Correctness proof for x86 generation: main proof.
Require Import Coqlib.
Require Import Maps.
Require Import Errors.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Events.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Op.
Require Import Locations.
Require Import Mach.
Require Import Machsem.
Require Import Machtyping.
Require Import Conventions.
Require Import Asm.
Require Import Asmgen.
Require Import Asmgenretaddr.
Require Import Asmgenproof1.
Section PRESERVATION.
Variable prog:
Mach.program.
Variable tprog:
Asm.program.
Hypothesis TRANSF:
transf_program prog =
Errors.OK tprog.
Let ge :=
Genv.globalenv prog.
Let tge :=
Genv.globalenv tprog.
Lemma symbols_preserved:
forall id,
Genv.find_symbol tge id =
Genv.find_symbol ge id.
Proof.
Lemma functions_translated:
forall b f,
Genv.find_funct_ptr ge b =
Some f ->
exists tf,
Genv.find_funct_ptr tge b =
Some tf /\
transf_fundef f =
Errors.OK tf.
Proof
(
Genv.find_funct_ptr_transf_partial transf_fundef _ TRANSF).
Lemma functions_transl:
forall fb f tf,
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
transf_function f =
OK tf ->
Genv.find_funct_ptr tge fb =
Some (
Internal tf).
Proof.
intros.
exploit functions_translated;
eauto.
intros [
tf' [
A B]].
monadInv B.
rewrite H0 in EQ;
inv EQ;
auto.
Qed.
Lemma varinfo_preserved:
forall b,
Genv.find_var_info tge b =
Genv.find_var_info ge b.
Proof.
Properties of control flow
Lemma find_instr_in:
forall c pos i,
find_instr pos c =
Some i ->
In i c.
Proof.
induction c;
simpl.
intros;
discriminate.
intros until i.
case (
zeq pos 0);
intros.
left;
congruence.
right;
eauto.
Qed.
Lemma find_instr_tail:
forall c1 i c2 pos,
code_tail pos c1 (
i ::
c2) ->
find_instr pos c1 =
Some i.
Proof.
induction c1;
simpl;
intros.
inv H.
destruct (
zeq pos 0).
subst pos.
inv H.
auto.
generalize (
code_tail_pos _ _ _ H4).
intro.
omegaContradiction.
inv H.
congruence.
replace (
pos0 + 1 - 1)
with pos0 by omega.
eauto.
Qed.
Remark code_tail_bounds:
forall fn ofs i c,
code_tail ofs fn (
i ::
c) -> 0 <=
ofs <
list_length_z fn.
Proof.
Lemma code_tail_next:
forall fn ofs i c,
code_tail ofs fn (
i ::
c) ->
code_tail (
ofs + 1)
fn c.
Proof.
assert (
forall ofs fn c,
code_tail ofs fn c ->
forall i c',
c =
i ::
c' ->
code_tail (
ofs + 1)
fn c').
induction 1;
intros.
subst c.
constructor.
constructor.
constructor.
eauto.
eauto.
Qed.
Lemma code_tail_next_int:
forall fn ofs i c,
list_length_z fn <=
Int.max_unsigned ->
code_tail (
Int.unsigned ofs)
fn (
i ::
c) ->
code_tail (
Int.unsigned (
Int.add ofs Int.one))
fn c.
Proof.
Lemma transf_function_no_overflow:
forall f tf,
transf_function f =
OK tf ->
list_length_z tf <=
Int.max_unsigned.
Proof.
transl_code_at_pc pc fn c holds if the code pointer pc points
within the IA32 code generated by translating Mach function fn,
and c is the tail of the generated code at the position corresponding
to the code pointer pc.
Inductive transl_code_at_pc:
val ->
block ->
Mach.function ->
Mach.code ->
bool ->
Asm.code ->
Asm.code ->
Prop :=
transl_code_at_pc_intro:
forall b ofs f c ep tf tc,
Genv.find_funct_ptr ge b =
Some (
Internal f) ->
transf_function f =
OK tf ->
transl_code f c ep =
OK tc ->
code_tail (
Int.unsigned ofs)
tf tc ->
transl_code_at_pc (
Vptr b ofs)
b f c ep tf tc.
The following lemmas show that straight-line executions
(predicate exec_straight) correspond to correct PPC executions
(predicate exec_steps) under adequate transl_code_at_pc hypotheses.
Lemma exec_straight_steps_1:
forall fn c rs m c'
rs'
m',
exec_straight tge fn c rs m c'
rs'
m' ->
list_length_z fn <=
Int.max_unsigned ->
forall b ofs,
rs#
PC =
Vptr b ofs ->
Genv.find_funct_ptr tge b =
Some (
Internal fn) ->
code_tail (
Int.unsigned ofs)
fn c ->
plus step tge (
State rs m)
E0 (
State rs'
m').
Proof.
Lemma exec_straight_steps_2:
forall fn c rs m c'
rs'
m',
exec_straight tge fn c rs m c'
rs'
m' ->
list_length_z fn <=
Int.max_unsigned ->
forall b ofs,
rs#
PC =
Vptr b ofs ->
Genv.find_funct_ptr tge b =
Some (
Internal fn) ->
code_tail (
Int.unsigned ofs)
fn c ->
exists ofs',
rs'#
PC =
Vptr b ofs'
/\
code_tail (
Int.unsigned ofs')
fn c'.
Proof.
Lemma exec_straight_exec:
forall fb f c ep tf tc c'
rs m rs'
m',
transl_code_at_pc (
rs PC)
fb f c ep tf tc ->
exec_straight tge tf tc rs m c'
rs'
m' ->
plus step tge (
State rs m)
E0 (
State rs'
m').
Proof.
Lemma exec_straight_at:
forall fb f c ep tf tc c'
ep'
tc'
rs m rs'
m',
transl_code_at_pc (
rs PC)
fb f c ep tf tc ->
transl_code f c'
ep' =
OK tc' ->
exec_straight tge tf tc rs m tc'
rs'
m' ->
transl_code_at_pc (
rs'
PC)
fb f c'
ep'
tf tc'.
Proof.
Correctness of the return addresses predicted by
Asmgen.return_address_offset.
Remark code_tail_no_bigger:
forall pos c1 c2,
code_tail pos c1 c2 -> (
length c2 <=
length c1)%
nat.
Proof.
induction 1; simpl; omega.
Qed.
Remark code_tail_unique:
forall fn c pos pos',
code_tail pos fn c ->
code_tail pos'
fn c ->
pos =
pos'.
Proof.
induction fn;
intros until pos';
intros ITA CT;
inv ITA;
inv CT;
auto.
generalize (
code_tail_no_bigger _ _ _ H3);
simpl;
intro;
omega.
generalize (
code_tail_no_bigger _ _ _ H3);
simpl;
intro;
omega.
f_equal.
eauto.
Qed.
Lemma return_address_offset_correct:
forall b ofs fb f c tf tc ofs',
transl_code_at_pc (
Vptr b ofs)
fb f c false tf tc ->
return_address_offset f c ofs' ->
ofs' =
ofs.
Proof.
The find_label function returns the code tail starting at the
given label. A connection with code_tail is then established.
Fixpoint find_label (
lbl:
label) (
c:
code) {
struct c} :
option code :=
match c with
|
nil =>
None
|
instr ::
c' =>
if is_label lbl instr then Some c'
else find_label lbl c'
end.
Lemma label_pos_code_tail:
forall lbl c pos c',
find_label lbl c =
Some c' ->
exists pos',
label_pos lbl pos c =
Some pos'
/\
code_tail (
pos' -
pos)
c c'
/\
pos <
pos' <=
pos +
list_length_z c.
Proof.
induction c.
simpl;
intros.
discriminate.
simpl;
intros until c'.
case (
is_label lbl a).
intro EQ;
injection EQ;
intro;
subst c'.
exists (
pos + 1).
split.
auto.
split.
replace (
pos + 1 -
pos)
with (0 + 1)
by omega.
constructor.
constructor.
rewrite list_length_z_cons.
generalize (
list_length_z_pos c).
omega.
intros.
generalize (
IHc (
pos + 1)
c'
H).
intros [
pos' [
A [
B C]]].
exists pos'.
split.
auto.
split.
replace (
pos' -
pos)
with ((
pos' - (
pos + 1)) + 1)
by omega.
constructor.
auto.
rewrite list_length_z_cons.
omega.
Qed.
The following lemmas show that the translation from Mach to Asm
preserves labels, in the sense that the following diagram commutes:
translation
Mach code ------------------------ Asm instr sequence
| |
| Mach.find_label lbl find_label lbl |
| |
v v
Mach code tail ------------------- Asm instr seq tail
translation
The proof demands many boring lemmas showing that Asm constructor
functions do not introduce new labels.
.
Variable lbl:
label.
Remark mk_mov_label:
forall rd rs k c,
mk_mov rd rs k =
OK c ->
find_label lbl c =
find_label lbl k.
Proof.
unfold mk_mov; intros.
destruct rd; try discriminate; destruct rs; inv H; auto.
Qed.
Remark mk_shift_label:
forall f r1 r2 k c,
mk_shift f r1 r2 k =
OK c ->
(
forall r,
is_label lbl (
f r) =
false) ->
find_label lbl c =
find_label lbl k.
Proof.
unfold mk_shift;
intros.
destruct (
ireg_eq r2 ECX).
monadInv H;
simpl;
rewrite H0;
auto.
destruct (
ireg_eq r1 ECX);
monadInv H;
simpl;
rewrite H0;
auto.
Qed.
Remark mk_mov2_label:
forall r1 r2 r3 r4 k,
find_label lbl (
mk_mov2 r1 r2 r3 r4 k) =
find_label lbl k.
Proof.
intros;
unfold mk_mov2.
destruct (
ireg_eq r1 r2);
auto.
destruct (
ireg_eq r3 r4);
auto.
destruct (
ireg_eq r3 r2);
auto.
destruct (
ireg_eq r1 r4);
auto.
Qed.
Remark mk_div_label:
forall f r1 r2 k c,
mk_div f r1 r2 k =
OK c ->
(
forall r,
is_label lbl (
f r) =
false) ->
find_label lbl c =
find_label lbl k.
Proof.
unfold mk_div;
intros.
destruct (
ireg_eq r1 EAX).
destruct (
ireg_eq r2 EDX);
monadInv H;
simpl;
rewrite H0;
auto.
monadInv H;
simpl.
rewrite mk_mov2_label.
simpl;
rewrite H0;
auto.
Qed.
Remark mk_mod_label:
forall f r1 r2 k c,
mk_mod f r1 r2 k =
OK c ->
(
forall r,
is_label lbl (
f r) =
false) ->
find_label lbl c =
find_label lbl k.
Proof.
unfold mk_mod;
intros.
destruct (
ireg_eq r1 EAX).
destruct (
ireg_eq r2 EDX);
monadInv H;
simpl;
rewrite H0;
auto.
monadInv H;
simpl.
rewrite mk_mov2_label.
simpl;
rewrite H0;
auto.
Qed.
Remark mk_shrximm_label:
forall r n k c,
mk_shrximm r n k =
OK c ->
find_label lbl c =
find_label lbl k.
Proof.
intros. monadInv H; auto.
Qed.
Remark mk_intconv_label:
forall f r1 r2 k c,
mk_intconv f r1 r2 k =
OK c ->
(
forall r r',
is_label lbl (
f r r') =
false) ->
find_label lbl c =
find_label lbl k.
Proof.
unfold mk_intconv;
intros.
destruct (
low_ireg r2);
inv H;
simpl;
rewrite H0;
auto.
Qed.
Remark mk_smallstore_label:
forall f addr r k c,
mk_smallstore f addr r k =
OK c ->
(
forall r addr,
is_label lbl (
f r addr) =
false) ->
find_label lbl c =
find_label lbl k.
Proof.
unfold mk_smallstore;
intros.
destruct (
low_ireg r).
monadInv H;
simpl;
rewrite H0;
auto.
destruct (
addressing_mentions addr ECX);
monadInv H;
simpl;
rewrite H0;
auto.
Qed.
Remark loadind_label:
forall base ofs ty dst k c,
loadind base ofs ty dst k =
OK c ->
find_label lbl c =
find_label lbl k.
Proof.
unfold loadind;
intros.
destruct ty.
monadInv H;
auto.
destruct (
preg_of dst);
inv H;
auto.
Qed.
Remark storeind_label:
forall base ofs ty src k c,
storeind src base ofs ty k =
OK c ->
find_label lbl c =
find_label lbl k.
Proof.
unfold storeind;
intros.
destruct ty.
monadInv H;
auto.
destruct (
preg_of src);
inv H;
auto.
Qed.
Remark mk_setcc_label:
forall xc rd k,
find_label lbl (
mk_setcc xc rd k) =
find_label lbl k.
Proof.
intros.
destruct xc;
simpl;
auto;
destruct (
ireg_eq rd EDX);
auto.
Qed.
Remark mk_jcc_label:
forall xc lbl'
k,
find_label lbl (
mk_jcc xc lbl'
k) =
find_label lbl k.
Proof.
intros. destruct xc; auto.
Qed.
Ltac ArgsInv :=
match goal with
| [
H:
Error _ =
OK _ |-
_ ] =>
discriminate
| [
H:
match ?
args with nil =>
_ |
_ ::
_ =>
_ end =
OK _ |-
_ ] =>
destruct args;
ArgsInv
| [
H:
bind _ _ =
OK _ |-
_ ] =>
monadInv H;
ArgsInv
|
_ =>
idtac
end.
Remark transl_cond_label:
forall cond args k c,
transl_cond cond args k =
OK c ->
find_label lbl c =
find_label lbl k.
Proof.
unfold transl_cond;
intros.
destruct cond;
ArgsInv;
auto.
destruct (
Int.eq_dec i Int.zero);
auto.
destruct c0;
auto.
destruct c0;
auto.
Qed.
Remark transl_op_label:
forall op args r k c,
transl_op op args r k =
OK c ->
find_label lbl c =
find_label lbl k.
Proof.
Remark transl_load_label:
forall chunk addr args dest k c,
transl_load chunk addr args dest k =
OK c ->
find_label lbl c =
find_label lbl k.
Proof.
intros. monadInv H. destruct chunk; monadInv EQ0; auto.
Qed.
Remark transl_store_label:
forall chunk addr args src k c,
transl_store chunk addr args src k =
OK c ->
find_label lbl c =
find_label lbl k.
Proof.
intros.
monadInv H.
destruct chunk;
monadInv EQ0;
auto;
eapply mk_smallstore_label;
eauto.
Qed.
Lemma transl_instr_label:
forall f i ep k c,
transl_instr f i ep k =
OK c ->
find_label lbl c =
if Mach.is_label lbl i then Some k else find_label lbl k.
Proof.
Lemma transl_code_label:
forall f c ep tc,
transl_code f c ep =
OK tc ->
match Mach.find_label lbl c with
|
None =>
find_label lbl tc =
None
|
Some c' =>
exists tc',
find_label lbl tc =
Some tc' /\
transl_code f c'
false =
OK tc'
end.
Proof.
Lemma transl_find_label:
forall f tf,
transf_function f =
OK tf ->
match Mach.find_label lbl f.(
fn_code)
with
|
None =>
find_label lbl tf =
None
|
Some c =>
exists tc,
find_label lbl tf =
Some tc /\
transl_code f c false =
OK tc
end.
Proof.
End TRANSL_LABEL.
A valid branch in a piece of Mach code translates to a valid ``go to''
transition in the generated PPC code.
Lemma find_label_goto_label:
forall f tf lbl rs m c'
b ofs,
Genv.find_funct_ptr ge b =
Some (
Internal f) ->
transf_function f =
OK tf ->
rs PC =
Vptr b ofs ->
Mach.find_label lbl f.(
fn_code) =
Some c' ->
exists tc',
exists rs',
goto_label tf lbl rs m =
Next rs'
m
/\
transl_code_at_pc (
rs'
PC)
b f c'
false tf tc'
/\
forall r,
r <>
PC ->
rs'#
r =
rs#
r.
Proof.
Proof of semantic preservation
Semantic preservation is proved using simulation diagrams
of the following form.
st1 --------------- st2
| |
t| *|t
| |
v v
st1'--------------- st2'
The invariant is the
match_states predicate below, which includes:
-
The PPC code pointed by the PC register is the translation of
the current Mach code sequence.
-
Mach register values and PPC register values agree.
:
list Machsem.stackframe ->
Prop :=
|
match_stack_nil:
match_stack nil
|
match_stack_cons:
forall fb sp ra c s f tf tc,
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
transl_code_at_pc ra fb f c false tf tc ->
sp <>
Vundef ->
ra <>
Vundef ->
match_stack s ->
match_stack (
Stackframe fb sp ra c ::
s).
Inductive match_states:
Machsem.state ->
Asm.state ->
Prop :=
|
match_states_intro:
forall s fb sp c ep ms m m'
rs f tf tc
(
STACKS:
match_stack s)
(
FIND:
Genv.find_funct_ptr ge fb =
Some (
Internal f))
(
MEXT:
Mem.extends m m')
(
AT:
transl_code_at_pc (
rs PC)
fb f c ep tf tc)
(
AG:
agree ms sp rs)
(
DXP:
ep =
true ->
rs#
EDX =
parent_sp s),
match_states (
Machsem.State s fb sp c ms m)
(
Asm.State rs m')
|
match_states_call:
forall s fb ms m m'
rs
(
STACKS:
match_stack s)
(
MEXT:
Mem.extends m m')
(
AG:
agree ms (
parent_sp s)
rs)
(
ATPC:
rs PC =
Vptr fb Int.zero)
(
ATLR:
rs RA =
parent_ra s),
match_states (
Machsem.Callstate s fb ms m)
(
Asm.State rs m')
|
match_states_return:
forall s ms m m'
rs
(
STACKS:
match_stack s)
(
MEXT:
Mem.extends m m')
(
AG:
agree ms (
parent_sp s)
rs)
(
ATPC:
rs PC =
parent_ra s),
match_states (
Machsem.Returnstate s ms m)
(
Asm.State rs m').
Lemma exec_straight_steps:
forall s fb f rs1 i c ep tf tc m1'
m2 m2'
sp ms2,
match_stack s ->
Mem.extends m2 m2' ->
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
transl_code_at_pc (
rs1 PC)
fb f (
i ::
c)
ep tf tc ->
(
forall k c,
transl_instr f i ep k =
OK c ->
exists rs2,
exec_straight tge tf c rs1 m1'
k rs2 m2'
/\
agree ms2 sp rs2
/\ (
edx_preserved ep i =
true ->
rs2#
EDX =
parent_sp s)) ->
exists st',
plus step tge (
State rs1 m1')
E0 st' /\
match_states (
Machsem.State s fb sp c ms2 m2)
st'.
Proof.
intros.
inversion H2.
subst.
monadInv H7.
exploit H3;
eauto.
intros [
rs2 [
A [
B C]]].
exists (
State rs2 m2');
split.
eapply exec_straight_exec;
eauto.
econstructor;
eauto.
eapply exec_straight_at;
eauto.
Qed.
Lemma exec_straight_steps_goto:
forall s fb f rs1 i c ep tf tc m1'
m2 m2'
sp ms2 lbl c',
match_stack s ->
Mem.extends m2 m2' ->
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
Mach.find_label lbl f.(
fn_code) =
Some c' ->
transl_code_at_pc (
rs1 PC)
fb f (
i ::
c)
ep tf tc ->
edx_preserved ep i =
false ->
(
forall k c,
transl_instr f i ep k =
OK c ->
exists jmp,
exists k',
exists rs2,
exec_straight tge tf c rs1 m1' (
jmp ::
k')
rs2 m2'
/\
agree ms2 sp rs2
/\
exec_instr tge tf jmp rs2 m2' =
goto_label tf lbl rs2 m2') ->
exists st',
plus step tge (
State rs1 m1')
E0 st' /\
match_states (
Machsem.State s fb sp c'
ms2 m2)
st'.
Proof.
Lemma parent_sp_def:
forall s,
match_stack s ->
parent_sp s <>
Vundef.
Proof.
induction 1; simpl. congruence. auto. Qed.
Lemma parent_ra_def:
forall s,
match_stack s ->
parent_ra s <>
Vundef.
Proof.
induction 1; simpl. unfold Vzero. congruence. auto. Qed.
Lemma lessdef_parent_sp:
forall s v,
match_stack s ->
Val.lessdef (
parent_sp s)
v ->
v =
parent_sp s.
Proof.
Lemma lessdef_parent_ra:
forall s v,
match_stack s ->
Val.lessdef (
parent_ra s)
v ->
v =
parent_ra s.
Proof.
We need to show that, in the simulation diagram, we cannot
take infinitely many Mach transitions that correspond to zero
transitions on the PPC side. Actually, all Mach transitions
correspond to at least one Asm transition, except the
transition from Machsem.Returnstate to Machsem.State.
So, the following integer measure will suffice to rule out
the unwanted behaviour.
Definition measure (
s:
Machsem.state) :
nat :=
match s with
|
Machsem.State _ _ _ _ _ _ => 0%
nat
|
Machsem.Callstate _ _ _ _ => 0%
nat
|
Machsem.Returnstate _ _ _ => 1%
nat
end.
We show the simulation diagram by case analysis on the Mach transition
on the left. Since the proof is large, we break it into one lemma
per transition.
Definition exec_instr_prop (
s1:
Machsem.state) (
t:
trace) (
s2:
Machsem.state) :
Prop :=
forall s1' (
MS:
match_states s1 s1'),
(
exists s2',
plus step tge s1'
t s2' /\
match_states s2 s2')
\/ (
measure s2 <
measure s1 /\
t =
E0 /\
match_states s2 s1')%
nat.
Lemma exec_Mlabel_prop:
forall (
s :
list stackframe) (
fb :
block) (
sp :
val)
(
lbl :
Mach.label) (
c :
list Mach.instruction) (
ms :
Mach.regset)
(
m :
mem),
exec_instr_prop (
Machsem.State s fb sp (
Mlabel lbl ::
c)
ms m)
E0
(
Machsem.State s fb sp c ms m).
Proof.
Lemma exec_Mgetstack_prop:
forall (
s :
list stackframe) (
fb :
block) (
sp :
val) (
ofs :
int)
(
ty :
typ) (
dst :
mreg) (
c :
list Mach.instruction)
(
ms :
Mach.regset) (
m :
mem) (
v :
val),
load_stack m sp ty ofs =
Some v ->
exec_instr_prop (
Machsem.State s fb sp (
Mgetstack ofs ty dst ::
c)
ms m)
E0
(
Machsem.State s fb sp c (
Regmap.set dst v ms)
m).
Proof.
Lemma exec_Msetstack_prop:
forall (
s :
list stackframe) (
fb :
block) (
sp :
val) (
src :
mreg)
(
ofs :
int) (
ty :
typ) (
c :
list Mach.instruction)
(
ms :
mreg ->
val) (
m m' :
mem),
store_stack m sp ty ofs (
ms src) =
Some m' ->
exec_instr_prop (
Machsem.State s fb sp (
Msetstack src ofs ty ::
c)
ms m)
E0
(
Machsem.State s fb sp c (
undef_setstack ms)
m').
Proof.
Lemma exec_Mgetparam_prop:
forall (
s :
list stackframe) (
fb :
block) (
f:
Mach.function) (
sp :
val)
(
ofs :
int) (
ty :
typ) (
dst :
mreg) (
c :
list Mach.instruction)
(
ms :
Mach.regset) (
m :
mem) (
v :
val),
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
load_stack m sp Tint f.(
fn_link_ofs) =
Some (
parent_sp s) ->
load_stack m (
parent_sp s)
ty ofs =
Some v ->
exec_instr_prop (
Machsem.State s fb sp (
Mgetparam ofs ty dst ::
c)
ms m)
E0
(
Machsem.State s fb sp c (
Regmap.set dst v (
Regmap.set IT1 Vundef ms))
m).
Proof.
Lemma exec_Mop_prop:
forall (
s :
list stackframe) (
fb :
block) (
sp :
val) (
op :
operation)
(
args :
list mreg) (
res :
mreg) (
c :
list Mach.instruction)
(
ms :
mreg ->
val) (
m :
mem) (
v :
val),
eval_operation ge sp op ms ##
args m =
Some v ->
exec_instr_prop (
Machsem.State s fb sp (
Mop op args res ::
c)
ms m)
E0
(
Machsem.State s fb sp c (
Regmap.set res v (
undef_op op ms))
m).
Proof.
Lemma exec_Mload_prop:
forall (
s :
list stackframe) (
fb :
block) (
sp :
val)
(
chunk :
memory_chunk) (
addr :
addressing) (
args :
list mreg)
(
dst :
mreg) (
c :
list Mach.instruction) (
ms :
mreg ->
val)
(
m :
mem) (
a v :
val),
eval_addressing ge sp addr ms ##
args =
Some a ->
Mem.loadv chunk m a =
Some v ->
exec_instr_prop (
Machsem.State s fb sp (
Mload chunk addr args dst ::
c)
ms m)
E0 (
Machsem.State s fb sp c (
Regmap.set dst v (
undef_temps ms))
m).
Proof.
Lemma exec_Mstore_prop:
forall (
s :
list stackframe) (
fb :
block) (
sp :
val)
(
chunk :
memory_chunk) (
addr :
addressing) (
args :
list mreg)
(
src :
mreg) (
c :
list Mach.instruction) (
ms :
mreg ->
val)
(
m m' :
mem) (
a :
val),
eval_addressing ge sp addr ms ##
args =
Some a ->
Mem.storev chunk m a (
ms src) =
Some m' ->
exec_instr_prop (
Machsem.State s fb sp (
Mstore chunk addr args src ::
c)
ms m)
E0
(
Machsem.State s fb sp c (
undef_temps ms)
m').
Proof.
Lemma exec_Mcall_prop:
forall (
s :
list stackframe) (
fb :
block) (
sp :
val)
(
sig :
signature) (
ros :
mreg +
ident) (
c :
Mach.code)
(
ms :
Mach.regset) (
m :
mem) (
f :
function) (
f' :
block)
(
ra :
int),
find_function_ptr ge ros ms =
Some f' ->
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
return_address_offset f c ra ->
exec_instr_prop (
Machsem.State s fb sp (
Mcall sig ros ::
c)
ms m)
E0
(
Callstate (
Stackframe fb sp (
Vptr fb ra)
c ::
s)
f'
ms m).
Proof.
Lemma agree_change_sp:
forall ms sp rs sp',
agree ms sp rs ->
sp' <>
Vundef ->
agree ms sp' (
rs#
ESP <-
sp').
Proof.
Lemma exec_Mtailcall_prop:
forall (
s :
list stackframe) (
fb stk :
block) (
soff :
int)
(
sig :
signature) (
ros :
mreg +
ident) (
c :
list Mach.instruction)
(
ms :
Mach.regset) (
m :
mem) (
f:
Mach.function) (
f' :
block)
m',
find_function_ptr ge ros ms =
Some f' ->
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
load_stack m (
Vptr stk soff)
Tint f.(
fn_link_ofs) =
Some (
parent_sp s) ->
load_stack m (
Vptr stk soff)
Tint f.(
fn_retaddr_ofs) =
Some (
parent_ra s) ->
Mem.free m stk 0
f.(
fn_stacksize) =
Some m' ->
exec_instr_prop
(
Machsem.State s fb (
Vptr stk soff) (
Mtailcall sig ros ::
c)
ms m)
E0
(
Callstate s f'
ms m').
Proof.
Lemma exec_Mgoto_prop:
forall (
s :
list stackframe) (
fb :
block) (
f :
function) (
sp :
val)
(
lbl :
Mach.label) (
c :
list Mach.instruction) (
ms :
Mach.regset)
(
m :
mem) (
c' :
Mach.code),
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
Mach.find_label lbl (
fn_code f) =
Some c' ->
exec_instr_prop (
Machsem.State s fb sp (
Mgoto lbl ::
c)
ms m)
E0
(
Machsem.State s fb sp c'
ms m).
Proof.
intros;
red;
intros;
inv MS.
assert (
f0 =
f)
by congruence.
subst f0.
inv AT.
monadInv H4.
exploit find_label_goto_label;
eauto.
intros [
tc' [
rs' [
GOTO [
AT2 INV]]]].
left;
exists (
State rs'
m');
split.
apply plus_one.
econstructor;
eauto.
eapply functions_transl;
eauto.
eapply find_instr_tail;
eauto.
simpl;
eauto.
econstructor;
eauto.
eapply agree_exten;
eauto with ppcgen.
congruence.
Qed.
Lemma exec_Mbuiltin_prop:
forall (
s :
list stackframe) (
f :
block) (
sp :
val)
(
ms :
Mach.regset) (
m :
mem) (
ef :
external_function)
(
args :
list mreg) (
res :
mreg) (
b :
list Mach.instruction)
(
t :
trace) (
v :
val) (
m' :
mem),
external_call ef ge ms ##
args m t v m' ->
exec_instr_prop (
Machsem.State s f sp (
Mbuiltin ef args res ::
b)
ms m)
t
(
Machsem.State s f sp b (
Regmap.set res v (
undef_temps ms))
m').
Proof.
Lemma exec_Mannot_prop:
forall (
s :
list stackframe) (
f :
block) (
sp :
val)
(
ms :
Mach.regset) (
m :
mem) (
ef :
external_function)
(
args :
list Mach.annot_param) (
b :
list Mach.instruction)
(
vargs:
list val) (
t :
trace) (
v :
val) (
m' :
mem),
Machsem.annot_arguments ms m sp args vargs ->
external_call ef ge vargs m t v m' ->
exec_instr_prop (
Machsem.State s f sp (
Mannot ef args ::
b)
ms m)
t
(
Machsem.State s f sp b ms m').
Proof.
Lemma exec_Mcond_true_prop:
forall (
s :
list stackframe) (
fb :
block) (
f :
function) (
sp :
val)
(
cond :
condition) (
args :
list mreg) (
lbl :
Mach.label)
(
c :
list Mach.instruction) (
ms :
mreg ->
val) (
m :
mem)
(
c' :
Mach.code),
eval_condition cond ms ##
args m =
Some true ->
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
Mach.find_label lbl (
fn_code f) =
Some c' ->
exec_instr_prop (
Machsem.State s fb sp (
Mcond cond args lbl ::
c)
ms m)
E0
(
Machsem.State s fb sp c' (
undef_temps ms)
m).
Proof.
Lemma exec_Mcond_false_prop:
forall (
s :
list stackframe) (
fb :
block) (
sp :
val)
(
cond :
condition) (
args :
list mreg) (
lbl :
Mach.label)
(
c :
list Mach.instruction) (
ms :
mreg ->
val) (
m :
mem),
eval_condition cond ms ##
args m =
Some false ->
exec_instr_prop (
Machsem.State s fb sp (
Mcond cond args lbl ::
c)
ms m)
E0
(
Machsem.State s fb sp c (
undef_temps ms)
m).
Proof.
Lemma exec_Mjumptable_prop:
forall (
s :
list stackframe) (
fb :
block) (
f :
function) (
sp :
val)
(
arg :
mreg) (
tbl :
list Mach.label) (
c :
list Mach.instruction)
(
rs :
mreg ->
val) (
m :
mem) (
n :
int) (
lbl :
Mach.label)
(
c' :
Mach.code),
rs arg =
Vint n ->
list_nth_z tbl (
Int.unsigned n) =
Some lbl ->
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
Mach.find_label lbl (
fn_code f) =
Some c' ->
exec_instr_prop
(
Machsem.State s fb sp (
Mjumptable arg tbl ::
c)
rs m)
E0
(
Machsem.State s fb sp c' (
undef_temps rs)
m).
Proof.
Lemma exec_Mreturn_prop:
forall (
s :
list stackframe) (
fb stk :
block) (
soff :
int)
(
c :
list Mach.instruction) (
ms :
Mach.regset) (
m :
mem) (
f:
Mach.function)
m',
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
load_stack m (
Vptr stk soff)
Tint f.(
fn_link_ofs) =
Some (
parent_sp s) ->
load_stack m (
Vptr stk soff)
Tint f.(
fn_retaddr_ofs) =
Some (
parent_ra s) ->
Mem.free m stk 0
f.(
fn_stacksize) =
Some m' ->
exec_instr_prop (
Machsem.State s fb (
Vptr stk soff) (
Mreturn ::
c)
ms m)
E0
(
Returnstate s ms m').
Proof.
Lemma exec_function_internal_prop:
forall (
s :
list stackframe) (
fb :
block) (
ms :
Mach.regset)
(
m :
mem) (
f :
function) (
m1 m2 m3 :
mem) (
stk :
block),
Genv.find_funct_ptr ge fb =
Some (
Internal f) ->
Mem.alloc m 0 (
fn_stacksize f) = (
m1,
stk) ->
let sp :=
Vptr stk Int.zero in
store_stack m1 sp Tint f.(
fn_link_ofs) (
parent_sp s) =
Some m2 ->
store_stack m2 sp Tint f.(
fn_retaddr_ofs) (
parent_ra s) =
Some m3 ->
exec_instr_prop (
Machsem.Callstate s fb ms m)
E0
(
Machsem.State s fb sp (
fn_code f) (
undef_temps ms)
m3).
Proof.
Lemma exec_function_external_prop:
forall (
s :
list stackframe) (
fb :
block) (
ms :
Mach.regset)
(
m :
mem) (
t0 :
trace) (
ms' :
RegEq.t ->
val)
(
ef :
external_function) (
args :
list val) (
res :
val) (
m':
mem),
Genv.find_funct_ptr ge fb =
Some (
External ef) ->
external_call ef ge args m t0 res m' ->
Machsem.extcall_arguments ms m (
parent_sp s) (
ef_sig ef)
args ->
ms' =
Regmap.set (
loc_result (
ef_sig ef))
res ms ->
exec_instr_prop (
Machsem.Callstate s fb ms m)
t0 (
Machsem.Returnstate s ms'
m').
Proof.
Lemma exec_return_prop:
forall (
s :
list stackframe) (
fb :
block) (
sp ra :
val)
(
c :
Mach.code) (
ms :
Mach.regset) (
m :
mem),
exec_instr_prop (
Machsem.Returnstate (
Stackframe fb sp ra c ::
s)
ms m)
E0
(
Machsem.State s fb sp c ms m).
Proof.
intros; red; intros; inv MS. inv STACKS. simpl in *.
right. split. omega. split. auto.
econstructor; eauto. rewrite ATPC; eauto.
congruence.
Qed.
Theorem transf_instr_correct:
forall s1 t s2,
Machsem.step ge s1 t s2 ->
exec_instr_prop s1 t s2.
Proof
(
Machsem.step_ind ge exec_instr_prop
exec_Mlabel_prop
exec_Mgetstack_prop
exec_Msetstack_prop
exec_Mgetparam_prop
exec_Mop_prop
exec_Mload_prop
exec_Mstore_prop
exec_Mcall_prop
exec_Mtailcall_prop
exec_Mbuiltin_prop
exec_Mannot_prop
exec_Mgoto_prop
exec_Mcond_true_prop
exec_Mcond_false_prop
exec_Mjumptable_prop
exec_Mreturn_prop
exec_function_internal_prop
exec_function_external_prop
exec_return_prop).
Lemma transf_initial_states:
forall st1,
Machsem.initial_state prog st1 ->
exists st2,
Asm.initial_state tprog st2 /\
match_states st1 st2.
Proof.
Lemma transf_final_states:
forall st1 st2 r,
match_states st1 st2 ->
Machsem.final_state st1 r ->
Asm.final_state st2 r.
Proof.
intros.
inv H0.
inv H.
constructor.
auto.
compute in H1.
generalize (
preg_val _ _ _ AX AG).
rewrite H1.
intros LD;
inv LD.
auto.
Qed.
Theorem transf_program_correct:
forward_simulation (
Machsem.semantics prog) (
Asm.semantics tprog).
Proof.
End PRESERVATION.