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d6b647ad
Commit
d6b647ad
authored
Jan 05, 2017
by
Robbert Krebbers
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Refactor proofs for invariant opening.
parent
66f8aa36
Pipeline
#3588
passed with stage
in 10 minutes and 40 seconds
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2 changed files
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25 additions
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31 deletions
+25
31
theories/base_logic/lib/invariants.v
theories/base_logic/lib/invariants.v
+25
30
theories/base_logic/lib/wsat.v
theories/base_logic/lib/wsat.v
+0
1
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theories/base_logic/lib/invariants.v
View file @
d6b647ad
...
...
@@ 28,44 +28,39 @@ Qed.
Global
Instance
inv_persistent
N
P
:
PersistentP
(
inv
N
P
).
Proof
.
rewrite
inv_eq
/
inv
;
apply
_
.
Qed
.
Lemma
fresh_inv_name
(
E
:
gset
positive
)
N
:
∃
i
,
i
∉
E
∧
i
∈
↑
N
.
Proof
.
exists
(
coPpick
(
↑
N
∖
coPset
.
of_gset
E
)).
rewrite

coPset
.
elem_of_of_gset
(
comm
and
)

elem_of_difference
.
apply
coPpick_elem_of
=>
Hfin
.
eapply
nclose_infinite
,
(
difference_finite_inv
_
_
),
Hfin
.
apply
of_gset_finite
.
Qed
.
Lemma
inv_alloc
N
E
P
:
▷
P
={
E
}=
∗
inv
N
P
.
Proof
.
rewrite
inv_eq
/
inv_def
fupd_eq
/
fupd_def
.
iIntros
"HP [Hw $]"
.
iMod
(
ownI_alloc
(
∈
↑
N
)
P
with
"[HP Hw]"
)
as
(
i
)
"(% & $ & ?)"
;
auto
.

intros
Ef
.
exists
(
coPpick
(
↑
N
∖
coPset
.
of_gset
Ef
)).
rewrite

coPset
.
elem_of_of_gset
comm

elem_of_difference
.
apply
coPpick_elem_of
=>
Hfin
.
eapply
nclose_infinite
,
(
difference_finite_inv
_
_
),
Hfin
.
apply
of_gset_finite
.

by
iFrame
.

rewrite
/
uPred_except_0
;
eauto
.
iMod
(
ownI_alloc
(
∈
↑
N
)
P
with
"[$HP $Hw]"
)
as
(
i
)
"(% & $ & ?)"
;
auto
using
fresh_inv_name
.
Qed
.
Lemma
inv_alloc_open
N
E
P
:
↑
N
⊆
E
→
True
={
E
,
E
∖↑
N
}=
∗
inv
N
P
∗
(
▷
P
={
E
∖↑
N
,
E
}=
∗
True
).
Proof
.
rewrite
inv_eq
/
inv_def
fupd_eq
/
fupd_def
.
iIntros
(
Sub
)
"[Hw HE]"
.
iMod
(
ownI_alloc_open
(
∈
↑
N
)
P
with
"Hw"
)
as
(
i
)
"(% & Hw & #Hi & HD)"
.

intros
Ef
.
exists
(
coPpick
(
↑
N
∖
coPset
.
of_gset
Ef
)).
rewrite

coPset
.
elem_of_of_gset
comm

elem_of_difference
.
apply
coPpick_elem_of
=>
Hfin
.
eapply
nclose_infinite
,
(
difference_finite_inv
_
_
),
Hfin
.
apply
of_gset_finite
.

iAssert
(
ownE
{[
i
]}
∗
ownE
(
↑
N
∖
{[
i
]})
∗
ownE
(
E
∖
↑
N
))%
I
with
"[HE]"
as
"(HEi & HEN\i & HE\N)"
.
{
rewrite

?ownE_op
;
[
set_solver

set_solver
].
rewrite
assoc_L
.
rewrite
<!
union_difference_L
;
try
done
;
set_solver
.
}
iModIntro
.
rewrite
/
uPred_except_0
.
iRight
.
iFrame
.
iSplitL
"Hw HEi"
.
+
by
iApply
"Hw"
.
+
iSplitL
"Hi"
;
[
eauto
].
iIntros
"HP [Hw HE\N]"
.
iDestruct
(
ownI_close
with
"[$Hw $Hi $HP $HD]"
)
as
"[? HEi]"
.
iModIntro
.
iRight
.
iFrame
.
iSplitL
;
[
done
].
iCombine
"HEi"
"HEN\i"
as
"HEN"
.
iCombine
"HEN"
"HE\N"
as
"HE"
.
rewrite

?ownE_op
;
[
set_solver

set_solver
].
rewrite
<!
union_difference_L
;
try
done
;
set_solver
.
rewrite
inv_eq
/
inv_def
fupd_eq
/
fupd_def
.
iIntros
(
Sub
)
"[Hw HE]"
.
iMod
(
ownI_alloc_open
(
∈
↑
N
)
P
with
"Hw"
)
as
(
i
)
"(% & Hw & #Hi & HD)"
;
auto
using
fresh_inv_name
.
iAssert
(
ownE
{[
i
]}
∗
ownE
(
↑
N
∖
{[
i
]})
∗
ownE
(
E
∖
↑
N
))%
I
with
"[HE]"
as
"(HEi & HEN\i & HE\N)"
.
{
rewrite

?ownE_op
;
[
set_solver
..].
rewrite
assoc_L
!
union_difference_L
//.
set_solver
.
}
do
2
iModIntro
.
iFrame
"HE\N"
.
iSplitL
"Hw HEi"
;
first
by
iApply
"Hw"
.
iSplitL
"Hi"
;
first
by
eauto
.
iIntros
"HP [Hw HE\N]"
.
iDestruct
(
ownI_close
with
"[$Hw $Hi $HP $HD]"
)
as
"[$ HEi]"
.
do
2
iModIntro
.
iSplitL
;
[
done
].
iCombine
"HEi"
"HEN\i"
as
"HEN"
;
iCombine
"HEN"
"HE\N"
as
"HE"
.
rewrite

?ownE_op
;
[
set_solver
..].
rewrite
!
union_difference_L
//
;
set_solver
.
Qed
.
Lemma
inv_open
E
N
P
:
...
...
theories/base_logic/lib/wsat.v
View file @
d6b647ad
...
...
@@ 165,5 +165,4 @@ Proof.
iApply
(
big_sepM_insert
_
I
)
;
first
done
.
iFrame
"HI"
.
by
iRight
.
Qed
.
End
wsat
.
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