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Simon Spies
Iris
Commits
4e756c6d
Commit
4e756c6d
authored
Mar 29, 2016
by
Robbert Krebbers
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Finite powerset RA.
parent
4d5474e2
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4e756c6d
From
iris
.
algebra
Require
Export
gmap
.
From
iris
.
algebra
Require
Import
excl
.
From
iris
.
prelude
Require
Import
mapset
.
Definition
gsetC
K
`
{
Countable
K
}
:
=
gmapC
K
(
exclC
unitC
).
Definition
to_gsetC
`
{
Countable
K
}
(
X
:
gset
K
)
:
gsetC
K
:
=
to_gmap
(
Excl
())
X
.
Section
gset
.
Context
`
{
Countable
K
}.
Implicit
Types
X
Y
:
gset
K
.
Lemma
to_gsetC_empty
:
to_gsetC
(
∅
:
gset
K
)
=
∅
.
Proof
.
apply
to_gmap_empty
.
Qed
.
Lemma
to_gsetC_union
X
Y
:
X
⊥
Y
→
to_gsetC
X
⋅
to_gsetC
Y
=
to_gsetC
(
X
∪
Y
).
Proof
.
intros
HXY
;
apply
:
map_eq
=>
i
;
rewrite
/
to_gsetC
/=.
rewrite
lookup_op
!
lookup_to_gmap
.
repeat
case_option_guard
;
set_solver
.
Qed
.
Lemma
to_gsetC_valid
X
:
✓
to_gsetC
X
.
Proof
.
intros
i
.
rewrite
/
to_gsetC
lookup_to_gmap
.
by
case_option_guard
.
Qed
.
Lemma
to_gsetC_valid_op
X
Y
:
✓
(
to_gsetC
X
⋅
to_gsetC
Y
)
↔
X
⊥
Y
.
Proof
.
split
;
last
(
intros
;
rewrite
to_gsetC_union
//
;
apply
to_gsetC_valid
).
intros
HXY
i
??
;
move
:
(
HXY
i
)
;
rewrite
lookup_op
!
lookup_to_gmap
.
rewrite
!
option_guard_True
//.
Qed
.
Context
`
{
Fresh
K
(
gset
K
),
!
FreshSpec
K
(
gset
K
)}.
Lemma
updateP_alloc_strong
(
Q
:
gsetC
K
→
Prop
)
(
I
:
gset
K
)
X
:
(
∀
i
,
i
∉
X
→
i
∉
I
→
Q
(
to_gsetC
({[
i
]}
∪
X
)))
→
to_gsetC
X
~~>
:
Q
.
Proof
.
intros
;
apply
updateP_alloc_strong
with
I
(
Excl
())
;
[
done
|]=>
i
.
rewrite
/
to_gsetC
lookup_to_gmap_None
-
to_gmap_union_singleton
;
eauto
.
Qed
.
Lemma
updateP_alloc
(
Q
:
gsetC
K
→
Prop
)
X
:
(
∀
i
,
i
∉
X
→
Q
(
to_gsetC
({[
i
]}
∪
X
)))
→
to_gsetC
X
~~>
:
Q
.
Proof
.
move
=>??.
eapply
updateP_alloc_strong
with
(
I
:
=
∅
)
;
by
eauto
.
Qed
.
Lemma
updateP_alloc_strong'
(
I
:
gset
K
)
X
:
to_gsetC
X
~~>
:
λ
Y
:
gsetC
K
,
∃
i
,
Y
=
to_gsetC
({[
i
]}
∪
X
)
∧
i
∉
I
∧
i
∉
X
.
Proof
.
eauto
using
updateP_alloc_strong
.
Qed
.
Lemma
updateP_alloc'
X
:
to_gsetC
X
~~>
:
λ
Y
:
gsetC
K
,
∃
i
,
Y
=
to_gsetC
({[
i
]}
∪
X
)
∧
i
∉
X
.
Proof
.
eauto
using
updateP_alloc
.
Qed
.
End
gset
.
\ No newline at end of file
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