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ca8e04a3
Commit
ca8e04a3
authored
Sep 21, 2016
by
Robbert Krebbers
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Reorganize frame preserving updates for lists.
parent
324db1dc
Pipeline
#2722
passed with stage
in 9 minutes and 25 seconds
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1
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-14
algebra/list.v
algebra/list.v
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algebra/list.v
View file @
ca8e04a3
...
...
@@ -338,25 +338,64 @@ Section properties.
Proof
.
by
rewrite
!
persistent_total
list_core_singletonM
=>
->.
Qed
.
(* Update *)
Lemma
list_
middle_updateP
(
P
:
A
→
Prop
)
(
Q
:
list
A
→
Prop
)
l1
x
l2
:
x
~~>
:
P
→
(
∀
y
,
P
y
→
Q
(
l1
++
y
::
l2
))
→
l1
++
x
::
l2
~~>
:
Q
.
Lemma
list_
singleton_updateP
(
P
:
A
→
Prop
)
(
Q
:
list
A
→
Prop
)
x
:
x
~~>
:
P
→
(
∀
y
,
P
y
→
Q
[
y
])
→
[
x
]
~~>
:
Q
.
Proof
.
intros
Hx
%
option_updateP'
HP
.
apply
cmra_total_updateP
=>
n
mf
;
rewrite
list_lookup_validN
=>
Hm
.
destruct
(
Hx
n
(
Some
(
mf
!!
length
l1
)))
as
([
y
|]&
H1
&
H2
)
;
simpl
in
*
;
try
done
.
{
move
:
(
Hm
(
length
l1
)).
by
rewrite
list_lookup_op
list_lookup_middle
.
}
exists
(
l1
++
y
::
l2
)
;
split
;
auto
.
rewrite
!
cmra_total_updateP
=>
Hup
HQ
n
lf
/
list_lookup_validN
Hv
.
destruct
(
Hup
n
(
from_option
id
∅
(
lf
!!
0
)))
as
(
y
&?&
Hv'
).
{
move
:
(
Hv
0
).
by
destruct
lf
;
rewrite
/=
?right_id
.
}
exists
[
y
]
;
split
;
first
by
auto
.
apply
list_lookup_validN
=>
i
.
destruct
(
lt_eq_lt_dec
i
(
length
l1
))
as
[[?|?]|?]
;
subst
.
-
move
:
(
Hm
i
)
;
by
rewrite
!
list_lookup_op
!
lookup_app_l
.
-
by
rewrite
list_lookup_op
list_lookup_middle
.
-
move
:
(
Hm
i
).
rewrite
!(
cons_middle
_
l1
l2
)
!
assoc
.
rewrite
!
list_lookup_op
!
lookup_app_r
!
app_length
//=
;
lia
.
move
:
(
Hv
i
)
Hv'
.
by
destruct
i
,
lf
;
rewrite
/=
?right_id
.
Qed
.
Lemma
list_singleton_updateP'
(
P
:
A
→
Prop
)
x
:
x
~~>
:
P
→
[
x
]
~~>
:
λ
k
,
∃
y
,
k
=
[
y
]
∧
P
y
.
Proof
.
eauto
using
list_singleton_updateP
.
Qed
.
Lemma
list_singleton_update
x
y
:
x
~~>
y
→
[
x
]
~~>
[
y
].
Proof
.
rewrite
!
cmra_update_updateP
;
eauto
using
list_singleton_updateP
with
subst
.
Qed
.
Lemma
app_updateP
(
P1
P2
Q
:
list
A
→
Prop
)
l1
l2
:
l1
~~>
:
P1
→
l2
~~>
:
P2
→
(
∀
k1
k2
,
P1
k1
→
P2
k2
→
length
l1
=
length
k1
∧
Q
(
k1
++
k2
))
→
l1
++
l2
~~>
:
Q
.
Proof
.
rewrite
!
cmra_total_updateP
=>
Hup1
Hup2
HQ
n
lf
.
rewrite
list_op_app
app_validN
=>
-[??].
destruct
(
Hup1
n
(
take
(
length
l1
)
lf
))
as
(
k1
&?&?)
;
auto
.
destruct
(
Hup2
n
(
drop
(
length
l1
)
lf
))
as
(
k2
&?&?)
;
auto
.
exists
(
k1
++
k2
).
rewrite
list_op_app
app_validN
.
by
destruct
(
HQ
k1
k2
)
as
[<-
?].
Qed
.
Lemma
app_update
l1
l2
k1
k2
:
length
l1
=
length
k1
→
l1
~~>
k1
→
l2
~~>
k2
→
l1
++
l2
~~>
k1
++
k2
.
Proof
.
rewrite
!
cmra_update_updateP
;
eauto
using
app_updateP
with
subst
.
Qed
.
Lemma
cons_updateP
(
P1
:
A
→
Prop
)
(
P2
Q
:
list
A
→
Prop
)
x
l
:
x
~~>
:
P1
→
l
~~>
:
P2
→
(
∀
y
k
,
P1
y
→
P2
k
→
Q
(
y
::
k
))
→
x
::
l
~~>
:
Q
.
Proof
.
intros
.
eapply
(
app_updateP
_
_
_
[
x
])
;
naive_solver
eauto
using
list_singleton_updateP'
.
Qed
.
Lemma
cons_updateP'
(
P1
:
A
→
Prop
)
(
P2
:
list
A
→
Prop
)
x
l
:
x
~~>
:
P1
→
l
~~>
:
P2
→
x
::
l
~~>
:
λ
k
,
∃
y
k'
,
k
=
y
::
k'
∧
P1
y
∧
P2
k'
.
Proof
.
eauto
10
using
cons_updateP
.
Qed
.
Lemma
cons_update
x
y
l
k
:
x
~~>
y
→
l
~~>
k
→
x
::
l
~~>
y
::
k
.
Proof
.
rewrite
!
cmra_update_updateP
;
eauto
using
cons_updateP
with
subst
.
Qed
.
Lemma
list_middle_updateP
(
P
:
A
→
Prop
)
(
Q
:
list
A
→
Prop
)
l1
x
l2
:
x
~~>
:
P
→
(
∀
y
,
P
y
→
Q
(
l1
++
y
::
l2
))
→
l1
++
x
::
l2
~~>
:
Q
.
Proof
.
intros
.
eapply
app_updateP
.
-
by
apply
cmra_update_updateP
.
-
by
eapply
cons_updateP'
,
cmra_update_updateP
.
-
naive_solver
.
Qed
.
Lemma
list_middle_update
l1
l2
x
y
:
x
~~>
y
→
l1
++
x
::
l2
~~>
l1
++
y
::
l2
.
Proof
.
rewrite
!
cmra_update_updateP
=>
H
;
eauto
using
list_middle_updateP
with
subst
.
rewrite
!
cmra_update_updateP
=>
?
;
eauto
using
list_middle_updateP
with
subst
.
Qed
.
Lemma
list_middle_local_update
l1
l2
x
y
ml
:
...
...
@@ -379,7 +418,6 @@ Section properties.
+
rewrite
!(
cons_middle
_
l1
l2
)
!
assoc
.
rewrite
!
list_lookup_opM
!
lookup_app_r
!
app_length
//=
;
lia
.
Qed
.
Lemma
list_singleton_local_update
i
x
y
ml
:
x
~l
~>
y
@
ml
≫
=
(!!
i
)
→
{[
i
:
=
x
]}
~l
~>
{[
i
:
=
y
]}
@
ml
.
Proof
.
intros
;
apply
list_middle_local_update
.
by
rewrite
replicate_length
.
Qed
.
...
...
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