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Gaëtan Gilbert
Iris
Commits
2269c6c0
Commit
2269c6c0
authored
5 years ago
by
Ralf Jung
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Merge branch 'ralf/sem-inv' into 'master'
Semantic invariants See merge request
iris/iris!319
parents
a2efc56b
3db87055
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CHANGELOG.md
+5
-0
5 additions, 0 deletions
CHANGELOG.md
theories/base_logic/lib/invariants.v
+171
-110
171 additions, 110 deletions
theories/base_logic/lib/invariants.v
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176 additions
and
110 deletions
CHANGELOG.md
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2269c6c0
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@@ -5,6 +5,11 @@ Coq development, but not every API-breaking change is listed. Changes marked
## Iris master
**Changes in the theory of Iris itself:**
*
[#] Redefine invariants as "semantic invariants" so that they support
splitting and other forms of weakening.
**Changes in Coq:**
*
A new tactic
`iStopProof`
to turn the proof mode entailment into an ordinary
...
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theories/base_logic/lib/invariants.v
+
171
−
110
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2269c6c0
...
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@@ -6,123 +6,184 @@ From iris.base_logic.lib Require Import wsat.
Set
Default
Proof
Using
"Type"
.
Import
uPred
.
(**
Derived forms and lemmas about them.
*)
(**
Semantic Invariants
*)
Definition
inv_def
`{
!
invG
Σ
}
(
N
:
namespace
)
(
P
:
iProp
Σ
)
:
iProp
Σ
:=
(
∃
i
P'
,
⌜
i
∈
(
↑
N
:
coPset
)
⌝
∧
▷
□
(
P'
↔
P
)
∧
ownI
i
P'
)
%
I
.
(
□
∀
E
,
⌜↑
N
⊆
E
⌝
→
|
=
{
E
,
E
∖
↑
N
}=>
▷
P
∗
(
▷
P
=
{
E
∖
↑
N
,
E
}
=∗
True
)
)
%
I
.
Definition
inv_aux
:
seal
(
@
inv_def
)
.
by
eexists
.
Qed
.
Definition
inv
{
Σ
i
}
:=
inv_aux
.(
unseal
)
Σ
i
.
Definition
inv_eq
:
@
inv
=
@
inv_def
:=
inv_aux
.(
seal_eq
)
.
Instance
:
Params
(
@
inv
)
3
:=
{}
.
Typeclasses
Opaque
inv
.
(** * Invariants *)
Section
inv
.
Context
`{
!
invG
Σ
}
.
Implicit
Types
i
:
positive
.
Implicit
Types
N
:
namespace
.
Implicit
Types
P
Q
R
:
iProp
Σ
.
Global
Instance
inv_contractive
N
:
Contractive
(
inv
N
)
.
Proof
.
rewrite
inv_eq
.
solve_contractive
.
Qed
.
Global
Instance
inv_ne
N
:
NonExpansive
(
inv
N
)
.
Proof
.
apply
contractive_ne
,
_
.
Qed
.
Global
Instance
inv_proper
N
:
Proper
((
⊣⊢
)
==>
(
⊣⊢
))
(
inv
N
)
.
Proof
.
apply
ne_proper
,
_
.
Qed
.
Global
Instance
inv_persistent
N
P
:
Persistent
(
inv
N
P
)
.
Proof
.
rewrite
inv_eq
/
inv
;
apply
_
.
Qed
.
Lemma
inv_iff
N
P
Q
:
▷
□
(
P
↔
Q
)
-∗
inv
N
P
-∗
inv
N
Q
.
Proof
.
iIntros
"#HPQ"
.
rewrite
inv_eq
.
iDestruct
1
as
(
i
P'
)
"(?&#HP&?)"
.
iExists
i
,
P'
.
iFrame
.
iNext
;
iAlways
;
iSplit
.
-
iIntros
"HP'"
.
iApply
"HPQ"
.
by
iApply
"HP"
.
-
iIntros
"HQ"
.
iApply
"HP"
.
by
iApply
"HPQ"
.
Qed
.
Lemma
fresh_inv_name
(
E
:
gset
positive
)
N
:
∃
i
,
i
∉
E
∧
i
∈
(
↑
N
:
coPset
)
.
Proof
.
exists
(
coPpick
(
↑
N
∖
gset_to_coPset
E
))
.
rewrite
-
elem_of_gset_to_coPset
(
comm
and
)
-
elem_of_difference
.
apply
coPpick_elem_of
=>
Hfin
.
eapply
nclose_infinite
,
(
difference_finite_inv
_
_),
Hfin
.
apply
gset_to_coPset_finite
.
Qed
.
Lemma
inv_alloc
N
E
P
:
▷
P
=
{
E
}
=∗
inv
N
P
.
Proof
.
rewrite
inv_eq
/
inv_def
uPred_fupd_eq
.
iIntros
"HP [Hw $]"
.
iMod
(
ownI_alloc
(.
∈
(
↑
N
:
coPset
))
P
with
"[$HP $Hw]"
)
as
(
i
?)
"[$ ?]"
;
auto
using
fresh_inv_name
.
do
2
iModIntro
.
iExists
i
,
P
.
rewrite
-
(
iff_refl
True
%
I
)
.
auto
.
Qed
.
Lemma
inv_alloc_open
N
E
P
:
↑
N
⊆
E
→
(|
=
{
E
,
E
∖↑
N
}=>
inv
N
P
∗
(
▷
P
=
{
E
∖↑
N
,
E
}
=∗
True
))
%
I
.
Proof
.
rewrite
inv_eq
/
inv_def
uPred_fupd_eq
.
iIntros
(
Sub
)
"[Hw HE]"
.
iMod
(
ownI_alloc_open
(.
∈
(
↑
N
:
coPset
))
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"
.
{
iExists
i
,
P
.
rewrite
-
(
iff_refl
True
%
I
)
.
auto
.
}
iIntros
"HP [Hw HE\N]"
.
iDestruct
(
ownI_close
with
"[$Hw $Hi $HP $HD]"
)
as
"[$ HEi]"
.
do
2
iModIntro
.
iSplitL
;
[|
done
]
.
iCombine
"HEi HEN\i HE\N"
as
"HEN"
.
rewrite
-
?ownE_op
;
[|
set_solver
..]
.
rewrite
assoc_L
-!
union_difference_L
//
;
set_solver
.
Qed
.
Lemma
inv_open
E
N
P
:
↑
N
⊆
E
→
inv
N
P
=
{
E
,
E
∖↑
N
}
=∗
▷
P
∗
(
▷
P
=
{
E
∖↑
N
,
E
}
=∗
True
)
.
Proof
.
rewrite
inv_eq
/
inv_def
uPred_fupd_eq
/
uPred_fupd_def
.
iDestruct
1
as
(
i
P'
)
"(Hi & #HP' & #HiP)"
.
iDestruct
"Hi"
as
%
?
%
elem_of_subseteq_singleton
.
rewrite
{
1
4
}(
union_difference_L
(
↑
N
)
E
)
//
ownE_op
;
last
set_solver
.
rewrite
{
1
5
}(
union_difference_L
{[
i
]}
(
↑
N
))
//
ownE_op
;
last
set_solver
.
iIntros
"(Hw & [HE $] & $) !> !>"
.
iDestruct
(
ownI_open
i
with
"[$Hw $HE $HiP]"
)
as
"($ & HP & HD)"
.
iDestruct
(
"HP'"
with
"HP"
)
as
"$"
.
iIntros
"HP [Hw $] !> !>"
.
iApply
(
ownI_close
_
P'
)
.
iFrame
"HD Hw HiP"
.
iApply
"HP'"
.
iFrame
.
Qed
.
Lemma
inv_open_strong
E
N
P
:
↑
N
⊆
E
→
inv
N
P
=
{
E
,
E
∖↑
N
}
=∗
▷
P
∗
∀
E'
,
▷
P
=
{
E'
,
↑
N
∪
E'
}
=∗
True
.
Proof
.
iIntros
(?)
"Hinv"
.
iPoseProof
(
inv_open
(
↑
N
)
N
P
with
"Hinv"
)
as
"H"
;
first
done
.
rewrite
difference_diag_L
.
iPoseProof
(
fupd_mask_frame_r
_
_
(
E
∖
↑
N
)
with
"H"
)
as
"H"
;
first
set_solver
.
rewrite
left_id_L
-
union_difference_L
//.
iMod
"H"
as
"[$ H]"
;
iModIntro
.
iIntros
(
E'
)
"HP"
.
iPoseProof
(
fupd_mask_frame_r
_
_
E'
with
"(H HP)"
)
as
"H"
;
first
set_solver
.
by
rewrite
left_id_L
.
Qed
.
Global
Instance
into_inv_inv
N
P
:
IntoInv
(
inv
N
P
)
N
:=
{}
.
Global
Instance
into_acc_inv
E
N
P
:
IntoAcc
(
X
:=
unit
)
(
inv
N
P
)
(
↑
N
⊆
E
)
True
(
fupd
E
(
E
∖↑
N
))
(
fupd
(
E
∖↑
N
)
E
)
(
λ
_,
▷
P
)
%
I
(
λ
_,
▷
P
)
%
I
(
λ
_,
None
)
%
I
.
Proof
.
rewrite
/
IntoAcc
/
accessor
exist_unit
.
iIntros
(?)
"#Hinv _"
.
iApply
inv_open
;
done
.
Qed
.
Lemma
inv_open_timeless
E
N
P
`{
!
Timeless
P
}
:
↑
N
⊆
E
→
inv
N
P
=
{
E
,
E
∖↑
N
}
=∗
P
∗
(
P
=
{
E
∖↑
N
,
E
}
=∗
True
)
.
Proof
.
iIntros
(?)
"Hinv"
.
iMod
(
inv_open
with
"Hinv"
)
as
"[>HP Hclose]"
;
auto
.
iIntros
"!> {$HP} HP"
.
iApply
"Hclose"
;
auto
.
Qed
.
Context
`{
!
invG
Σ
}
.
Implicit
Types
i
:
positive
.
Implicit
Types
N
:
namespace
.
Implicit
Types
E
:
coPset
.
Implicit
Types
P
Q
R
:
iProp
Σ
.
(** ** Internal model of invariants *)
Definition
own_inv
(
N
:
namespace
)
(
P
:
iProp
Σ
)
:
iProp
Σ
:=
(
∃
i
P'
,
⌜
i
∈
(
↑
N
:
coPset
)
⌝
∧
▷
□
(
P'
↔
P
)
∧
ownI
i
P'
)
%
I
.
Lemma
own_inv_open
E
N
P
:
↑
N
⊆
E
→
own_inv
N
P
=
{
E
,
E
∖↑
N
}
=∗
▷
P
∗
(
▷
P
=
{
E
∖↑
N
,
E
}
=∗
True
)
.
Proof
.
rewrite
uPred_fupd_eq
/
uPred_fupd_def
.
iDestruct
1
as
(
i
P'
)
"(Hi & #HP' & #HiP)"
.
iDestruct
"Hi"
as
%
?
%
elem_of_subseteq_singleton
.
rewrite
{
1
4
}(
union_difference_L
(
↑
N
)
E
)
//
ownE_op
;
last
set_solver
.
rewrite
{
1
5
}(
union_difference_L
{[
i
]}
(
↑
N
))
//
ownE_op
;
last
set_solver
.
iIntros
"(Hw & [HE $] & $) !> !>"
.
iDestruct
(
ownI_open
i
with
"[$Hw $HE $HiP]"
)
as
"($ & HP & HD)"
.
iDestruct
(
"HP'"
with
"HP"
)
as
"$"
.
iIntros
"HP [Hw $] !> !>"
.
iApply
(
ownI_close
_
P'
)
.
iFrame
"HD Hw HiP"
.
iApply
"HP'"
.
iFrame
.
Qed
.
Lemma
fresh_inv_name
(
E
:
gset
positive
)
N
:
∃
i
,
i
∉
E
∧
i
∈
(
↑
N
:
coPset
)
.
Proof
.
exists
(
coPpick
(
↑
N
∖
gset_to_coPset
E
))
.
rewrite
-
elem_of_gset_to_coPset
(
comm
and
)
-
elem_of_difference
.
apply
coPpick_elem_of
=>
Hfin
.
eapply
nclose_infinite
,
(
difference_finite_inv
_
_),
Hfin
.
apply
gset_to_coPset_finite
.
Qed
.
Lemma
own_inv_alloc
N
E
P
:
▷
P
=
{
E
}
=∗
own_inv
N
P
.
Proof
.
rewrite
uPred_fupd_eq
.
iIntros
"HP [Hw $]"
.
iMod
(
ownI_alloc
(.
∈
(
↑
N
:
coPset
))
P
with
"[$HP $Hw]"
)
as
(
i
?)
"[$ ?]"
;
auto
using
fresh_inv_name
.
do
2
iModIntro
.
iExists
i
,
P
.
rewrite
-
(
iff_refl
True
%
I
)
.
auto
.
Qed
.
Lemma
own_inv_alloc_open
N
E
P
:
↑
N
⊆
E
→
(|
=
{
E
,
E
∖↑
N
}=>
own_inv
N
P
∗
(
▷
P
=
{
E
∖↑
N
,
E
}
=∗
True
))
%
I
.
Proof
.
rewrite
uPred_fupd_eq
.
iIntros
(
Sub
)
"[Hw HE]"
.
iMod
(
ownI_alloc_open
(.
∈
(
↑
N
:
coPset
))
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"
.
{
iExists
i
,
P
.
rewrite
-
(
iff_refl
True
%
I
)
.
auto
.
}
iIntros
"HP [Hw HE\N]"
.
iDestruct
(
ownI_close
with
"[$Hw $Hi $HP $HD]"
)
as
"[$ HEi]"
.
do
2
iModIntro
.
iSplitL
;
[|
done
]
.
iCombine
"HEi HEN\i HE\N"
as
"HEN"
.
rewrite
-
?ownE_op
;
[|
set_solver
..]
.
rewrite
assoc_L
-!
union_difference_L
//
;
set_solver
.
Qed
.
Lemma
own_inv_to_inv
M
P
:
own_inv
M
P
-∗
inv
M
P
.
Proof
.
iIntros
"#I"
.
rewrite
inv_eq
.
iIntros
(
E
H
)
.
iPoseProof
(
own_inv_open
with
"I"
)
as
"H"
;
eauto
.
Qed
.
(** ** Public API of invariants *)
Global
Instance
inv_contractive
N
:
Contractive
(
inv
N
)
.
Proof
.
rewrite
inv_eq
.
solve_contractive
.
Qed
.
Global
Instance
inv_ne
N
:
NonExpansive
(
inv
N
)
.
Proof
.
apply
contractive_ne
,
_
.
Qed
.
Global
Instance
inv_proper
N
:
Proper
(
equiv
==>
equiv
)
(
inv
N
)
.
Proof
.
apply
ne_proper
,
_
.
Qed
.
Global
Instance
inv_persistent
N
P
:
Persistent
(
inv
N
P
)
.
Proof
.
rewrite
inv_eq
.
apply
_
.
Qed
.
Lemma
inv_acc
N
P
Q
:
inv
N
P
-∗
▷
□
(
P
-∗
Q
∗
(
Q
-∗
P
))
-∗
inv
N
Q
.
Proof
.
rewrite
inv_eq
.
iIntros
"#HI #Acc !>"
(
E
H
)
.
iMod
(
"HI"
$!
E
H
)
as
"[HP Hclose]"
.
iDestruct
(
"Acc"
with
"HP"
)
as
"[$ HQP]"
.
iIntros
"!> HQ"
.
iApply
"Hclose"
.
iApply
"HQP"
.
done
.
Qed
.
Lemma
inv_iff
N
P
Q
:
▷
□
(
P
↔
Q
)
-∗
inv
N
P
-∗
inv
N
Q
.
Proof
.
iIntros
"#HPQ #HI"
.
iApply
(
inv_acc
with
"HI"
)
.
iIntros
"!> !# HP"
.
iSplitL
"HP"
.
-
by
iApply
"HPQ"
.
-
iIntros
"HQ"
.
by
iApply
"HPQ"
.
Qed
.
Lemma
inv_alloc
N
E
P
:
▷
P
=
{
E
}
=∗
inv
N
P
.
Proof
.
iIntros
"HP"
.
iApply
own_inv_to_inv
.
iApply
(
own_inv_alloc
N
E
with
"HP"
)
.
Qed
.
Lemma
inv_alloc_open
N
E
P
:
↑
N
⊆
E
→
(|
=
{
E
,
E
∖↑
N
}=>
inv
N
P
∗
(
▷
P
=
{
E
∖↑
N
,
E
}
=∗
True
))
%
I
.
Proof
.
iIntros
(?)
.
iMod
own_inv_alloc_open
as
"[HI $]"
;
first
done
.
iApply
own_inv_to_inv
.
done
.
Qed
.
Lemma
inv_open
E
N
P
:
↑
N
⊆
E
→
inv
N
P
=
{
E
,
E
∖↑
N
}
=∗
▷
P
∗
(
▷
P
=
{
E
∖↑
N
,
E
}
=∗
True
)
.
Proof
.
rewrite
inv_eq
/
inv_def
;
iIntros
(?)
"#HI"
.
by
iApply
"HI"
.
Qed
.
(** ** Proof mode integration *)
Global
Instance
into_inv_inv
N
P
:
IntoInv
(
inv
N
P
)
N
:=
{}
.
Global
Instance
into_acc_inv
N
P
E
:
IntoAcc
(
X
:=
unit
)
(
inv
N
P
)
(
↑
N
⊆
E
)
True
(
fupd
E
(
E
∖
↑
N
))
(
fupd
(
E
∖
↑
N
)
E
)
(
λ
_
:
(),
(
▷
P
)
%
I
)
(
λ
_
:
(),
(
▷
P
)
%
I
)
(
λ
_
:
(),
None
)
.
Proof
.
rewrite
inv_eq
/
IntoAcc
/
accessor
bi
.
exist_unit
.
iIntros
(?)
"#Hinv _"
.
iApply
"Hinv"
;
done
.
Qed
.
(** ** Derived properties *)
Lemma
inv_open_strong
E
N
P
:
↑
N
⊆
E
→
inv
N
P
=
{
E
,
E
∖↑
N
}
=∗
▷
P
∗
∀
E'
,
▷
P
=
{
E'
,
↑
N
∪
E'
}
=∗
True
.
Proof
.
iIntros
(?)
"Hinv"
.
iPoseProof
(
inv_open
(
↑
N
)
N
P
with
"Hinv"
)
as
"H"
;
first
done
.
rewrite
difference_diag_L
.
iPoseProof
(
fupd_mask_frame_r
_
_
(
E
∖
↑
N
)
with
"H"
)
as
"H"
;
first
set_solver
.
rewrite
left_id_L
-
union_difference_L
//.
iMod
"H"
as
"[$ H]"
;
iModIntro
.
iIntros
(
E'
)
"HP"
.
iPoseProof
(
fupd_mask_frame_r
_
_
E'
with
"(H HP)"
)
as
"H"
;
first
set_solver
.
by
rewrite
left_id_L
.
Qed
.
Lemma
inv_open_timeless
E
N
P
`{
!
Timeless
P
}
:
↑
N
⊆
E
→
inv
N
P
=
{
E
,
E
∖↑
N
}
=∗
P
∗
(
P
=
{
E
∖↑
N
,
E
}
=∗
True
)
.
Proof
.
iIntros
(?)
"Hinv"
.
iMod
(
inv_open
with
"Hinv"
)
as
"[>HP Hclose]"
;
auto
.
iIntros
"!> {$HP} HP"
.
iApply
"Hclose"
;
auto
.
Qed
.
Lemma
inv_sep_l
N
P
Q
:
inv
N
(
P
∗
Q
)
-∗
inv
N
P
.
Proof
.
iIntros
"#HI"
.
iApply
inv_acc
;
eauto
.
iIntros
"!> !# [$ $] $"
.
Qed
.
Lemma
inv_sep_r
N
P
Q
:
inv
N
(
P
∗
Q
)
-∗
inv
N
Q
.
Proof
.
rewrite
(
comm
_
P
Q
)
.
eapply
inv_sep_l
.
Qed
.
Lemma
inv_sep
N
P
Q
:
inv
N
(
P
∗
Q
)
-∗
inv
N
P
∗
inv
N
Q
.
Proof
.
iIntros
"#H"
.
iPoseProof
(
inv_sep_l
with
"H"
)
as
"$"
.
iPoseProof
(
inv_sep_r
with
"H"
)
as
"$"
.
Qed
.
End
inv
.
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