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65a383bd
Commit
65a383bd
authored
Feb 13, 2016
by
Ralf Jung
Browse files
strengthen auth to also provide validity of the current total element
parent
0a593561
Changes
1
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Inline
Side-by-side
program_logic/auth.v
View file @
65a383bd
...
...
@@ -16,7 +16,7 @@ Section auth.
forall
a
b
,
(
✓
Auth
(
Excl
a
)
b
:
iPropG
Λ
Σ
)
⊑
(
∃
b'
,
a
≡
b
⋅
b'
).
Definition
auth_inv
(
γ
:
gname
)
:
iPropG
Λ
Σ
:
=
(
∃
a
,
own
AuthI
γ
(
●
a
)
★
φ
a
)%
I
.
(
∃
a
,
(
■✓
a
∧
own
AuthI
γ
(
●
a
)
)
★
φ
a
)%
I
.
Definition
auth_own
(
γ
:
gname
)
(
a
:
A
)
:
iPropG
Λ
Σ
:
=
own
AuthI
γ
(
◯
a
).
Definition
auth_ctx
(
γ
:
gname
)
:
iPropG
Λ
Σ
:
=
inv
N
(
auth_inv
γ
).
...
...
@@ -29,6 +29,7 @@ Section auth.
rewrite
sep_exist_l
.
apply
exist_elim
=>
γ
.
rewrite
-(
exist_intro
γ
).
transitivity
(
▷
auth_inv
γ
★
auth_own
γ
a
)%
I
.
{
rewrite
/
auth_inv
-
later_intro
-(
exist_intro
a
).
rewrite
const_equiv
//
left_id
.
rewrite
[(
_
★
φ
_
)%
I
]
comm
-
assoc
.
apply
sep_mono
;
first
done
.
rewrite
/
auth_own
-
own_op
auth_both_op
.
done
.
}
rewrite
(
inv_alloc
N
)
/
auth_ctx
pvs_frame_r
.
apply
pvs_mono
.
...
...
@@ -39,19 +40,23 @@ Section auth.
True
⊑
pvs
E
E
(
auth_own
γ
∅
).
Proof
.
by
rewrite
own_update_empty
/
auth_own
.
Qed
.
Context
{
H
φ
:
∀
n
,
Proper
(
dist
n
==>
dist
n
)
φ
}.
Context
{
φ
_ne
:
∀
n
,
Proper
(
dist
n
==>
dist
n
)
φ
}.
Local
Instance
φ
_proper
:
Proper
((
≡
)
==>
(
≡
))
φ
:
=
ne_proper
_
.
Lemma
auth_opened
E
a
γ
:
(
▷
auth_inv
γ
★
auth_own
γ
a
)
⊑
pvs
E
E
(
∃
a'
,
▷φ
(
a
⋅
a'
)
★
own
AuthI
γ
(
●
(
a
⋅
a'
)
⋅
◯
a
)).
(
▷
auth_inv
γ
★
auth_own
γ
a
)
⊑
pvs
E
E
(
∃
a'
,
■✓
(
a
⋅
a'
)
★
▷φ
(
a
⋅
a'
)
★
own
AuthI
γ
(
●
(
a
⋅
a'
)
⋅
◯
a
)).
Proof
.
rewrite
/
auth_inv
.
rewrite
later_exist
sep_exist_r
.
apply
exist_elim
=>
b
.
rewrite
later_sep
[(
▷
own
_
_
_
)%
I
]
pvs_timeless
!
pvs_frame_r
.
apply
pvs_mono
.
rewrite
/
auth_own
[(
_
★
▷φ
_
)%
I
]
comm
-
assoc
-
own_op
.
rewrite
own_valid_r
auth_valid
!
sep_exist_l
/=.
apply
exist_elim
=>
a'
.
rewrite
later_sep
[(
▷
(
_
∧
_
))%
I
]
pvs_timeless
!
pvs_frame_r
.
apply
pvs_mono
.
rewrite
always_and_sep_l
-!
assoc
.
apply
const_elim_sep_l
=>
Hv
.
rewrite
/
auth_own
[(
▷φ
_
★
_
)%
I
]
comm
assoc
-
own_op
.
rewrite
own_valid_r
auth_valid
sep_exist_l
sep_exist_r
/=.
apply
exist_elim
=>
a'
.
rewrite
[
∅
⋅
_
]
left_id
-(
exist_intro
a'
).
apply
(
eq_rewrite
b
(
a
⋅
a'
)
(
λ
x
,
▷φ
x
★
own
AuthI
γ
(
●
x
⋅
◯
a
))%
I
)
;
first
by
solve_ne
.
{
by
rewrite
!
sep_elim_r
.
}
(
λ
x
,
■✓
x
★
▷φ
x
★
own
AuthI
γ
(
●
x
⋅
◯
a
))%
I
).
{
by
move
=>
n
?
?
/
timeless_iff
->.
}
{
apply
sep_elim_l'
,
sep_elim_r'
.
done
.
(* FIXME why does "eauto using I not work? *)
}
rewrite
const_equiv
//
left_id
comm
.
apply
sep_mono
;
first
done
.
by
rewrite
sep_elim_l
.
Qed
.
...
...
@@ -64,7 +69,7 @@ Section auth.
intros
HL
Hv
.
rewrite
/
auth_inv
/
auth_own
-(
exist_intro
(
L
a
⋅
a'
)).
rewrite
later_sep
[(
_
★
▷φ
_
)%
I
]
comm
-
assoc
.
rewrite
-
pvs_frame_l
.
apply
sep_mono
;
first
done
.
rewrite
-
later_intro
-
own_op
.
rewrite
const_equiv
//
left_id
-
later_intro
-
own_op
.
by
apply
own_update
,
(
auth_local_update_l
L
).
Qed
.
...
...
@@ -72,20 +77,22 @@ Section auth.
step-indices. However, since A is timeless, that should not be
a restriction. *)
Lemma
auth_fsa
{
X
:
Type
}
{
FSA
}
(
FSAs
:
FrameShiftAssertion
(
A
:
=
X
)
FSA
)
`
{!
LocalUpdate
Lv
L
}
E
P
(
Q
:
X
→
iPropG
Λ
Σ
)
R
γ
a
:
`
{!
LocalUpdate
Lv
L
}
E
P
(
Q
:
X
→
iPropG
Λ
Σ
)
γ
a
:
nclose
N
⊆
E
→
R
⊑
auth_ctx
γ
→
R
⊑
(
auth_own
γ
a
★
(
∀
a'
,
▷φ
(
a
⋅
a'
)
-
★
P
⊑
auth_ctx
γ
→
P
⊑
(
auth_own
γ
a
★
(
∀
a'
,
■✓
(
a
⋅
a'
)
★
▷φ
(
a
⋅
a'
)
-
★
FSA
(
E
∖
nclose
N
)
(
λ
x
,
■
(
Lv
a
∧
✓
(
L
a
⋅
a'
))
★
▷φ
(
L
a
⋅
a'
)
★
(
auth_own
γ
(
L
a
)
-
★
Q
x
))))
→
R
⊑
FSA
E
Q
.
P
⊑
FSA
E
Q
.
Proof
.
rewrite
/
auth_ctx
=>
HN
Hinv
Hinner
.
eapply
inv_fsa
;
[
eassumption
..|].
rewrite
Hinner
=>{
Hinner
Hinv
R
}.
eapply
inv_fsa
;
[
eassumption
..|].
rewrite
Hinner
=>{
Hinner
Hinv
P
}.
apply
wand_intro_l
.
rewrite
assoc
auth_opened
!
pvs_frame_r
!
sep_exist_r
.
apply
fsa_strip_pvs
;
first
done
.
apply
exist_elim
=>
a'
.
rewrite
(
forall_elim
a'
).
rewrite
[(
▷
_
★
_
)%
I
]
comm
.
rewrite
-[((
_
★
▷
_
)
★
_
)%
I
]
assoc
wand_elim_r
fsa_frame_l
.
(* Getting this wand eliminated is really annoying. *)
rewrite
[(
■
_
★
_
)%
I
]
comm
-!
assoc
[(
▷φ
_
★
_
★
_
)%
I
]
assoc
[(
▷φ
_
★
_
)%
I
]
comm
.
rewrite
wand_elim_r
fsa_frame_l
.
apply
fsa_mono_pvs
;
first
done
.
intros
x
.
rewrite
comm
-!
assoc
.
apply
const_elim_sep_l
=>-[
HL
Hv
].
rewrite
assoc
[(
_
★
(
_
-
★
_
))%
I
]
comm
-
assoc
.
...
...
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