bmatch bc m b'
ab ->
bmatch bc m'
b'
ab.
Proof.
Lemma store_zeros_same:
forall p m b pos n m',
store_zeros m b pos n =
Some m' ->
smatch bc m b p ->
smatch bc m'
b p.
Proof.
intros until n.
functional induction (
store_zeros m b pos n);
intros.
-
inv H.
auto.
-
eapply IHo;
eauto.
change p with (
vplub (
I Int.zero)
p).
eapply smatch_store;
eauto.
constructor.
-
discriminate.
Qed.
Lemma store_zeros_other:
forall b'
ab m b p n m',
store_zeros m b p n =
Some m' ->
b' <>
b ->
bmatch bc m b'
ab ->
bmatch bc m'
b'
ab.
Proof.
Definition initial_mem_match (
bc:
block_classification) (
m:
mem) (
g:
genv) :=
forall b v,
Genv.find_var_info g b =
Some v ->
v.(
gvar_volatile) =
false ->
v.(
gvar_readonly) =
true ->
bmatch bc m b (
store_init_data_list (
ablock_init Pbot) 0
v.(
gvar_init)).
Lemma alloc_global_match:
forall m g idg m',
Genv.genv_next g =
Mem.nextblock m ->
initial_mem_match bc m g ->
Genv.alloc_global ge m idg =
Some m' ->
initial_mem_match bc m' (
Genv.add_global g idg).
Proof.
Lemma alloc_globals_match:
forall gl m g m',
Genv.genv_next g =
Mem.nextblock m ->
initial_mem_match bc m g ->
Genv.alloc_globals ge m gl =
Some m' ->
initial_mem_match bc m' (
Genv.add_globals g gl).
Proof.
Definition romem_consistent (
g:
genv) (
rm:
romem) :=
forall id b ab,
Genv.find_symbol g id =
Some b ->
rm!
id =
Some ab ->
exists v,
Genv.find_var_info g b =
Some v
/\
v.(
gvar_readonly) =
true
/\
v.(
gvar_volatile) =
false
/\
ab =
store_init_data_list (
ablock_init Pbot) 0
v.(
gvar_init).
Lemma alloc_global_consistent:
forall g rm idg,
romem_consistent g rm ->
romem_consistent (
Genv.add_global g idg) (
alloc_global rm idg).
Proof.
Lemma alloc_globals_consistent:
forall gl g rm,
romem_consistent g rm ->
romem_consistent (
Genv.add_globals g gl) (
List.fold_left alloc_global gl rm).
Proof.
End INIT.
Theorem initial_mem_matches:
forall m,
Genv.init_mem prog =
Some m ->
exists bc,
genv_match bc ge
/\
bc_below bc (
Mem.nextblock m)
/\
bc_nostack bc
/\
romatch bc m (
romem_for_program prog)
/\ (
forall b,
Mem.valid_block m b ->
bc b <>
BCinvalid).
Proof.
End INITIAL.
Require Import Axioms.
Theorem sound_initial:
forall prog st,
initial_state prog st ->
sound_state prog st.
Proof.
Hint Resolve areg_sound aregs_sound:
va.
Interface with other optimizations
Definition avalue (
a:
VA.t) (
r:
reg) :
aval :=
match a with
|
VA.Bot =>
Vbot
|
VA.State ae am =>
AE.get r ae
end.
Lemma avalue_sound:
forall prog s f sp pc e m r,
sound_state prog (
State s f (
Vptr sp Int.zero)
pc e m) ->
exists bc,
vmatch bc e#
r (
avalue (
analyze (
romem_for_program prog)
f)!!
pc r)
/\
genv_match bc (
Genv.globalenv prog)
/\
bc sp =
BCstack.
Proof.
intros. inv H. exists bc; split; auto. rewrite AN. apply EM.
Qed.
Definition aaddr (
a:
VA.t) (
r:
reg) :
aptr :=
match a with
|
VA.Bot =>
Pbot
|
VA.State ae am =>
aptr_of_aval (
AE.get r ae)
end.
Lemma aaddr_sound:
forall prog s f sp pc e m r b ofs,
sound_state prog (
State s f (
Vptr sp Int.zero)
pc e m) ->
e#
r =
Vptr b ofs ->
exists bc,
pmatch bc b ofs (
aaddr (
analyze (
romem_for_program prog)
f)!!
pc r)
/\
genv_match bc (
Genv.globalenv prog)
/\
bc sp =
BCstack.
Proof.
intros.
inv H.
exists bc;
split;
auto.
unfold aaddr;
rewrite AN.
apply match_aptr_of_aval.
rewrite <-
H0.
apply EM.
Qed.
Definition aaddressing (
a:
VA.t) (
addr:
addressing) (
args:
list reg) :
aptr :=
match a with
|
VA.Bot =>
Pbot
|
VA.State ae am =>
aptr_of_aval (
eval_static_addressing addr (
aregs ae args))
end.
Lemma aaddressing_sound:
forall prog s f sp pc e m addr args b ofs,
sound_state prog (
State s f (
Vptr sp Int.zero)
pc e m) ->
eval_addressing (
Genv.globalenv prog) (
Vptr sp Int.zero)
addr e##
args =
Some (
Vptr b ofs) ->
exists bc,
pmatch bc b ofs (
aaddressing (
analyze (
romem_for_program prog)
f)!!
pc addr args)
/\
genv_match bc (
Genv.globalenv prog)
/\
bc sp =
BCstack.
Proof.