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George Pirlea
Iris
Commits
ebb452d3
Commit
ebb452d3
authored
Jan 26, 2019
by
Robbert Krebbers
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Alternative definition of basic updates.
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_CoqProject
_CoqProject
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theories/base_logic/bupd_alt.v
theories/base_logic/bupd_alt.v
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_CoqProject
View file @
ebb452d3
...
...
@@ 52,6 +52,7 @@ theories/base_logic/bi.v
theories/base_logic/derived.v
theories/base_logic/proofmode.v
theories/base_logic/base_logic.v
theories/base_logic/bupd_alt.v
theories/base_logic/lib/iprop.v
theories/base_logic/lib/own.v
theories/base_logic/lib/saved_prop.v
...
...
theories/base_logic/bupd_alt.v
0 → 100644
View file @
ebb452d3
From
iris
.
base_logic
Require
Export
base_logic
.
From
iris
.
proofmode
Require
Import
tactics
.
(** This file contains an alternative version of basic updates, that is
expression in terms of just the plain modality [■]. *)
Definition
bupd_alt
`
{
BiPlainly
PROP
}
(
P
:
PROP
)
:
PROP
:
=
(
∀
R
,
(
P

∗
■
R
)

∗
■
R
)%
I
.
(** This definition is stated for any BI with a plain modality. The above
definition is akin to the continuation monad, where one should think of [■ R]
being the final result that one wants to get out of the basic update in the end
of the day (via [bupd_alt (■ P) ⊢ ■ P]).
We show that:
1. [bupd_alt] enjoys the usual rules of the basic update modality.
2. [bupd_alt] entails any other modality that enjoys the laws of a basic update
modality (see [bupd_bupd_alt]).
3. The ordinary basic update modality [==>] on [uPred] entails [bupd_alt]
(see [bupd_alt_bupd]). This result is proven in the model of [uPred].
The first two points are shown for any BI with a plain modality. *)
Section
bupd_alt
.
Context
`
{
BiPlainly
PROP
}.
Implicit
Types
P
Q
R
:
PROP
.
Notation
bupd_alt
:
=
(@
bupd_alt
PROP
_
).
Global
Instance
bupd_alt_ne
:
NonExpansive
bupd_alt
.
Proof
.
solve_proper
.
Qed
.
Global
Instance
bupd_alt_proper
:
Proper
((
≡
)
==>
(
≡
))
bupd_alt
.
Proof
.
solve_proper
.
Qed
.
Global
Instance
bupd_alt_mono'
:
Proper
((
⊢
)
==>
(
⊢
))
bupd_alt
.
Proof
.
solve_proper
.
Qed
.
Global
Instance
bupd_alt_flip_mono'
:
Proper
(
flip
(
⊢
)
==>
flip
(
⊢
))
bupd_alt
.
Proof
.
solve_proper
.
Qed
.
(** The laws of the basic update modality hold *)
Lemma
bupd_alt_intro
P
:
P
⊢
bupd_alt
P
.
Proof
.
iIntros
"HP"
(
R
)
"H"
.
by
iApply
"H"
.
Qed
.
Lemma
bupd_alt_mono
P
Q
:
(
P
⊢
Q
)
→
bupd_alt
P
⊢
bupd_alt
Q
.
Proof
.
by
intros
>.
Qed
.
Lemma
bupd_alt_trans
P
:
bupd_alt
(
bupd_alt
P
)
⊢
bupd_alt
P
.
Proof
.
iIntros
"HP"
(
R
)
"H"
.
iApply
"HP"
.
iIntros
"HP"
.
by
iApply
"HP"
.
Qed
.
Lemma
bupd_alt_frame_r
P
Q
:
bupd_alt
P
∗
Q
⊢
bupd_alt
(
P
∗
Q
).
Proof
.
iIntros
"[HP HQ]"
(
R
)
"H"
.
iApply
"HP"
.
iIntros
"HP"
.
iApply
(
"H"
with
"[$]"
).
Qed
.
Lemma
bupd_alt_plainly
P
:
bupd_alt
(
■
P
)
⊢
■
P
.
Proof
.
iIntros
"H"
.
iApply
(
"H"
$!
P
with
"[]"
)
;
auto
.
Qed
.
(** Any modality conforming with [BiBUpdPlainly] entails the alternative
definition *)
Lemma
bupd_bupd_alt
`
{!
BiBUpd
PROP
,
BiBUpdPlainly
PROP
}
P
:
(==>
P
)
⊢
bupd_alt
P
.
Proof
.
iIntros
"HP"
(
R
)
"H"
.
by
iMod
(
"H"
with
"HP"
)
as
"?"
.
Qed
.
(** We get the usual rule for frame preserving updates if we have an [own]
connective satisfying the following rule w.r.t. interaction with plainly. *)
Context
{
M
:
ucmraT
}
(
own
:
M
→
PROP
).
Context
(
own_updateP_plainly
:
∀
x
Φ
R
,
x
~~>
:
Φ
→
own
x
∗
(
∀
y
,
⌜Φ
y
⌝

∗
own
y

∗
■
R
)
⊢
■
R
).
Lemma
own_updateP
x
(
Φ
:
M
→
Prop
)
:
x
~~>
:
Φ
→
own
x
⊢
bupd_alt
(
∃
y
,
⌜Φ
y
⌝
∧
own
y
).
Proof
.
iIntros
(
Hup
)
"Hx"
;
iIntros
(
R
)
"H"
.
iApply
(
own_updateP_plainly
with
"[$Hx H]"
)
;
first
done
.
iIntros
(
y
?)
"Hy"
.
iApply
"H"
;
auto
.
Qed
.
End
bupd_alt
.
(** The alternative definition entails the ordinary basic update *)
Lemma
bupd_alt_bupd
{
M
}
(
P
:
uPred
M
)
:
bupd_alt
P
⊢
==>
P
.
Proof
.
rewrite
/
bupd_alt
.
uPred
.
unseal
;
split
=>
n
x
Hx
H
k
y
?
Hxy
.
unshelve
refine
(
H
{
uPred_holds
k
_
:
=
∃
x'
:
M
,
✓
{
k
}
(
x'
⋅
y
)
∧
P
k
x'
}
k
y
_
_
_
).

intros
n1
n2
x1
x2
(
z
&?&?)
_
?.
eauto
using
cmra_validN_le
,
uPred_mono
.

done
.

done
.

intros
k'
z
??
HP
.
exists
z
.
by
rewrite
(
comm
op
).
Qed
.
Lemma
bupd_alt_bupd_iff
{
M
}
(
P
:
uPred
M
)
:
bupd_alt
P
⊣
⊢
==>
P
.
Proof
.
apply
(
anti_symm
_
).
apply
bupd_alt_bupd
.
apply
bupd_bupd_alt
.
Qed
.
(** The law about the interaction between [uPred_ownM] and plainly holds. *)
Lemma
ownM_updateP
{
M
:
ucmraT
}
x
(
Φ
:
M
→
Prop
)
(
R
:
uPred
M
)
:
x
~~>
:
Φ
→
uPred_ownM
x
∗
(
∀
y
,
⌜Φ
y
⌝

∗
uPred_ownM
y

∗
■
R
)
⊢
■
R
.
Proof
.
uPred
.
unseal
=>
Hup
;
split
;
intros
n
z
Hv
(?&
z2
&?&[
z1
?]&
HR
)
;
ofe_subst
.
destruct
(
Hup
n
(
Some
(
z1
⋅
z2
)))
as
(
y
&?&?)
;
simpl
in
*.
{
by
rewrite
assoc
.
}
refine
(
HR
y
n
z1
_
_
_
n
y
_
_
_
)
;
auto
.

rewrite
comm
.
by
eapply
cmra_validN_op_r
.

by
rewrite
(
comm
_
_
y
)
(
comm
_
z2
).

apply
(
reflexivity
(
R
:
=
includedN
_
)).
Qed
.
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