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7c762be1
Commit
7c762be1
authored
Sep 27, 2016
by
Robbert Krebbers
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CMRA structure on uPred.
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algebra/upred.v
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7c762be1
...
...
@@ -1486,3 +1486,66 @@ Hint Resolve sep_elim_l' sep_elim_r' sep_mono : I.
Hint
Immediate
True_intro
False_elim
:
I
.
Hint
Immediate
iff_refl
eq_refl'
:
I
.
End
uPred
.
(* CMRA structure on uPred *)
Section
cmra
.
Context
{
M
:
ucmraT
}.
Instance
uPred_valid'
:
Valid
(
uPred
M
)
:
=
λ
P
,
∀
n
x
,
✓
{
n
}
x
→
P
n
x
.
Instance
uPred_validN'
:
ValidN
(
uPred
M
)
:
=
λ
n
P
,
∀
n'
x
,
n'
≤
n
→
✓
{
n'
}
x
→
P
n'
x
.
Instance
uPred_op
:
Op
(
uPred
M
)
:
=
uPred_sep
.
Instance
uPred_pcore
:
PCore
(
uPred
M
)
:
=
λ
_
,
Some
True
%
I
.
Instance
uPred_validN_ne
n
:
Proper
(
dist
n
==>
iff
)
(
uPred_validN'
n
).
Proof
.
intros
P
Q
HPQ
;
split
=>
H
n'
x
??
;
by
apply
HPQ
,
H
.
Qed
.
Lemma
uPred_validN_alt
n
(
P
:
uPred
M
)
:
✓
{
n
}
P
→
P
≡
{
n
}
≡
True
%
I
.
Proof
.
uPred
.
unseal
=>
HP
;
split
=>
n'
x
??
;
split
;
[
done
|].
intros
_
.
by
apply
HP
.
Qed
.
Lemma
uPred_cmra_validN_op_l
n
P
Q
:
✓
{
n
}
(
P
★
Q
)%
I
→
✓
{
n
}
P
.
Proof
.
uPred
.
unseal
.
intros
HPQ
n'
x
??.
destruct
(
HPQ
n'
x
)
as
(
x1
&
x2
&->&?&?)
;
auto
.
eapply
uPred_mono
with
x1
;
eauto
using
cmra_includedN_l
.
Qed
.
Definition
uPred_cmra_mixin
:
CMRAMixin
(
uPred
M
).
Proof
.
apply
cmra_total_mixin
;
try
apply
_
||
by
eauto
.
-
intros
n
P
Q
??.
by
cofe_subst
.
-
intros
P
;
split
.
+
intros
HP
n
n'
x
?.
apply
HP
.
+
intros
HP
n
x
.
by
apply
(
HP
n
).
-
intros
n
P
HP
n'
x
?.
apply
HP
;
auto
.
-
intros
P
.
by
rewrite
left_id
.
-
intros
P
Q
_
.
exists
True
%
I
.
by
rewrite
left_id
.
-
intros
n
P
Q
.
apply
uPred_cmra_validN_op_l
.
-
intros
n
P
Q1
Q2
HP
HPQ
.
exists
True
%
I
,
P
;
split_and
!.
+
by
rewrite
left_id
.
+
move
:
HP
;
by
rewrite
HPQ
=>
/
uPred_cmra_validN_op_l
/
uPred_validN_alt
.
+
move
:
HP
;
rewrite
HPQ
=>
/
uPred_cmra_validN_op_l
/
uPred_validN_alt
=>
->.
by
rewrite
left_id
.
Qed
.
Canonical
Structure
uPredR
:
=
CMRAT
(
uPred
M
)
uPred_cofe_mixin
uPred_cmra_mixin
.
Instance
uPred_empty
:
Empty
(
uPred
M
)
:
=
True
%
I
.
Definition
uPred_ucmra_mixin
:
UCMRAMixin
(
uPred
M
).
Proof
.
split
;
last
done
.
-
by
rewrite
/
empty
/
uPred_empty
uPred_pure_eq
.
-
intros
P
.
by
rewrite
left_id
.
Qed
.
Canonical
Structure
uPredUR
:
=
UCMRAT
(
uPred
M
)
uPred_cofe_mixin
uPred_cmra_mixin
uPred_ucmra_mixin
.
End
cmra
.
Arguments
uPredR
:
clear
implicits
.
Arguments
uPredUR
:
clear
implicits
.
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