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Glen Mével
iristimeproofs
Commits
e19aa532
Commit
e19aa532
authored
Nov 15, 2018
by
Glen Mével
Browse files
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Prove soundness of Combined.v
parent
21f2b119
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2
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2 changed files
with
551 additions
and
63 deletions
+551
63
theories/Combined.v
theories/Combined.v
+546
62
theories/union_find/Proof.v
theories/union_find/Proof.v
+5
1
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theories/Combined.v
View file @
e19aa532
(
*
TEMPORARY
:
*
This
file
is
far
from
being
complete
.
It
only
addresses
the
case
when
*
max_tc
<
max_tr
,
and
soundness
is
not
done
yet
.
Definitions
will
change
.
*
It
does
provide
a
(
hopefully
)
workable
interface
for
proving
programs
with
*
time
credits
and
time
receipts
,
including
proof

mode
tactics
.
*
)
From
iris_time
.
heap_lang
Require
Import
proofmode
notation
adequacy
lang
.
From
iris
.
base_logic
Require
Import
invariants
.
...
...
@@ 65,6 +58,8 @@ Class tctrHeapPreG Σ := {
tctrHeapPreG_inG_TRdup
:>
inG
Σ
(
authR
mnatUR
)
;
}
.
Class
UseTC
:=
useTC
:
bool
.
Class
tctrHeapG
Σ
:=
{
tctrHeapG_heapG
:>
heapG
Σ
;
tctrHeapG_inG_TC
:>
inG
Σ
(
authR
natUR
)
;
...
...
@@ 74,6 +69,7 @@ Class tctrHeapG Σ := {
tctrHeapG_name_TC
:
gname
;
tctrHeapG_name_TR
:
gname
;
tctrHeapG_name_TRdup
:
gname
;
tctrHeapG_useTC
:>
UseTC
;
}
.
Local
Notation
γ
:=
tctrHeapG_name_TC
.
...
...
@@ 89,39 +85,46 @@ Local Notation ℓ := tick_counter.
(
*
This
code
is
irrelevant
for
tick_spec
but
has
to
be
unsafe
for
proving
*
the
safety
theorem
:
*
)
Definition
fail
:
val
:=
λ
:
<>
,
#()
#().
Global
Instance
credits_tick
{
_
:
TickCounter
}
:
Tick
:=
generic_tick
fail
.
Definition
loop
:
val
:=
rec:
"f"
<>
:=
"f"
#().
Global
Instance
tctr_tick
{
_
:
TickCounter
}
{
_
:
UseTC
}
:
Tick
:=
generic_tick
(
if
useTC
then
fail
else
loop
).
Section
Definitions
.
Context
`
{
tctrHeapG
Σ
}
.
Context
(
max_tc
max_tr
:
nat
).
Context
{
Ineq
:
max_tc
<
max_tr
}
.
Definition
TC
(
n
:
nat
)
:
iProp
Σ
:=
own
γ
(
◯
nat
n
)
.
if
useTC
then
own
γ
(
◯
nat
n
)
else
True
%
I
.
Lemma
TC_plus
m
n
:
TC
(
m
+
n
)
≡
(
TC
m
∗
TC
n
)
%
I
.
Proof
using
.
by
rewrite
/
TC
auth_frag_op
own_op
.
Qed
.
Proof
using
.
rewrite
/
TC
;
destruct
useTC
.

by
rewrite
auth_frag_op
own_op
.

by
iStartProof
.
Qed
.
Lemma
TC_succ
n
:
TC
(
S
n
)
≡
(
TC
1
%
nat
∗
TC
n
)
%
I
.
Proof
using
.
by
rewrite
(
eq_refl
:
S
n
=
1
+
n
)
%
nat
TC_plus
.
Qed
.
Lemma
TC_weaken
(
n
₁
n
₂
:
nat
)
:
(
n
₂
≤
n
₁
)
%
nat
→
TC
n
₁

∗
TC
n
₂
.
Proof
using
.
apply
own_auth_nat_weaken
.
Qed
.
Proof
using
.
rewrite
/
TC
;
destruct
useTC
.

by
apply
own_auth_nat_weaken
.

done
.
Qed
.
Lemma
TC_timeless
n
:
Timeless
(
TC
n
).
Proof
using
.
exact
_.
Qed
.
Proof
using
.
rewrite
/
TC
;
destruct
useTC
;
exact
_.
Qed
.
(
*
We
give
higher
priorities
to
the
(
+
)
instances
so
that
the
(
S
n
)
instances
are
not
chosen
when
m
is
a
constant
.
*
)
...
...
@@ 156,7 +159,6 @@ Section Definitions.
Timeless
(
TR
n
).
Proof
using
.
exact
_.
Qed
.
(
*
We
give
higher
priorities
to
the
(
+
)
instances
so
that
the
(
S
n
)
instances
are
not
chosen
when
m
is
a
constant
.
*
)
Global
Instance
into_sep_TR_plus
m
n
:
IntoSep
(
TR
(
m
+
n
))
(
TR
m
)
(
TR
n
)

0.
...
...
@@ 190,20 +192,23 @@ Section Definitions.
Definition
tctrN
:=
nroot
.
@
"combinedTimeCreditTimeReceipt"
.
Context
(
max_tr
:
nat
).
Definition
TCTR_invariant
:
iProp
Σ
:=
inv
tctrN
(
∃
(
m
:
nat
),
⌜
(
m
≤
max_tc
)
%
nat
⌝
⌜
(
m
<
max_tr
)
%
nat
⌝
∗
ℓ
↦
#
m
∗
own
γ
(
●
nat
m
)
∗
own
γ
1
(
●
nat
(
max_t
c

m
))
∗
own
γ
2
(
●
mnat
(
max_t
c

m
))
∗
(
if
useTC
then
own
γ
(
●
nat
m
)
else
True
)
∗
own
γ
1
(
●
nat
(
max_t
r

1

m
))
∗
own
γ
2
(
●
mnat
(
max_t
r

1

m
))
)
%
I
.
Lemma
zero_TC
E
:
↑
tctrN
⊆
E
→
TCTR_invariant
={
E
}=
∗
TC
0.
Proof
using
.
rewrite
/
TC
/
TCTR_invariant
;
destruct
useTC
;
last
by
auto
.
iIntros
(
?
)
"#Htickinv"
.
iInv
tctrN
as
(
m
)
">(? & ? & H● & ?)"
"InvClose"
.
iDestruct
(
own_auth_nat_null
with
"H●"
)
as
"[H● $]"
.
...
...
@@ 215,7 +220,7 @@ Section Definitions.
TCTR_invariant
={
E
}=
∗
TR
0.
Proof
using
.
iIntros
(
?
)
"#Htickinv"
.
iInv
tctrN
as
(
m
)
"
>(? & ? & ? & Hγ1● &
?)"
"InvClose"
.
iInv
tctrN
as
(
m
)
"
(>? & >? & ? & >Hγ1● & >
?)"
"InvClose"
.
iDestruct
(
own_auth_nat_null
with
"Hγ1●"
)
as
"[Hγ1● $]"
.
iApply
"InvClose"
;
eauto
with
iFrame
.
Qed
.
...
...
@@ 225,7 +230,7 @@ Section Definitions.
TCTR_invariant
={
E
}=
∗
TRdup
0.
Proof
using
.
iIntros
(
?
)
"#Htickinv"
.
iInv
tctrN
as
(
m
)
"
>(? & ? & ? & ? &
Hγ2●)"
"InvClose"
.
iInv
tctrN
as
(
m
)
"
(>? & >? & ? & >? & >
Hγ2●)"
"InvClose"
.
iDestruct
(
own_auth_mnat_null
with
"Hγ2●"
)
as
"[Hγ2● $]"
.
iApply
"InvClose"
;
eauto
with
iFrame
.
Qed
.
...
...
@@ 233,16 +238,16 @@ Section Definitions.
Lemma
TR_nmax_absurd
(
E
:
coPset
)
:
↑
tctrN
⊆
E
→
TCTR_invariant

∗
TR
max_tr
={
E
}=
∗
False
.
Proof
using
Ineq
.
Proof
using
.
iIntros
(
?
)
"#Inv Hγ1◯"
.
iInv
tctrN
as
(
m
)
"
>(I & _ & _ &
Hγ1● & _)"
"InvClose"
;
iDestruct
"I"
as
%
I
.
iInv
tctrN
as
(
m
)
"
(>I & _ & _ & >
Hγ1● & _)"
"InvClose"
;
iDestruct
"I"
as
%
I
.
iDestruct
(
own_auth_nat_le
with
"Hγ1● Hγ1◯"
)
as
%
I
'
.
exfalso
;
lia
.
Qed
.
Lemma
TR_lt_nmax
n
(
E
:
coPset
)
:
↑
tctrN
⊆
E
→
TCTR_invariant

∗
TR
n
={
E
}=
∗
TR
n
∗
⌜
n
<
max_tr
⌝
%
nat
.
Proof
using
Ineq
.
Proof
using
.
iIntros
(
?
)
"#Inv Hγ1◯"
.
destruct
(
le_lt_dec
max_tr
n
)
as
[
I

I
]
;
last
by
iFrame
.
iDestruct
(
TR_weaken
n
max_tr
with
"Hγ1◯"
)
as
"Hγ1◯"
;
first
done
.
...
...
@@ 253,16 +258,16 @@ Section Definitions.
Lemma
TRdup_nmax_absurd
(
E
:
coPset
)
:
↑
tctrN
⊆
E
→
TCTR_invariant

∗
TRdup
max_tr
={
E
}=
∗
False
.
Proof
using
Ineq
.
Proof
using
.
iIntros
(
?
)
"#Inv Hγ2◯"
.
iInv
tctrN
as
(
m
)
"
>(I & _ & _ & _ &
Hγ2●)"
"InvClose"
;
iDestruct
"I"
as
%
I
.
iInv
tctrN
as
(
m
)
"
(>I & _ & _ & _ & >
Hγ2●)"
"InvClose"
;
iDestruct
"I"
as
%
I
.
iDestruct
(
own_auth_mnat_le
with
"Hγ2● Hγ2◯"
)
as
%
I
'
.
exfalso
;
lia
.
Qed
.
Lemma
TRdup_lt_nmax
n
(
E
:
coPset
)
:
↑
tctrN
⊆
E
→
TCTR_invariant

∗
TRdup
n
={
E
}=
∗
TRdup
n
∗
⌜
n
<
max_tr
⌝
%
nat
.
Proof
using
Ineq
.
Proof
using
.
iIntros
(
?
)
"#Inv Hγ1◯"
.
destruct
(
le_lt_dec
max_tr
n
)
as
[
I

I
]
;
last
by
iFrame
.
iDestruct
(
TRdup_weaken
n
max_tr
with
"Hγ1◯"
)
as
"Hγ1◯"
;
first
done
.
...
...
@@ 273,59 +278,77 @@ Section Definitions.
Lemma
TR_TRdup
(
E
:
coPset
)
n
:
↑
tctrN
⊆
E
→
TCTR_invariant

∗
TR
n
={
E
}=
∗
TR
n
∗
TRdup
n
.
Proof
using
max_tr
.
Proof
using
.
iIntros
(
?
)
"#Inv Hγ1◯"
.
iInv
tctrN
as
(
m
)
"
>(I & Hℓ & Hγ● & Hγ1● &
Hγ2●)"
"InvClose"
.
iInv
tctrN
as
(
m
)
"
(>I & >Hℓ & Hγ● & >Hγ1● & >
Hγ2●)"
"InvClose"
.
iDestruct
(
own_auth_nat_le
with
"Hγ1● Hγ1◯"
)
as
%
I
'
.
iDestruct
(
auth_mnat_update_read_auth
with
"Hγ2●"
)
as
">[Hγ2● Hγ2◯]"
.
iAssert
(
TR
n
∗
TRdup
n
)
%
I
with
"[$Hγ1◯ Hγ2◯]"
as
"$"
.
{
rewrite
(
_
:
(
max_t
c

m
)
%
nat
=
(
max_tc

m
)
%
nat
`max
`
n
)
;
last
lia
.
rewrite
(
_
:
(
max_t
r

1

m
)
%
nat
=
(
max_tr

1

m
)
%
nat
`max
`
n
)
;
last
lia
.
iDestruct
"Hγ2◯"
as
"[_ $]"
.
}
iApply
"InvClose"
.
auto
with
iFrame
.
Qed
.
Theorem
loop_spec
s
E
(
Φ
:
val
→
iProp
Σ
)
:
WP
loop
#()
@
s
;
E
{{
Φ
}}%
I
.
Proof
.
iL
ö
b
as
"IH"
.
wp_rec
.
iExact
"IH"
.
Qed
.
Theorem
tick_spec
s
E
(
v
:
val
)
n
:
↑
tctrN
⊆
E
→
TCTR_invariant

∗
{{{
▷
TC
1
∗
▷
TRdup
n
}}}
tick
v
@
s
;
E
{{{
RET
v
;
TR
1
∗
TRdup
(
n
+
1
)
}}}
.
Proof
using
max_tr
.
Proof
using
.
intros
?
.
iIntros
"#Inv"
(
Ψ
)
"!# [Hγ◯ Hγ2◯] HΨ"
.
iL
ö
b
as
"IH"
.
wp_lam
.
(
*
open
the
invariant
,
in
order
to
read
the
value
m
of
location
ℓ
:
*
)
wp_bind
(
!
_
)
%
E
;
iInv
tctrN
as
(
m
)
"
>(I & Hℓ & Hγ● & Hγ1● &
Hγ2●)"
"InvClose"
.
iInv
tctrN
as
(
m
)
"
(>I & >Hℓ & Hγ● & >Hγ1● & >
Hγ2●)"
"InvClose"
.
wp_load
.
(
*
deduce
that
m
≥
1
,
because
we
hold
a
time
credit
:
*
)
iDestruct
(
own_auth_nat_le
with
"Hγ● Hγ◯"
)
as
%
J
.
(
*
in
the
case
where
we
are
using
time
credits
,
deduce
that
m
≥
1
,
because
we
hold
a
time
credit
:
*
)
iAssert
(
⌜
useTC
=
true
⌝

∗
⌜
1
≤
m
⌝
)
%
I
%
nat
with
"[Hγ● Hγ◯]"
as
"#J"
.
{
rewrite
/
TC
;
iIntros
">"
.
by
iApply
(
own_auth_nat_le
with
"Hγ● Hγ◯"
).
}
(
*
close
the
invariant
:
*
)
iMod
(
"InvClose"
with
"[ I Hℓ Hγ● Hγ1● Hγ2● ]"
)
as
"_"
;
[
by
auto
with
iFrame

iModIntro
].
wp_let
.
(
*
test
whether
m
≤
0
:
*
)
wp_op
;
destruct
(
bool_decide
(
m
≤
0
))
eqn
:
J
'
;
wp_if
.
(
*
—
if
m
≤
0
then
this
is
absurd
,
because
we
hold
a
time
credit
:
*
)

apply
Is_true_eq_left
,
bool_decide_spec
in
J
'
.
wp_op
.
case_bool_decide
as
J
'
;
wp_if
.
(
*
—
if
m
≤
0
:
*
)

destruct
useTC
.
(
*
if
we
use
TC
,
then
m
≤
0
is
absurd
,
because
we
hold
a
time
credit
:
*
)
+
iDestruct
(
"J"
with
"[//]"
)
as
%
J
.
exfalso
;
lia
.
(
*
if
we
don
’
t
,
then
we
loop
indefinitely
,
which
gives
us
any
spec
we
want
(
because
we
are
reasoning
in
partial
correctness
)
:
*
)
+
iApply
loop_spec
.
(
*
—
otherwise
:
*
)

clear
J
'
.
wp_op
.

wp_op
.
(
*
open
the
invariant
again
,
in
order
to
perform
CAS
on
location
ℓ
:
*
)
wp_bind
(
CAS
_
_
_
)
%
E
;
(
*
iInv
timeCreditN
as
(
n
'
)
">(Hℓ & Hγ●)"
"InvClose"
.
*
)
iInv
tctrN
as
(
m
'
)
">(I & Hℓ & Hγ● & Hγ1● & Hγ2●)"
"InvClose"
.
(
*
test
whether
the
value
ℓ
is
still
k
,
by
comparing
m
with
m
'
:
*
)
iInv
tctrN
as
(
m
'
)
"(>I & >Hℓ & Hγ● & >Hγ1● & >Hγ2●)"
"InvClose"
.
(
*
test
whether
the
value
ℓ
is
still
m
,
by
comparing
m
with
m
'
:
*
)
destruct
(
nat_eq_dec
m
m
'
)
as
[
<

Hneq
].
(
*
—
if
it
holds
,
then
CAS
succeeds
and
ℓ
is
decremented
:
*
)
+
wp_cas_suc
.
(
*
reform
the
invariant
with
m
−
1
instead
of
m
:
*
)
replace
(
Z
.
of_nat
m

1
)
with
(
Z
.
of_nat
(
m

1
)
%
nat
)
by
lia
.
iMod
(
auth_nat_update_decr
_
_
_
1
with
"Hγ● Hγ◯"
)
as
"[Hγ● Hγ◯]"
;
first
done
.
iAssert
(
==>
if
useTC
then
own
γ
(
●
nat
(
m

1
))
else
True
)
%
I
with
"[Hγ● Hγ◯]"
as
">Hγ●"
.
{
rewrite
/
TC
;
destruct
useTC
;
last
done
.
by
iMod
(
auth_nat_update_decr
_
_
_
1
with
"Hγ● Hγ◯"
)
as
"[$ _]"
.
}
iMod
(
auth_nat_update_incr
_
_
1
with
"Hγ1●"
)
as
"[Hγ1● Hγ1◯]"
;
simpl
.
iMod
(
auth_mnat_update_incr
'
_
_
_
1
with
"Hγ2● Hγ2◯"
)
as
"[Hγ2● Hγ2◯]"
;
simpl
.
iDestruct
"I"
as
%
I
.
replace
(
max_t
c

m
+
1
)
%
nat
with
(
max_tc

(
m

1
))
%
nat
by
lia
.
assert
((
m

1
)
≤
max_tc
)
%
nat
by
lia
.
replace
(
max_t
r

1

m
+
1
)
%
nat
with
(
max_tr

1

(
m

1
))
%
nat
by
lia
.
assert
((
m

1
)
<
max_tr
)
%
nat
by
lia
.
(
*
close
the
invariant
:
*
)
iMod
(
"InvClose"
with
"[ Hℓ Hγ● Hγ1● Hγ2● ]"
)
as
"_"
;
[
by
auto
with
iFrame

iModIntro
].
(
*
finish
:
*
)
...
...
@@ 342,13 +365,13 @@ Section Definitions.
Theorem
tick_spec_simple
v
n
:
TCTR_invariant

∗
{{{
TC
1
∗
TRdup
n
}}}
tick
v
{{{
RET
v
;
TR
1
∗
TRdup
(
n
+
1
)
}}}
.
Proof
using
max_tr
.
Proof
using
.
iIntros
"#Hinv"
(
Ψ
)
"!# [ HTC HTR ] HΨ"
.
by
iApply
(
tick_spec
with
"Hinv [$HTC $HTR] HΨ"
).
Qed
.
Lemma
TC_implementation
:
TCTR_invariant

∗
TCTR_interface
TC
max_tr
TR
TRdup
.
Proof
using
Ineq
.
Proof
using
.
iIntros
"#Hinv"
.
repeat
iSplitR
.

iPureIntro
.
by
apply
TC_timeless
.

by
iApply
(
zero_TC
with
"Hinv"
).
...
...
@@ 398,7 +421,7 @@ Proof.
iDestruct
(
envs_lookup_sound
_
i
false
with
"HΔ"
)
as
"[? _]"
=>
//. }
iDestruct
(
into_laterN_env_sound
with
"HΔ"
)
as
"HΔ1"
=>
//.
iDestruct
(
envs_simple_replace_singleton_sound
with
"HΔ1"
)
as
"[HTC HΔ2]"
=>
//=.
iDestruct
"HTC"
as
"[
>
HTC HTC']"
.
iDestruct
(
"HΔ2"
with
"HTC'"
)
as
"HΔ2"
.
iDestruct
"HTC"
as
"[HTC HTC']"
.
iDestruct
(
"HΔ2"
with
"HTC'"
)
as
"HΔ2"
.
iApply
wp_bind
.
destruct
HTR
as
[(
j
&
n
'
&
HTR1
&
HTR2
)

>
],
HTRdup
as
[(
j
'
&
pe
''
&
p
&
HTRDup1
&
HTRDup2
)

>
].
...
...
@@ 453,13 +476,474 @@ Ltac wp_tick ::=

_
=>
fail
"wp_tick: not a 'wp'"
end
.
Example
test
`
{
tctrHeapG
Σ
}
:
TCTR_invariant
100

∗
TC
37

∗
TRdup
63
%
nat

∗
TR
0

∗
WP
tick
#
0
{{
_
,
True
}}
.
Proof
using
.
iIntros
"#Inv HTC HTRdup HTR"
.
wp_tick
.
Abort
.
(
*
*
Simulation
*
)
Section
Simulation
.
(
*
Simulation
in
the
“
successful
”
case
.
*
)
Context
{
Hloc
:
TickCounter
}
.
Context
{
useTC
:
UseTC
}
.
Lemma
exec_tick_success
n
v
σ
:
prim_exec
(
tick
v
)
(
<
[
ℓ
:=
#(
S
n
)]
>
σ
)
v
(
<
[
ℓ
:=
#
n
]
>
σ
)
[].
Proof
.
by
apply
exec_tick_success
.
Qed
.
Definition
simulation_exec_success
:=
simulation_exec_success
(
if
useTC
then
fail
else
loop
).
Definition
simulation_exec_success
'
:=
simulation_exec_success
'
(
if
useTC
then
fail
else
loop
).
(
*
Simulation
in
the
“
failing
”
case
.
*
)
Context
(
HuseTC
:
useTC
=
true
).
Lemma
not_safe_tick
v
σ
:
¬
safe
(
tick
v
)
(
<
[
ℓ
:=
#
0
]
>
σ
).
Proof
.
assert
(
prim_exec
(
tick
v
)
(
<
[
ℓ
:=
#
0
]
>
σ
)
(#()
#())
(
<
[
ℓ
:=
#
0
]
>
σ
)
[])
as
Hexec
.
{
unlock
tick
tctr_tick
generic_tick
fail
;
rewrite
HuseTC
.
eapply
prim_exec_cons_nofork
.
{
by
prim_step
.
}
simpl
.
eapply
prim_exec_cons_nofork
.
{
prim_step
.
apply
lookup_insert
.
}
simpl
.
eapply
prim_exec_cons_nofork
.
{
prim_step
.
}
simpl
.
eapply
prim_exec_cons_nofork
.
{
by
prim_step
.
}
simpl
.
eapply
prim_exec_cons_nofork
.
{
by
prim_step
.
}
simpl
.
eapply
prim_exec_cons_nofork
.
{
by
prim_step
.
}
simpl
.
eapply
prim_exec_cons_nofork
.
{
by
prim_step
.
}
simpl
.
apply
prim_exec_nil
.
}
assert
(
¬
safe
(#()
#())
(
<
[
ℓ
:=
#
0
]
>
σ
))
as
Hnotsafe
.
{
apply
stuck_not_safe
,
head_stuck_stuck
,
ectxi_language_sub_redexes_are_values
.

split
;
first
done
.
inversion
1.

intros
[]
;
try
discriminate
1
;
inversion
1
;
by
eauto
.
}
intros
Hsafe
.
eapply
Hnotsafe
,
safe_exec_nofork
,
prim_exec_exec
;
eassumption
.
Qed
.
Local
Ltac
not_safe_tick
:=
lazymatch
goal
with


¬
safe
?
e
_
=>
reshape_expr
false
e
ltac
:
(
fun
K
e
'
=>
apply
not_safe_fill
'
with
K
e
'
;
first
reflexivity
;
eapply
not_safe_tick
)
end
;
done
.
Lemma
simulation_head_step_failure
e1
σ
1
κ
e2
σ
2
efs
:
head_step
e1
σ
1
κ
e2
σ
2
efs
→
¬
safe
«
e1
»
S
«σ
1
,
0
»
.
Proof
.
destruct
1
as
[(
*
RecS
*
)
f
x
e
σ













(
*
ForkS
*
)
e1
σ






]
;
simpl_trans
;
try
not_safe_tick
.
(
*
RecS
f
x
e
σ
:
*
)

eapply
not_safe_prim_step
;
last
prim_step
.
not_safe_tick
.
(
*
ForkS
e
σ
:
*
)

eapply
not_safe_prim_step
;
last
prim_step
.
not_safe_tick
.
Qed
.
Lemma
simulation_prim_step_failure
e1
σ
1
κ
e2
σ
2
efs
:
prim_step
e1
σ
1
κ
e2
σ
2
efs
→
¬
safe
«
e1
»
S
«σ
1
,
0
»
.
Proof
.
intros
[
K
e1
'
e2
'
>
>
H
].
rewrite
translation_fill
.
by
eapply
not_safe_fill
,
simulation_head_step_failure
.
Qed
.
Lemma
simulation_step_failure
t1
σ
1
κ
t2
σ
2
:
step
(
t1
,
σ
1
)
κ
(
t2
,
σ
2
)
→
∃
e1
,
e1
∈
t1
∧
¬
safe
«
e1
»
S
«σ
1
,
0
»
.
Proof
.
intros
[
e1
σ
1_
e2
σ
2_
efs
t
t
'
E1
E2
Hstep
]
;
injection
E1
as
>
<
;
injection
E2
as
>
<
.
exists
e1
;
split
;
first
set_solver
.
by
eapply
simulation_prim_step_failure
.
Qed
.
Local
Lemma
simulation_exec_failure_now
n
t1
σ
1
t2
σ
2
:
nsteps
erased_step
(
S
n
)
(
t1
,
σ
1
)
(
t2
,
σ
2
)
→
∃
e1
,
e1
∈
t1
∧
¬
safe
«
e1
»
S
«σ
1
,
0
»
.
Proof
.
make_eq
(
S
n
)
as
Sn
En
.
make_eq
(
t1
,
σ
1
)
as
config1
E1
.
destruct
1
as
[
?

?
?
[]
?
[
?
?
]
_
],
E1
.

discriminate
En
.

by
eapply
simulation_step_failure
.
Qed
.
Lemma
simulation_exec_failure
m
n
e1
σ
1
t2
σ
2
:
σ
2
!!
ℓ
=
None
→
nsteps
erased_step
(
m
+
S
n
)
([
e1
],
σ
1
)
(
t2
,
σ
2
)
→
¬
safe
«
e1
»
S
«σ
1
,
m
»
.
Proof
.
intros
H
ℓ
Hnsteps
.
assert
(
∃
t3
σ
3
,
rtc
erased_step
(
T
«
[
e1
]
»
,
S
«σ
1
,
m
»
)
(
T
«
t3
»
,
S
«σ
3
,
0
»
)
∧
∃
e3
,
e3
∈
t3
∧
¬
safe
«
e3
»
S
«σ
3
,
0
»
)
as
(
t3
&
σ
3
&
Hsteps
&
e3
&
E3
&
Hsafe
).
{
apply
nsteps_split
in
Hnsteps
as
((
t3
,
σ
3
)
&
Hnsteps1to3
&
Hnsteps3to2
).
exists
t3
,
σ
3
;
repeat
split
.

assert
(
σ
3
!!
ℓ
=
None
)
by
(
eapply
loc_fresh_in_dom_nsteps
;
cycle
1
;
eassumption
).
replace
m
with
(
m
+
0
)
%
nat
by
lia
.
apply
simulation_exec_success
;
assumption
.

by
eapply
simulation_exec_failure_now
.
}
apply
(
elem_of_list_fmap_1
translation
)
in
E3
.
eapply
not_safe_exec
;
eassumption
.
Qed
.
End
Simulation
.
(
*
*
Soundness
*
)
Section
Soundness
.
(
**
Part
of
the
proof
of
soundness
which
is
specific
to
the
case
where
**
time
credits
are
used
(
m
<
max_tr
).
**
)
(
*
assuming
the
safety
of
the
translated
expression
,
*
a
proof
that
the
computation
time
of
the
original
expression
is
bounded
.
*
)
Local
Lemma
gt_sum
m
n
:
(
m
>
n
)
%
nat
→
∃
(
k
:
nat
),
m
=
(
n
+
S
k
)
%
nat
.
Proof
.
intro
.
exists
(
m

S
n
)
%
nat
.
lia
.
Qed
.
Lemma
safe_tctranslation__bounded
{
Hloc
:
TickCounter
}
{
useTC
:
UseTC
}
m
e
σ
t2
σ
2
n
:
useTC
=
true
→
σ
2
!!
ℓ
=
None
→
safe
«
e
»
S
«σ
,
m
»
→
nsteps
erased_step
n
([
e
],
σ
)
(
t2
,
σ
2
)
→
(
n
≤
m
)
%
nat
.
Proof
.
intros
HuseTC
Hfresh
Hsafe
Hnsteps
.
(
*
reasoning
by
contradiction
:
assume
(
n
>
m
),
ie
.
(
n
=
m
+
S
k
)
for
some
k
:
*
)
apply
not_gt
;
intros
[
k
>
]
%
gt_sum
.
(
*
apply
the
simulation
lemma
:
*
)
by
eapply
simulation_exec_failure
.
Qed
.
(
*
assuming
the
safety
of
the
translated
expression
,
*
a
proof
that
the
original
expression
is
safe
.
*
)
Lemma
safe_tctranslation__safe_here
{
Hloc
:
TickCounter
}
{
useTC
:
UseTC
}
m
e
σ
:
useTC
=
true
→
loc_fresh_in_expr
ℓ
e
→
safe
«
e
»
S
«σ
,
m
»
→
is_Some
(
to_val
e
)
∨
reducible
e
σ
.
Proof
.
intros
HuseTC
Hfresh
Hsafe
.
(
*
case
analysis
on
whether
e
is
a
value
…
*
)
destruct
(
to_val
e
)
as
[
v

]
eqn
:
Hnotval
.
(
*
—
if
e
is
a
value
,
then
we
get
the
result
immediately
:
*
)

left
.
eauto
.
(
*
—
if
e
is
not
a
value
,
then
we
show
that
it
is
reducible
:
*
)

right
.
(
*
we
decompose
e
into
a
maximal
evaluation
context
K
and
a
head

redex
:
*
)
pose
proof
(
not_val_fill_active_item
_
Hnotval
)
as
He
;
clear
Hnotval
.
destruct
He
as
[
(
K
&
x
&
>
)

[
(
K
&
e1
&
>
)

[
(
K
&
f
&
x
&
e1
&
>
)

(
K
&
ki
&
v
&
>
&
Hactive
)
]]].
(
*
—
either
e
=
Var
x
:
*
)
+
(
*
then
Var
x
is
stuck
*
)
exfalso
.
eapply
stuck_not_safe
;
[

done
].
rewrite
translation_fill
.
apply
stuck_fill
,
head_stuck_stuck
.
{
split
;
[
done

].
intros
????
Hstep
.
inversion
Hstep
.
}
apply
ectxi_language_sub_redexes_are_values
=>
[]
//.
(
*
—
either
e
=
K
[
Fork
e1
]
:
*
)
+
(
*
then
we
easily
derive
a
reduction
from
e
:
*
)
eexists
_
,
_
,
_
,
_.
apply
Ectx_step
'
,
ForkS
.
(
*
—
either
e
=
K
[
Rec
f
x
e1
]
:
*
)
+
(
*
then
we
easily
derive
a
reduction
from
e
:
*
)
eexists
_
,
_
,
_
,
_.
apply
Ectx_step
'
,
RecS
.
(
*
—
or
e
=
K
[
ki
[
v
]]
where
ki
is
an
active
item
:
*
)
+
(
*
it
is
enough
to
show
that
ki
[
v
]
is
reducible
:
*
)
apply
loc_fresh_in_expr_fill_inv
in
Hfresh
;
rewrite
>
translation_fill
in
Hsafe
;
apply
safe_fill_inv
in
Hsafe
;
apply
reducible_fill
;
clear
K
.
(
*
we
deduce
the
reducibility
of
ki
[
v
]
from
that
of
«
ki
»
[
«
v
»
]
:
*
)
eapply
active_item_translation_reducible
;
[
done

done

].
(
*
remind
that
«
ki
[
v
]
»
=
«
ki
»
[
tick
«
v
»
]
:
*
)
rewrite
>
translation_fill_item_active
in
Hsafe
;
last
done
.
(
*
we
have
that
«
ki
»
[
tick
«
v
»
]
reduces
to
«
ki
»
[
«
v
»
]
*
(
thanks
to
the
safety
hypothesis
,
m
≥
1
and
‘
tick
’
can
be
run
)
:
*
)
assert
(
prim_exec
(
fill_item
Ki
«
ki
»
(
tick
V
«
v
»
))
S
«σ
,
m
»
(
fill_item
Ki
«
ki
»
V
«
v
»
)
S
«σ
,
m

1
»
[]
)
as
Hsteps
%
prim_exec_exec
.
{
assert
(
fill
[
Ki
«
ki
»
]
=
fill_item
Ki
«
ki
»
)
as
E
by
reflexivity
;
destruct
E
.
apply
prim_exec_fill
.
apply
safe_fill_inv
in
Hsafe
.
(
*
This
is
where
the
unsafe
behavior
of
“
tick
”
is
used
:
*
)
destruct
m
as
[

m
]
;
first
(
exfalso
;
eapply
not_safe_tick
;
eassumption
).
replace
(
S
m

1
)
%
nat
with
m
by
lia
.
apply
exec_tick_success
.
}
(
*
using
the
safety
of
«
ki
»
[
tick
«
v
»
],
we
proceed
by
case
analysis
…
*
)
eapply
Hsafe
in
Hsteps
as
[
Hisval

Hred
]
;
auto
using
elem_of_list_here
.
(
*
—
either
«
ki
»
[
«
v
»
]
is
a
value
:
this
is
not
possible
because
ki
is
active
.
*
)
*
simpl
in
Hisval
.
rewrite
active_item_not_val
in
Hisval
;
[
by
apply
is_Some_None
in
Hisval

by
apply
is_active_translationKi
].
(
*
—
or
«
ki
»
[
«
v
»
]
reduces
to
something
:
this
is
precisely
what
we
need
.
*
)
*
exact
Hred
.
Qed
.
Lemma
safe_tctranslation__safe
{
Hloc
:
TickCounter
}
{
useTC
:
UseTC
}
m
e
σ
t2
σ
2
e2
:
useTC
=
true
→
loc_fresh_in_expr
ℓ
e2
→
σ
2
!!
ℓ
=
None
→
safe
«
e
»
S
«σ
,
m
»
→
rtc
erased_step
([
e
],
σ
)
(
t2
,
σ
2
)
→
e2
∈
t2
→
is_Some
(
to_val
e2
)
∨
reducible
e2
σ
2.
Proof
.
intros
HuseTC
H
ℓ
e
H
ℓσ
Hsafe
[
n
Hnsteps
]
%
rtc_nsteps
He2
.
assert
(
n
≤
m
)
%
nat
by
by
eapply
safe_tctranslation__bounded
.
assert
(
safe
«
e2
»
S
«σ
2
,
m

n
»
)
as
Hsafe2
.
{
eapply
safe_exec
.

eapply
elem_of_list_fmap_1
.
eassumption
.

eassumption
.

change
[
«
e
»
]
with
T
«
[
e
]
»
.
apply
simulation_exec_success
'
;
auto
.
}
by
eapply
safe_tctranslation__safe_here
.
Qed
.
(
*
assuming
the
adequacy
of
the
translated
expression
,
*
a
proof
that
the
original
expression
has
adequate
results
.
*
)
Lemma
adequate_tctranslation__adequate_result
{
Hloc
:
TickCounter
}
{
useTC
:
UseTC
}
m
φ
e
σ
t2
σ
2
v2
:
useTC
=
true
→
σ
2
!!
ℓ
=
None
→
adequate
NotStuck
«
e
»
S
«σ
,
m
»
(
λ
v
σ
,
φ
(
invtranslationV
v
))
→
rtc
erased_step
([
e
],
σ
)
(
Val
v2
::
t2
,
σ
2
)
→
φ
v2
.
Proof
.
intros
HuseTC
Hfresh
Hadq
[
n
Hnsteps
]
%
rtc_nsteps
.
assert
(
safe
«
e
»
S
«σ
,
m
»
)
as
Hsafe
by
by
eapply
safe_adequate
.
assert
(
n
≤
m
)
%
nat
by
by
eapply
safe_tctranslation__bounded
.
replace
(
φ
v2
)
with
((
φ
∘
invtranslationV
)
(
translationV
v2
))
by
(
simpl
;
by
rewrite
invtranslationV_translationV
).
eapply
(
adequate_result
_
_
_
(
λ
v
σ
,
φ
(
invtranslationV
v
)))
;
first
done
.
change
[
«
e
»
%
E
]
with
T
«
[
e
]
»
.
replace
(
Val
«
v2
»
::
_
)
with
(
T
«
Val
v2
::
t2
»
)
by
done
.
eapply
simulation_exec_success
'
;
eauto
.
Qed
.
(
*
now
let
’
s
combine
the
three
results
.
*
)
Lemma
adequate_tctranslation__adequate_and_bounded
m
φ
e
σ
:
let
useTC
:
UseTC
:=
true
in
(
∀
`
{
TickCounter
}
,
adequate
NotStuck
«
e
»
S
«σ
,
m
»
(
λ
v
σ
,
φ
(
invtranslationV
v
)))
→
adequate
NotStuck
e
σ
(
λ
v
σ
,
φ
v
)
∧
bounded_time
e
σ
m
.
Proof
.
intros
useTC
Hadq
.
split
;
first
split
.
(
*
(
1
)
adequate
result
:
*
)

intros
t2
σ
2
v2
Hsteps
.
(
*
build
a
location
ℓ
which
is
not
in
the
domain
of
σ
2
:
*
)
pose
(
Build_TickCounter
(
fresh
(
dom
(
gset
loc
)
σ
2
)))
as
Hloc
.
assert
(
σ
2
!!
ℓ
=
None
)
by
(
simpl
;
eapply
not_elem_of_dom
,
is_fresh
).
by
eapply
adequate_tctranslation__adequate_result
.
(
*
(
2
)
safety
:
*
)

intros
t2
σ
2
e2
_
Hsteps
He2
.
(
*
build
a
location
ℓ
which
is
fresh
in
e2
and
in
the
domain
of
σ
2
:
*
)
pose
(
set1
:=
loc_set_of_expr
e2
:
gset
loc
).
pose
(
set2
:=
dom
(
gset
loc
)
σ
2
:
gset
loc
).
pose
(
Build_TickCounter
(
fresh
(
set1
∪
set2
)))
as
Hloc
.
eassert
(
ℓ
∉
set1
∪
set2
)
as
[
H
ℓ
1
H
ℓ
2
]
%
not_elem_of_union
by
(
unfold
ℓ
;
apply
is_fresh
).
assert
(
loc_fresh_in_expr
ℓ
e2
)
by
by
apply
loc_not_in_set_is_fresh_in_expr
.
assert
(
σ
2
!!
ℓ
=
None
)
by
by
(
simpl
;
eapply
not_elem_of_dom
).
specialize
(
Hadq
Hloc
)
as
Hsafe
%
safe_adequate
.
by
eapply
safe_tctranslation__safe
.
(
*
(
3
)
bounded
time
:
*
)

intros
t2
σ
2
k
.
(
*
build
a
location
ℓ
which
is
not
in
the
domain
of
σ
2
:
*
)
pose
(
Build_TickCounter
(
fresh
(
dom
(
gset
loc
)
σ
2
)))
as
Hloc
.
assert
(
σ
2
!!
ℓ
=
None
)
by
(
unfold
ℓ
;
eapply
not_elem_of_dom
,
is_fresh
).
specialize
(
Hadq
Hloc
)
as
Hsafe
%
safe_adequate
.
by
eapply
safe_tctranslation__bounded
.
Qed
.
(
**
Part
of
the
proof
of
soundness
which
is
specific
to
the
case
where
**
time
credits
are
not
used
(
m
≥
max_tr
).
**
)
(
*
assuming
the
adequacy
of
the
translated
expression
,
*
a
proof
that
the
original
expression
is
n

adequate
.
*
)
Definition
adequate_trtranslation__nadequate
:=
adequate_translation__nadequate
loop
.
(
**
Part
of
the
proof
which
is
common
to
both
cases
.
**
)
(
*
from
a
Hoare
triple
in
Iris
,
*
derive
the
adequacy
of
the
translated
expression
.
*
)
Lemma
spec_tctranslation__adequate_translation
{
Σ
}
max_tr
m
ψ
e
:
(
0
<
max_tr
)
%
nat
→
(
∀
`
{
tctrHeapG
Σ
}
,
TCTR_invariant
max_tr

∗
{{{
TC
m
}}}
«
e
»
{{{
v
,
RET
v
;
⌜ψ
v
⌝
}}}
)
→
∀
`
{
tctrHeapPreG
Σ
}
`
{
TickCounter
}
σ
,
let
useTC
:
UseTC
:=
bool_decide
(
m
<
max_tr
)
%
nat
in
let
M
:
nat
:=
min
m
(
max_tr

1
)
in
adequate
NotStuck
«
e
»
S
«σ
,
M
»
(
λ
v
σ
,
ψ
v
).
Proof
.
intros
Imax_tr
Hspec
HpreG
Hloc
σ
useTC
M
.
assert
(
M
≤
m
)
%
nat
by
(
unfold
M
;
lia
).
assert
(
M
<
max_tr
)
%
nat
by
(
unfold
M
;
lia
).
(
*
apply
the
adequacy
results
.
*
)
apply
(
wp_adequacy
_
_
)
;
simpl
;
intros
HinvG
.
(
*
…
now
we
have
to
prove
a
WP
.
*
)
set
σ'
:=
S
«σ»
.
(
*
allocate
the
heap
,
including
cell
ℓ
(
on
which
we
need
to
keep
an
eye
)
:
*
)
iMod
(
own_alloc
(
●
to_gen_heap
(
<
[
ℓ
:=
#
M
]
>
σ'
)
⋅
◯
to_gen_heap
{
[
ℓ
:=
#
M
]
}
))
as
(
h
)
"[Hh● Hℓ◯]"
.
{
apply
auth_valid_discrete_2
;
split
.

rewrite

insert_delete
;
set
σ''
:=
delete
ℓ
σ'
.
unfold
to_gen_heap
;
rewrite
2
!
fmap_insert
fmap_empty
insert_empty
.
exists
(
to_gen_heap
σ''
).
rewrite
(
@
gmap
.
insert_singleton_op
_
_
_
_
(
to_gen_heap
σ''
))
//.
rewrite
lookup_fmap
;
apply
fmap_None
,
lookup_delete
.

apply
to_gen_heap_valid
.
}
(
*
allocate
the
ghost
state
associated
with
ℓ
:
*
)
iAssert
(
==>
∃
γ
,
(
if
useTC
then
own
γ
(
●
M
)
else
True
)
∗
(
if
useTC
then
own
γ
(
◯
m
)
else
True
)
)
%
I
as
"Hγalloc"
.
{
rewrite
/
useTC
;
case_bool_decide
;
last
done
.
rewrite
(
_
:
M
=
m
)
;
last
by
(
rewrite
/
M
;
lia
).
iMod
(
auth_nat_alloc
m
)
as
(
γ
)
"[Hγ● Hγ◯]"
.
auto
with
iFrame
.
}
iMod
"Hγalloc"
as
(
γ
)
"[Hγ● Hγ◯]"
.