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2c644a10
Commit
2c644a10
authored
Sep 26, 2016
by
Robbert Krebbers
Browse files
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Generic properties for commuting big ops.
parent
123a7c05
Changes
2
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2 changed files
with
151 additions
and
31 deletions
+151
-31
algebra/cmra_big_op.v
algebra/cmra_big_op.v
+61
-1
algebra/upred_big_op.v
algebra/upred_big_op.v
+90
-30
No files found.
algebra/cmra_big_op.v
View file @
2c644a10
...
...
@@ -185,7 +185,6 @@ Section list.
Qed
.
End
list
.
(** ** Big ops over finite maps *)
Section
gmap
.
Context
`
{
Countable
K
}
{
A
:
Type
}.
...
...
@@ -371,3 +370,64 @@ Section gset.
Qed
.
End
gset
.
End
big_op
.
Lemma
big_opL_commute
{
M1
M2
:
ucmraT
}
{
A
}
(
h
:
M1
→
M2
)
`
{!
Proper
((
≡
)
==>
(
≡
))
h
}
(
f
:
nat
→
A
→
M1
)
l
:
h
∅
≡
∅
→
(
∀
x
y
,
h
(
x
⋅
y
)
≡
h
x
⋅
h
y
)
→
h
([
⋅
list
]
k
↦
x
∈
l
,
f
k
x
)
≡
([
⋅
list
]
k
↦
x
∈
l
,
h
(
f
k
x
)).
Proof
.
intros
??.
revert
f
.
induction
l
as
[|
x
l
IH
]=>
f
.
-
by
rewrite
!
big_opL_nil
.
-
by
rewrite
!
big_opL_cons
-
IH
.
Qed
.
Lemma
big_opL_commute1
{
M1
M2
:
ucmraT
}
{
A
}
(
h
:
M1
→
M2
)
`
{!
Proper
((
≡
)
==>
(
≡
))
h
}
(
f
:
nat
→
A
→
M1
)
l
:
(
∀
x
y
,
h
(
x
⋅
y
)
≡
h
x
⋅
h
y
)
→
l
≠
[]
→
h
([
⋅
list
]
k
↦
x
∈
l
,
f
k
x
)
≡
([
⋅
list
]
k
↦
x
∈
l
,
h
(
f
k
x
)).
Proof
.
intros
??.
revert
f
.
induction
l
as
[|
x
[|
x'
l'
]
IH
]=>
f
//.
-
by
rewrite
!
big_opL_singleton
.
-
by
rewrite
!(
big_opL_cons
_
x
)
-
IH
.
Qed
.
Lemma
big_opM_commute
{
M1
M2
:
ucmraT
}
`
{
Countable
K
}
{
A
}
(
h
:
M1
→
M2
)
`
{!
Proper
((
≡
)
==>
(
≡
))
h
}
(
f
:
K
→
A
→
M1
)
m
:
h
∅
≡
∅
→
(
∀
x
y
,
h
(
x
⋅
y
)
≡
h
x
⋅
h
y
)
→
h
([
⋅
map
]
k
↦
x
∈
m
,
f
k
x
)
≡
([
⋅
map
]
k
↦
x
∈
m
,
h
(
f
k
x
)).
Proof
.
intros
.
rewrite
/
big_opM
.
induction
(
map_to_list
m
)
as
[|[
i
x
]
l
IH
]
;
csimpl
;
rewrite
-
?IH
;
auto
.
Qed
.
Lemma
big_opM_commute1
{
M1
M2
:
ucmraT
}
`
{
Countable
K
}
{
A
}
(
h
:
M1
→
M2
)
`
{!
Proper
((
≡
)
==>
(
≡
))
h
}
(
f
:
K
→
A
→
M1
)
m
:
(
∀
x
y
,
h
(
x
⋅
y
)
≡
h
x
⋅
h
y
)
→
m
≠
∅
→
h
([
⋅
map
]
k
↦
x
∈
m
,
f
k
x
)
≡
([
⋅
map
]
k
↦
x
∈
m
,
h
(
f
k
x
)).
Proof
.
rewrite
-
map_to_list_empty'
/
big_opM
=>
??.
induction
(
map_to_list
m
)
as
[|[
i
x
]
[|
i'
x'
]
IH
]
;
csimpl
in
*
;
rewrite
?right_id
-
?IH
//.
Qed
.
Lemma
big_opS_commute
{
M1
M2
:
ucmraT
}
`
{
Countable
A
}
(
h
:
M1
→
M2
)
`
{!
Proper
((
≡
)
==>
(
≡
))
h
}
(
f
:
A
→
M1
)
X
:
h
∅
≡
∅
→
(
∀
x
y
,
h
(
x
⋅
y
)
≡
h
x
⋅
h
y
)
→
h
([
⋅
set
]
x
∈
X
,
f
x
)
≡
([
⋅
set
]
x
∈
X
,
h
(
f
x
)).
Proof
.
intros
.
rewrite
/
big_opS
.
induction
(
elements
X
)
as
[|
x
l
IH
]
;
csimpl
;
rewrite
-
?IH
;
auto
.
Qed
.
Lemma
big_opS_commute1
{
M1
M2
:
ucmraT
}
`
{
Countable
A
}
(
h
:
M1
→
M2
)
`
{!
Proper
((
≡
)
==>
(
≡
))
h
}
(
f
:
A
→
M1
)
X
:
(
∀
x
y
,
h
(
x
⋅
y
)
≡
h
x
⋅
h
y
)
→
X
≢
∅
→
h
([
⋅
set
]
x
∈
X
,
f
x
)
≡
([
⋅
set
]
x
∈
X
,
h
(
f
x
)).
Proof
.
rewrite
-
elements_empty'
/
big_opS
=>
??.
induction
(
elements
X
)
as
[|
x
[|
x'
l
]
IH
]
;
csimpl
in
*
;
rewrite
?right_id
-
?IH
//.
Qed
.
algebra/upred_big_op.v
View file @
2c644a10
From
iris
.
algebra
Require
Export
upred
list
.
From
iris
.
algebra
Require
Export
upred
list
cmra_big_op
.
From
iris
.
prelude
Require
Import
gmap
fin_collections
functions
.
Import
uPred
.
...
...
@@ -267,21 +267,41 @@ Section list.
by
rewrite
-!
assoc
(
assoc
_
(
Ψ
_
_
))
[(
Ψ
_
_
★
_
)%
I
]
comm
-!
assoc
.
Qed
.
Lemma
big_sepL_later
Φ
l
:
▷
([
★
list
]
k
↦
x
∈
l
,
Φ
k
x
)
⊣
⊢
([
★
list
]
k
↦
x
∈
l
,
▷
Φ
k
x
).
Lemma
big_sepL_commute
(
Ψ
:
uPred
M
→
uPred
M
)
`
{!
Proper
((
≡
)
==>
(
≡
))
Ψ
}
Φ
l
:
Ψ
True
⊣
⊢
True
→
(
∀
P
Q
,
Ψ
(
P
★
Q
)
⊣
⊢
Ψ
P
★
Ψ
Q
)
→
Ψ
([
★
list
]
k
↦
x
∈
l
,
Φ
k
x
)
⊣
⊢
([
★
list
]
k
↦
x
∈
l
,
Ψ
(
Φ
k
x
)).
Proof
.
revert
Φ
.
induction
l
as
[|
x
l
IH
]=>
Φ
.
{
by
rewrite
!
big_sepL_nil
later_True
.
}
by
rewrite
!
big_sepL_cons
later_sep
IH
.
intros
??.
revert
Φ
.
induction
l
as
[|
x
l
IH
]=>
Φ
//.
by
rewrite
!
big_sepL_cons
-
IH
.
Qed
.
Lemma
big_sepL_op_commute
{
B
:
ucmraT
}
(
Ψ
:
B
→
uPred
M
)
`
{!
Proper
((
≡
)
==>
(
≡
))
Ψ
}
(
f
:
nat
→
A
→
B
)
l
:
Ψ
∅
⊣
⊢
True
→
(
∀
x
y
,
Ψ
(
x
⋅
y
)
⊣
⊢
Ψ
x
★
Ψ
y
)
→
Ψ
([
⋅
list
]
k
↦
x
∈
l
,
f
k
x
)
⊣
⊢
([
★
list
]
k
↦
x
∈
l
,
Ψ
(
f
k
x
)).
Proof
.
intros
??.
revert
f
.
induction
l
as
[|
x
l
IH
]=>
f
//.
by
rewrite
big_sepL_cons
big_opL_cons
-
IH
.
Qed
.
Lemma
big_sepL_op_commute1
{
B
:
ucmraT
}
(
Ψ
:
B
→
uPred
M
)
`
{!
Proper
((
≡
)
==>
(
≡
))
Ψ
}
(
f
:
nat
→
A
→
B
)
l
:
(
∀
x
y
,
Ψ
(
x
⋅
y
)
⊣
⊢
Ψ
x
★
Ψ
y
)
→
l
≠
[]
→
Ψ
([
⋅
list
]
k
↦
x
∈
l
,
f
k
x
)
⊣
⊢
([
★
list
]
k
↦
x
∈
l
,
Ψ
(
f
k
x
)).
Proof
.
intros
??.
revert
f
.
induction
l
as
[|
x
[|
x'
l'
]
IH
]=>
f
//.
{
by
rewrite
big_sepL_singleton
big_opL_singleton
.
}
by
rewrite
big_sepL_cons
big_opL_cons
-
IH
.
Qed
.
Lemma
big_sepL_later
Φ
l
:
▷
([
★
list
]
k
↦
x
∈
l
,
Φ
k
x
)
⊣
⊢
([
★
list
]
k
↦
x
∈
l
,
▷
Φ
k
x
).
Proof
.
apply
(
big_sepL_commute
_
)
;
auto
using
later_True
,
later_sep
.
Qed
.
Lemma
big_sepL_always
Φ
l
:
(
□
[
★
list
]
k
↦
x
∈
l
,
Φ
k
x
)
⊣
⊢
([
★
list
]
k
↦
x
∈
l
,
□
Φ
k
x
).
Proof
.
revert
Φ
.
induction
l
as
[|
x
l
IH
]=>
Φ
.
{
by
rewrite
!
big_sepL_nil
always_pure
.
}
by
rewrite
!
big_sepL_cons
always_sep
IH
.
Qed
.
Proof
.
apply
(
big_sepL_commute
_
)
;
auto
using
always_pure
,
always_sep
.
Qed
.
Lemma
big_sepL_always_if
p
Φ
l
:
□
?p
([
★
list
]
k
↦
x
∈
l
,
Φ
k
x
)
⊣
⊢
([
★
list
]
k
↦
x
∈
l
,
□
?p
Φ
k
x
).
...
...
@@ -430,21 +450,41 @@ Section gmap.
by
rewrite
IH
-!
assoc
(
assoc
_
(
Ψ
_
_
))
[(
Ψ
_
_
★
_
)%
I
]
comm
-!
assoc
.
Qed
.
Lemma
big_sepM_later
Φ
m
:
▷
([
★
map
]
k
↦
x
∈
m
,
Φ
k
x
)
⊣
⊢
([
★
map
]
k
↦
x
∈
m
,
▷
Φ
k
x
).
Lemma
big_sepM_commute
(
Ψ
:
uPred
M
→
uPred
M
)
`
{!
Proper
((
≡
)
==>
(
≡
))
Ψ
}
Φ
m
:
Ψ
True
⊣
⊢
True
→
(
∀
P
Q
,
Ψ
(
P
★
Q
)
⊣
⊢
Ψ
P
★
Ψ
Q
)
→
Ψ
([
★
map
]
k
↦
x
∈
m
,
Φ
k
x
)
⊣
⊢
([
★
map
]
k
↦
x
∈
m
,
Ψ
(
Φ
k
x
)).
Proof
.
rewrite
/
uPred_big_sepM
.
induction
(
map_to_list
m
)
as
[|[
i
x
]
l
IH
]
;
csimpl
;
rewrite
?later_True
//.
by
rewrite
later_sep
IH
.
intros
??.
rewrite
/
uPred_big_sepM
.
induction
(
map_to_list
m
)
as
[|[
i
x
]
l
IH
]
;
rewrite
//=
-
?IH
;
auto
.
Qed
.
Lemma
big_sepM_op_commute
{
B
:
ucmraT
}
(
Ψ
:
B
→
uPred
M
)
`
{!
Proper
((
≡
)
==>
(
≡
))
Ψ
}
(
f
:
K
→
A
→
B
)
m
:
Ψ
∅
⊣
⊢
True
→
(
∀
x
y
,
Ψ
(
x
⋅
y
)
⊣
⊢
Ψ
x
★
Ψ
y
)
→
Ψ
([
⋅
map
]
k
↦
x
∈
m
,
f
k
x
)
⊣
⊢
([
★
map
]
k
↦
x
∈
m
,
Ψ
(
f
k
x
)).
Proof
.
intros
??.
rewrite
/
big_opM
/
uPred_big_sepM
.
induction
(
map_to_list
m
)
as
[|[
i
x
]
l
IH
]
;
rewrite
//=
-
?IH
;
auto
.
Qed
.
Lemma
big_sepM_op_commute1
{
B
:
ucmraT
}
(
Ψ
:
B
→
uPred
M
)
`
{!
Proper
((
≡
)
==>
(
≡
))
Ψ
}
(
f
:
K
→
A
→
B
)
m
:
(
∀
x
y
,
Ψ
(
x
⋅
y
)
⊣
⊢
Ψ
x
★
Ψ
y
)
→
m
≠
∅
→
Ψ
([
⋅
map
]
k
↦
x
∈
m
,
f
k
x
)
⊣
⊢
([
★
map
]
k
↦
x
∈
m
,
Ψ
(
f
k
x
)).
Proof
.
rewrite
-
map_to_list_empty'
.
intros
??.
rewrite
/
big_opM
/
uPred_big_sepM
.
induction
(
map_to_list
m
)
as
[|[
i
x
]
[|
i'
x'
]
IH
]
;
csimpl
in
*
;
rewrite
?right_id
-
?IH
//.
Qed
.
Lemma
big_sepM_later
Φ
m
:
▷
([
★
map
]
k
↦
x
∈
m
,
Φ
k
x
)
⊣
⊢
([
★
map
]
k
↦
x
∈
m
,
▷
Φ
k
x
).
Proof
.
apply
(
big_sepM_commute
_
)
;
auto
using
later_True
,
later_sep
.
Qed
.
Lemma
big_sepM_always
Φ
m
:
(
□
[
★
map
]
k
↦
x
∈
m
,
Φ
k
x
)
⊣
⊢
([
★
map
]
k
↦
x
∈
m
,
□
Φ
k
x
).
Proof
.
rewrite
/
uPred_big_sepM
.
induction
(
map_to_list
m
)
as
[|[
i
x
]
l
IH
]
;
csimpl
;
rewrite
?always_pure
//.
by
rewrite
always_sep
IH
.
Qed
.
Proof
.
apply
(
big_sepM_commute
_
)
;
auto
using
always_pure
,
always_sep
.
Qed
.
Lemma
big_sepM_always_if
p
Φ
m
:
□
?p
([
★
map
]
k
↦
x
∈
m
,
Φ
k
x
)
⊣
⊢
([
★
map
]
k
↦
x
∈
m
,
□
?p
Φ
k
x
).
...
...
@@ -569,20 +609,40 @@ Section gset.
by
rewrite
IH
-!
assoc
(
assoc
_
(
Ψ
_
))
[(
Ψ
_
★
_
)%
I
]
comm
-!
assoc
.
Qed
.
Lemma
big_sepS_later
Φ
X
:
▷
([
★
set
]
y
∈
X
,
Φ
y
)
⊣
⊢
([
★
set
]
y
∈
X
,
▷
Φ
y
).
Lemma
big_sepS_commute
(
Ψ
:
uPred
M
→
uPred
M
)
`
{!
Proper
((
≡
)
==>
(
≡
))
Ψ
}
Φ
X
:
Ψ
True
⊣
⊢
True
→
(
∀
P
Q
,
Ψ
(
P
★
Q
)
⊣
⊢
Ψ
P
★
Ψ
Q
)
→
Ψ
([
★
set
]
x
∈
X
,
Φ
x
)
⊣
⊢
([
★
set
]
x
∈
X
,
Ψ
(
Φ
x
)).
Proof
.
rewrite
/
uPred_big_sepS
.
induction
(
elements
X
)
as
[|
x
l
IH
]
;
csimpl
;
first
by
rewrite
?later_True
.
by
rewrite
later_sep
IH
.
intros
??.
rewrite
/
uPred_big_sepS
.
induction
(
elements
X
)
as
[|
x
l
IH
]
;
rewrite
//=
-
?IH
;
auto
.
Qed
.
Lemma
big_sepS_always
Φ
X
:
□
([
★
set
]
y
∈
X
,
Φ
y
)
⊣
⊢
([
★
set
]
y
∈
X
,
□
Φ
y
).
Lemma
big_sepS_op_commute
{
B
:
ucmraT
}
(
Ψ
:
B
→
uPred
M
)
`
{!
Proper
((
≡
)
==>
(
≡
))
Ψ
}
(
f
:
A
→
B
)
X
:
Ψ
∅
⊣
⊢
True
→
(
∀
x
y
,
Ψ
(
x
⋅
y
)
⊣
⊢
Ψ
x
★
Ψ
y
)
→
Ψ
([
⋅
set
]
x
∈
X
,
f
x
)
⊣
⊢
([
★
set
]
x
∈
X
,
Ψ
(
f
x
)).
Proof
.
rewrite
/
uPred_big_sepS
.
induction
(
elements
X
)
as
[|
x
l
IH
]
;
csimpl
;
first
by
rewrite
?always_pure
.
by
rewrite
always_sep
IH
.
intros
??.
rewrite
/
big_opS
/
uPred_big_sepS
.
induction
(
elements
X
)
as
[|
x
l
IH
]
;
rewrite
//=
-
?IH
;
auto
.
Qed
.
Lemma
big_sepS_op_commute1
{
B
:
ucmraT
}
(
Ψ
:
B
→
uPred
M
)
`
{!
Proper
((
≡
)
==>
(
≡
))
Ψ
}
(
f
:
A
→
B
)
X
:
(
∀
x
y
,
Ψ
(
x
⋅
y
)
⊣
⊢
Ψ
x
★
Ψ
y
)
→
X
≢
∅
→
Ψ
([
⋅
set
]
x
∈
X
,
f
x
)
⊣
⊢
([
★
set
]
x
∈
X
,
Ψ
(
f
x
)).
Proof
.
rewrite
-
elements_empty'
.
intros
??.
rewrite
/
big_opS
/
uPred_big_sepS
.
induction
(
elements
X
)
as
[|
x
[|
x'
l
]
IH
]
;
csimpl
in
*
;
rewrite
?right_id
-
?IH
//.
Qed
.
Lemma
big_sepS_later
Φ
X
:
▷
([
★
set
]
y
∈
X
,
Φ
y
)
⊣
⊢
([
★
set
]
y
∈
X
,
▷
Φ
y
).
Proof
.
apply
(
big_sepS_commute
_
)
;
auto
using
later_True
,
later_sep
.
Qed
.
Lemma
big_sepS_always
Φ
X
:
□
([
★
set
]
y
∈
X
,
Φ
y
)
⊣
⊢
([
★
set
]
y
∈
X
,
□
Φ
y
).
Proof
.
apply
(
big_sepS_commute
_
)
;
auto
using
always_pure
,
always_sep
.
Qed
.
Lemma
big_sepS_always_if
q
Φ
X
:
□
?q
([
★
set
]
y
∈
X
,
Φ
y
)
⊣
⊢
([
★
set
]
y
∈
X
,
□
?q
Φ
y
).
Proof
.
destruct
q
;
simpl
;
auto
using
big_sepS_always
.
Qed
.
...
...
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