Correctness proof for Asm generation: machine-independent framework
Require Import Coqlib.
Require Intv.
Require Import AST.
Require Import Errors.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Events.
Require Import Smallstep.
Require Import Locations.
Require Import Mach.
Require Import Asm.
Require Import Asmgen.
Require Import Conventions.
Processor registers and register states
Global Hint Extern 2 (
_ <>
_) =>
congruence:
asmgen.
Lemma ireg_of_eq:
forall r r',
ireg_of r =
OK r' ->
preg_of r =
IR r'.
Proof.
Lemma freg_of_eq:
forall r r',
freg_of r =
OK r' ->
preg_of r =
FR r'.
Proof.
Lemma preg_of_injective:
forall r1 r2,
preg_of r1 =
preg_of r2 ->
r1 =
r2.
Proof.
destruct r1; destruct r2; simpl; intros; reflexivity || discriminate.
Qed.
Lemma preg_of_data:
forall r,
data_preg (
preg_of r) =
true.
Proof.
intros. destruct r; reflexivity.
Qed.
Global Hint Resolve preg_of_data:
asmgen.
Lemma data_diff:
forall r r',
data_preg r =
true ->
data_preg r' =
false ->
r <>
r'.
Proof.
congruence.
Qed.
Global Hint Resolve data_diff:
asmgen.
Lemma preg_of_not_SP:
forall r,
preg_of r <>
SP.
Proof.
intros.
unfold preg_of;
destruct r;
simpl;
congruence.
Qed.
Lemma preg_of_not_PC:
forall r,
preg_of r <>
PC.
Proof.
intros.
apply data_diff;
auto with asmgen.
Qed.
Global Hint Resolve preg_of_not_SP preg_of_not_PC:
asmgen.
Lemma nextinstr_pc:
forall rs, (
nextinstr rs)#
PC =
Val.offset_ptr rs#
PC Ptrofs.one.
Proof.
Lemma nextinstr_inv:
forall r rs,
r <>
PC -> (
nextinstr rs)#
r =
rs#
r.
Proof.
Lemma nextinstr_inv1:
forall r rs,
data_preg r =
true -> (
nextinstr rs)#
r =
rs#
r.
Proof.
intros.
apply nextinstr_inv.
red;
intro;
subst;
discriminate.
Qed.
Lemma nextinstr_set_preg:
forall rs m v,
(
nextinstr (
rs#(
preg_of m) <-
v))#
PC =
Val.offset_ptr rs#
PC Ptrofs.one.
Proof.
Lemma undef_regs_other:
forall r rl rs,
(
forall r',
In r'
rl ->
r <>
r') ->
undef_regs rl rs r =
rs r.
Proof.
induction rl;
simpl;
intros.
auto.
rewrite IHrl by auto.
rewrite Pregmap.gso;
auto.
Qed.
Fixpoint preg_notin (
r:
preg) (
rl:
list mreg) :
Prop :=
match rl with
|
nil =>
True
|
r1 ::
nil =>
r <>
preg_of r1
|
r1 ::
rl =>
r <>
preg_of r1 /\
preg_notin r rl
end.
Remark preg_notin_charact:
forall r rl,
preg_notin r rl <-> (
forall mr,
In mr rl ->
r <>
preg_of mr).
Proof.
induction rl; simpl; intros.
tauto.
destruct rl.
simpl. split. intros. intuition congruence. auto.
rewrite IHrl. split.
intros [A B]. intros. destruct H. congruence. auto.
auto.
Qed.
Lemma undef_regs_other_2:
forall r rl rs,
preg_notin r rl ->
undef_regs (
map preg_of rl)
rs r =
rs r.
Proof.
Agreement between Mach registers and processor registers
Record agree (
ms:
Mach.regset) (
sp:
val) (
rs:
Asm.regset) :
Prop :=
mkagree {
agree_sp:
rs#
SP =
sp;
agree_sp_def:
sp <>
Vundef;
agree_mregs:
forall r:
mreg,
Val.lessdef (
ms r) (
rs#(
preg_of r))
}.
Lemma preg_val:
forall ms sp rs r,
agree ms sp rs ->
Val.lessdef (
ms r)
rs#(
preg_of r).
Proof.
intros. destruct H. auto.
Qed.
Lemma preg_vals:
forall ms sp rs,
agree ms sp rs ->
forall l,
Val.lessdef_list (
map ms l) (
map rs (
map preg_of l)).
Proof.
induction l;
simpl.
constructor.
constructor.
eapply preg_val;
eauto.
auto.
Qed.
Lemma sp_val:
forall ms sp rs,
agree ms sp rs ->
sp =
rs#
SP.
Proof.
intros. destruct H; auto.
Qed.
Lemma ireg_val:
forall ms sp rs r r',
agree ms sp rs ->
ireg_of r =
OK r' ->
Val.lessdef (
ms r)
rs#
r'.
Proof.
Lemma freg_val:
forall ms sp rs r r',
agree ms sp rs ->
freg_of r =
OK r' ->
Val.lessdef (
ms r) (
rs#
r').
Proof.
Lemma agree_exten:
forall ms sp rs rs',
agree ms sp rs ->
(
forall r,
data_preg r =
true ->
rs'#
r =
rs#
r) ->
agree ms sp rs'.
Proof.
intros.
destruct H.
split;
auto.
rewrite H0;
auto.
auto.
intros.
rewrite H0;
auto.
apply preg_of_data.
Qed.
Preservation of register agreement under various assignments.
Lemma agree_set_mreg:
forall ms sp rs r v rs',
agree ms sp rs ->
Val.lessdef v (
rs'#(
preg_of r)) ->
(
forall r',
data_preg r' =
true ->
r' <>
preg_of r ->
rs'#
r' =
rs#
r') ->
agree (
Regmap.set r v ms)
sp rs'.
Proof.
Corollary agree_set_mreg_parallel:
forall ms sp rs r v v',
agree ms sp rs ->
Val.lessdef v v' ->
agree (
Regmap.set r v ms)
sp (
Pregmap.set (
preg_of r)
v'
rs).
Proof.
Lemma agree_set_other:
forall ms sp rs r v,
agree ms sp rs ->
data_preg r =
false ->
agree ms sp (
rs#
r <-
v).
Proof.
Lemma agree_nextinstr:
forall ms sp rs,
agree ms sp rs ->
agree ms sp (
nextinstr rs).
Proof.
Lemma agree_set_pair:
forall sp p v v'
ms rs,
agree ms sp rs ->
Val.lessdef v v' ->
agree (
Mach.set_pair p v ms)
sp (
set_pair (
map_rpair preg_of p)
v'
rs).
Proof.
Lemma agree_undef_nondata_regs:
forall ms sp rl rs,
agree ms sp rs ->
(
forall r,
In r rl ->
data_preg r =
false) ->
agree ms sp (
undef_regs rl rs).
Proof.
induction rl;
simpl;
intros.
auto.
apply IHrl.
apply agree_exten with rs;
auto.
intros.
apply Pregmap.gso.
red;
intros;
subst.
assert (
data_preg a =
false)
by auto.
congruence.
intros.
apply H0;
auto.
Qed.
Lemma agree_undef_regs:
forall ms sp rl rs rs',
agree ms sp rs ->
(
forall r',
data_preg r' =
true ->
preg_notin r'
rl ->
rs'#
r' =
rs#
r') ->
agree (
Mach.undef_regs rl ms)
sp rs'.
Proof.
Lemma agree_undef_regs2:
forall ms sp rl rs rs',
agree (
Mach.undef_regs rl ms)
sp rs ->
(
forall r',
data_preg r' =
true ->
preg_notin r'
rl ->
rs'#
r' =
rs#
r') ->
agree (
Mach.undef_regs rl ms)
sp rs'.
Proof.
Lemma agree_set_undef_mreg:
forall ms sp rs r v rl rs',
agree ms sp rs ->
Val.lessdef v (
rs'#(
preg_of r)) ->
(
forall r',
data_preg r' =
true ->
r' <>
preg_of r ->
preg_notin r'
rl ->
rs'#
r' =
rs#
r') ->
agree (
Regmap.set r v (
Mach.undef_regs rl ms))
sp rs'.
Proof.
Lemma agree_undef_caller_save_regs:
forall ms sp rs,
agree ms sp rs ->
agree (
Mach.undef_caller_save_regs ms)
sp (
Asm.undef_caller_save_regs rs).
Proof.
Lemma agree_change_sp:
forall ms sp rs sp',
agree ms sp rs ->
sp' <>
Vundef ->
agree ms sp' (
rs#
SP <-
sp').
Proof.
intros.
inv H.
split;
auto.
intros.
rewrite Pregmap.gso;
auto with asmgen.
Qed.
Connection between Mach and Asm calling conventions for external
functions.
Lemma extcall_arg_match:
forall ms sp rs m m'
l v,
agree ms sp rs ->
Mem.extends m m' ->
Mach.extcall_arg ms m sp l v ->
exists v',
Asm.extcall_arg rs m'
l v' /\
Val.lessdef v v'.
Proof.
intros.
inv H1.
exists (
rs#(
preg_of r));
split.
constructor.
eapply preg_val;
eauto.
unfold load_stack in H2.
exploit Mem.loadv_extends;
eauto.
intros [
v' [
A B]].
rewrite (
sp_val _ _ _ H)
in A.
exists v';
split;
auto.
econstructor.
eauto.
assumption.
Qed.
Lemma extcall_arg_pair_match:
forall ms sp rs m m'
p v,
agree ms sp rs ->
Mem.extends m m' ->
Mach.extcall_arg_pair ms m sp p v ->
exists v',
Asm.extcall_arg_pair rs m'
p v' /\
Val.lessdef v v'.
Proof.
Lemma extcall_args_match:
forall ms sp rs m m',
agree ms sp rs ->
Mem.extends m m' ->
forall ll vl,
list_forall2 (
Mach.extcall_arg_pair ms m sp)
ll vl ->
exists vl',
list_forall2 (
Asm.extcall_arg_pair rs m')
ll vl' /\
Val.lessdef_list vl vl'.
Proof.
induction 3;
intros.
exists (@
nil val);
split.
constructor.
constructor.
exploit extcall_arg_pair_match;
eauto.
intros [
v1' [
A B]].
destruct IHlist_forall2 as [
vl' [
C D]].
exists (
v1' ::
vl');
split;
constructor;
auto.
Qed.
Lemma extcall_arguments_match:
forall ms m m'
sp rs sg args,
agree ms sp rs ->
Mem.extends m m' ->
Mach.extcall_arguments ms m sp sg args ->
exists args',
Asm.extcall_arguments rs m'
sg args' /\
Val.lessdef_list args args'.
Proof.
Translation of arguments and results to builtins.
Remark builtin_arg_match:
forall ge (
rs:
regset)
sp m a v,
eval_builtin_arg ge (
fun r =>
rs (
preg_of r))
sp m a v ->
eval_builtin_arg ge rs sp m (
map_builtin_arg preg_of a)
v.
Proof.
induction 1; simpl; eauto with barg.
Qed.
Lemma builtin_args_match:
forall ge ms sp rs m m',
agree ms sp rs ->
Mem.extends m m' ->
forall al vl,
eval_builtin_args ge ms sp m al vl ->
exists vl',
eval_builtin_args ge rs sp m' (
map (
map_builtin_arg preg_of)
al)
vl'
/\
Val.lessdef_list vl vl'.
Proof.
induction 3;
intros;
simpl.
exists (@
nil val);
split;
constructor.
exploit (@
eval_builtin_arg_lessdef _ ge ms (
fun r =>
rs (
preg_of r)));
eauto.
intros;
eapply preg_val;
eauto.
intros (
v1' &
A &
B).
destruct IHlist_forall2 as [
vl' [
C D]].
exists (
v1' ::
vl');
split;
constructor;
auto.
apply builtin_arg_match;
auto.
Qed.
Lemma agree_set_res:
forall res ms sp rs v v',
agree ms sp rs ->
Val.lessdef v v' ->
agree (
Mach.set_res res v ms)
sp (
Asm.set_res (
map_builtin_res preg_of res)
v'
rs).
Proof.
Lemma set_res_other:
forall r res v rs,
data_preg r =
false ->
set_res (
map_builtin_res preg_of res)
v rs r =
rs r.
Proof.
induction res;
simpl;
intros.
-
apply Pregmap.gso.
red;
intros;
subst r.
rewrite preg_of_data in H;
discriminate.
-
auto.
-
rewrite IHres2,
IHres1;
auto.
Qed.
Correspondence between Mach code and Asm code
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.
The ``code tail'' of an instruction list c is the list of instructions
starting at PC pos.
Inductive code_tail:
Z ->
code ->
code ->
Prop :=
|
code_tail_0:
forall c,
code_tail 0
c c
|
code_tail_S:
forall pos i c1 c2,
code_tail pos c1 c2 ->
code_tail (
pos + 1) (
i ::
c1)
c2.
Lemma code_tail_pos:
forall pos c1 c2,
code_tail pos c1 c2 ->
pos >= 0.
Proof.
induction 1. lia. lia.
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.
extlia.
inv H.
congruence.
replace (
pos0 + 1 - 1)
with pos0 by lia.
eauto.
Qed.
Remark code_tail_bounds_1:
forall fn ofs c,
code_tail ofs fn c -> 0 <=
ofs <=
list_length_z fn.
Proof.
Remark code_tail_bounds_2:
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 <=
Ptrofs.max_unsigned ->
code_tail (
Ptrofs.unsigned ofs)
fn (
i ::
c) ->
code_tail (
Ptrofs.unsigned (
Ptrofs.add ofs Ptrofs.one))
fn c.
Proof.
transl_code_at_pc pc fb f c ep tf tc holds if the code pointer pc points
within the Asm code generated by translating Mach function f,
and tc is the tail of the generated code at the position corresponding
to the code pointer pc.
Inductive transl_code_at_pc (
ge:
Mach.genv):
val ->
block ->
Mach.function ->
Mach.code ->
bool ->
Asm.function ->
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 =
Errors.OK tf ->
transl_code f c ep =
OK tc ->
code_tail (
Ptrofs.unsigned ofs) (
fn_code tf)
tc ->
transl_code_at_pc ge (
Vptr b ofs)
b f c ep tf tc.
Equivalence between transl_code and transl_code'.
Local Open Scope error_monad_scope.
Lemma transl_code_rec_transl_code:
forall f il ep k,
transl_code_rec f il ep k = (
do c <-
transl_code f il ep;
k c).
Proof.
Lemma transl_code'
_transl_code:
forall f il ep,
transl_code'
f il ep =
transl_code f il ep.
Proof.
Predictor for return addresses in generated Asm code.
The return_address_offset predicate defined here is used in the
semantics for Mach to determine the return addresses that are
stored in activation records.
Consider a Mach function
f and a sequence
c of Mach instructions
representing the Mach code that remains to be executed after a
function call returns. The predicate
return_address_offset f c ofs
holds if
ofs is the integer offset of the PPC instruction
following the call in the Asm code obtained by translating the
code of
f. Graphically:
Mach function f |--------- Mcall ---------|
Mach code c | |--------|
| \ \
| \ \
| \ \
Asm code | |--------|
Asm function |------------- Pcall ---------|
<-------- ofs ------->
(
f:
Mach.function) (
c:
Mach.code) (
ofs:
ptrofs) :
Prop :=
forall tf tc,
transf_function f =
OK tf ->
transl_code f c false =
OK tc ->
code_tail (
Ptrofs.unsigned ofs) (
fn_code tf)
tc.
We now show that such an offset always exists if the Mach code c
is a suffix of f.(fn_code). This holds because the translation
from Mach to PPC is compositional: each Mach instruction becomes
zero, one or several PPC instructions, but the order of instructions
is preserved.
Lemma is_tail_code_tail:
forall c1 c2,
is_tail c1 c2 ->
exists ofs,
code_tail ofs c2 c1.
Proof.
induction 1. exists 0; constructor.
destruct IHis_tail as [ofs CT]. exists (ofs + 1); constructor; auto.
Qed.
Section RETADDR_EXISTS.
Hypothesis transl_instr_tail:
forall f i ep k c,
transl_instr f i ep k =
OK c ->
is_tail k c.
Hypothesis transf_function_inv:
forall f tf,
transf_function f =
OK tf ->
exists tc,
exists ep,
transl_code f (
Mach.fn_code f)
ep =
OK tc /\
is_tail tc (
fn_code tf).
Hypothesis transf_function_len:
forall f tf,
transf_function f =
OK tf ->
list_length_z (
fn_code tf) <=
Ptrofs.max_unsigned.
Lemma transl_code_tail:
forall f c1 c2,
is_tail c1 c2 ->
forall tc2 ep2,
transl_code f c2 ep2 =
OK tc2 ->
exists tc1,
exists ep1,
transl_code f c1 ep1 =
OK tc1 /\
is_tail tc1 tc2.
Proof.
induction 1;
simpl;
intros.
exists tc2;
exists ep2;
split;
auto with coqlib.
monadInv H0.
exploit IHis_tail;
eauto.
intros [
tc1 [
ep1 [
A B]]].
exists tc1;
exists ep1;
split.
auto.
apply is_tail_trans with x.
auto.
eapply transl_instr_tail;
eauto.
Qed.
Lemma return_address_exists:
forall f sg ros c,
is_tail (
Mcall sg ros ::
c)
f.(
Mach.fn_code) ->
exists ra,
return_address_offset f c ra.
Proof.
End RETADDR_EXISTS.
Remark code_tail_no_bigger:
forall pos c1 c2,
code_tail pos c1 c2 -> (
length c2 <=
length c1)%
nat.
Proof.
induction 1; simpl; lia.
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;
lia.
generalize (
code_tail_no_bigger _ _ _ H3);
simpl;
intro;
lia.
f_equal.
eauto.
Qed.
Lemma return_address_offset_correct:
forall ge b ofs fb f c tf tc ofs',
transl_code_at_pc ge (
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 lia.
constructor.
constructor.
rewrite list_length_z_cons.
generalize (
list_length_z_pos c).
lia.
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 lia.
constructor.
auto.
rewrite list_length_z_cons.
lia.
Qed.
Helper lemmas to reason about
-
the "code is tail of" property
-
correct translation of labels.
(
k c:
code) :
Prop :=
is_tail k c /\
forall lbl,
find_label lbl c =
find_label lbl k.
Lemma tail_nolabel_refl:
forall c,
tail_nolabel c c.
Proof.
Lemma tail_nolabel_trans:
forall c1 c2 c3,
tail_nolabel c2 c3 ->
tail_nolabel c1 c2 ->
tail_nolabel c1 c3.
Proof.
intros.
destruct H;
destruct H0;
split.
eapply is_tail_trans;
eauto.
intros.
rewrite H1;
auto.
Qed.
Definition nolabel (
i:
instruction) :=
match i with Plabel _ =>
False |
_ =>
True end.
Global Hint Extern 1 (
nolabel _) =>
exact I :
labels.
Lemma tail_nolabel_cons:
forall i c k,
nolabel i ->
tail_nolabel k c ->
tail_nolabel k (
i ::
c).
Proof.
intros. destruct H0. split.
constructor; auto.
intros. simpl. rewrite <- H1. destruct i; reflexivity || contradiction.
Qed.
Global Hint Resolve tail_nolabel_refl:
labels.
Ltac TailNoLabel :=
eauto with labels;
match goal with
| [ |-
tail_nolabel _ (
_ ::
_) ] =>
apply tail_nolabel_cons; [
auto;
exact I |
TailNoLabel]
| [
H:
Error _ =
OK _ |-
_ ] =>
discriminate
| [
H:
assertion_failed =
OK _ |-
_ ] =>
discriminate
| [
H:
OK _ =
OK _ |-
_ ] =>
inv H;
TailNoLabel
| [
H:
bind _ _ =
OK _ |-
_ ] =>
monadInv H;
TailNoLabel
| [
H: (
if ?
x then _ else _) =
OK _ |-
_ ] =>
destruct x;
TailNoLabel
| [
H:
match ?
x with nil =>
_ |
_ ::
_ =>
_ end =
OK _ |-
_ ] =>
destruct x;
TailNoLabel
|
_ =>
idtac
end.
Remark tail_nolabel_find_label:
forall lbl k c,
tail_nolabel k c ->
find_label lbl c =
find_label lbl k.
Proof.
intros. destruct H. auto.
Qed.
Remark tail_nolabel_is_tail:
forall k c,
tail_nolabel k c ->
is_tail k c.
Proof.
intros. destruct H. auto.
Qed.
Execution of straight-line code
Section STRAIGHTLINE.
Variable ge:
genv.
Variable fn:
function.
Straight-line code is composed of processor instructions that execute
in sequence (no branches, no function calls and returns).
The following inductive predicate relates the machine states
before and after executing a straight-line sequence of instructions.
Instructions are taken from the first list instead of being fetched
from memory.
Inductive exec_straight:
code ->
regset ->
mem ->
code ->
regset ->
mem ->
Prop :=
|
exec_straight_one:
forall i1 c rs1 m1 rs2 m2,
exec_instr ge fn i1 rs1 m1 =
Next rs2 m2 ->
rs2#
PC =
Val.offset_ptr rs1#
PC Ptrofs.one ->
exec_straight (
i1 ::
c)
rs1 m1 c rs2 m2
|
exec_straight_step:
forall i c rs1 m1 rs2 m2 c'
rs3 m3,
exec_instr ge fn i rs1 m1 =
Next rs2 m2 ->
rs2#
PC =
Val.offset_ptr rs1#
PC Ptrofs.one ->
exec_straight c rs2 m2 c'
rs3 m3 ->
exec_straight (
i ::
c)
rs1 m1 c'
rs3 m3.
Lemma exec_straight_trans:
forall c1 rs1 m1 c2 rs2 m2 c3 rs3 m3,
exec_straight c1 rs1 m1 c2 rs2 m2 ->
exec_straight c2 rs2 m2 c3 rs3 m3 ->
exec_straight c1 rs1 m1 c3 rs3 m3.
Proof.
Lemma exec_straight_two:
forall i1 i2 c rs1 m1 rs2 m2 rs3 m3,
exec_instr ge fn i1 rs1 m1 =
Next rs2 m2 ->
exec_instr ge fn i2 rs2 m2 =
Next rs3 m3 ->
rs2#
PC =
Val.offset_ptr rs1#
PC Ptrofs.one ->
rs3#
PC =
Val.offset_ptr rs2#
PC Ptrofs.one ->
exec_straight (
i1 ::
i2 ::
c)
rs1 m1 c rs3 m3.
Proof.
Lemma exec_straight_three:
forall i1 i2 i3 c rs1 m1 rs2 m2 rs3 m3 rs4 m4,
exec_instr ge fn i1 rs1 m1 =
Next rs2 m2 ->
exec_instr ge fn i2 rs2 m2 =
Next rs3 m3 ->
exec_instr ge fn i3 rs3 m3 =
Next rs4 m4 ->
rs2#
PC =
Val.offset_ptr rs1#
PC Ptrofs.one ->
rs3#
PC =
Val.offset_ptr rs2#
PC Ptrofs.one ->
rs4#
PC =
Val.offset_ptr rs3#
PC Ptrofs.one ->
exec_straight (
i1 ::
i2 ::
i3 ::
c)
rs1 m1 c rs4 m4.
Proof.
The following lemmas show that straight-line executions
(predicate exec_straight) correspond to correct Asm executions.
Lemma exec_straight_steps_1:
forall c rs m c'
rs'
m',
exec_straight c rs m c'
rs'
m' ->
list_length_z (
fn_code fn) <=
Ptrofs.max_unsigned ->
forall b ofs,
rs#
PC =
Vptr b ofs ->
Genv.find_funct_ptr ge b =
Some (
Internal fn) ->
code_tail (
Ptrofs.unsigned ofs) (
fn_code fn)
c ->
plus step ge (
State rs m)
E0 (
State rs'
m').
Proof.
Lemma exec_straight_steps_2:
forall c rs m c'
rs'
m',
exec_straight c rs m c'
rs'
m' ->
list_length_z (
fn_code fn) <=
Ptrofs.max_unsigned ->
forall b ofs,
rs#
PC =
Vptr b ofs ->
Genv.find_funct_ptr ge b =
Some (
Internal fn) ->
code_tail (
Ptrofs.unsigned ofs) (
fn_code fn)
c ->
exists ofs',
rs'#
PC =
Vptr b ofs'
/\
code_tail (
Ptrofs.unsigned ofs') (
fn_code fn)
c'.
Proof.
A variant that supports zero steps of execution
Inductive exec_straight_opt:
code ->
regset ->
mem ->
code ->
regset ->
mem ->
Prop :=
|
exec_straight_opt_refl:
forall c rs m,
exec_straight_opt c rs m c rs m
|
exec_straight_opt_intro:
forall c1 rs1 m1 c2 rs2 m2,
exec_straight c1 rs1 m1 c2 rs2 m2 ->
exec_straight_opt c1 rs1 m1 c2 rs2 m2.
Lemma exec_straight_opt_left:
forall c3 rs3 m3 c1 rs1 m1 c2 rs2 m2,
exec_straight c1 rs1 m1 c2 rs2 m2 ->
exec_straight_opt c2 rs2 m2 c3 rs3 m3 ->
exec_straight c1 rs1 m1 c3 rs3 m3.
Proof.
Lemma exec_straight_opt_right:
forall c3 rs3 m3 c1 rs1 m1 c2 rs2 m2,
exec_straight_opt c1 rs1 m1 c2 rs2 m2 ->
exec_straight c2 rs2 m2 c3 rs3 m3 ->
exec_straight c1 rs1 m1 c3 rs3 m3.
Proof.
Lemma exec_straight_opt_step:
forall i c rs1 m1 rs2 m2 c'
rs3 m3,
exec_instr ge fn i rs1 m1 =
Next rs2 m2 ->
rs2#
PC =
Val.offset_ptr rs1#
PC Ptrofs.one ->
exec_straight_opt c rs2 m2 c'
rs3 m3 ->
exec_straight (
i ::
c)
rs1 m1 c'
rs3 m3.
Proof.
Lemma exec_straight_opt_step_opt:
forall i c rs1 m1 rs2 m2 c'
rs3 m3,
exec_instr ge fn i rs1 m1 =
Next rs2 m2 ->
rs2#
PC =
Val.offset_ptr rs1#
PC Ptrofs.one ->
exec_straight_opt c rs2 m2 c'
rs3 m3 ->
exec_straight_opt (
i ::
c)
rs1 m1 c'
rs3 m3.
Proof.
End STRAIGHTLINE.
Properties of the Mach call stack
Section MATCH_STACK.
Variable ge:
Mach.genv.
Inductive match_stack:
list Mach.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 ge ra fb f c false tf tc ->
sp <>
Vundef ->
match_stack s ->
match_stack (
Stackframe fb sp ra c ::
s).
Lemma parent_sp_def:
forall s,
match_stack s ->
parent_sp s <>
Vundef.
Proof.
Lemma parent_ra_def:
forall s,
match_stack s ->
parent_ra s <>
Vundef.
Proof.
induction 1;
simpl.
unfold Vnullptr;
destruct Archi.ptr64;
congruence.
inv H0.
congruence.
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.
End MATCH_STACK.