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2ccdb104
Commit
2ccdb104
authored
Feb 02, 2016
by
Robbert Krebbers
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Basic properties of frame preserving updates and those for products.
parent
4882ecf8
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modures/cmra.v
modures/cmra.v
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modures/cmra.v
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2ccdb104
...
...
@@ -318,6 +318,15 @@ Proof.
*
by
intros
Hx
z
?
;
exists
y
;
split
;
[
done
|
apply
(
Hx
z
)].
*
by
intros
Hx
z
n
?
;
destruct
(
Hx
z
n
)
as
(?&<-&?).
Qed
.
Lemma
ra_updateP_id
(
P
:
A
→
Prop
)
x
:
P
x
→
x
⇝
:
P
.
Proof
.
by
intros
?
z
n
?
;
exists
x
.
Qed
.
Lemma
ra_updateP_compose
(
P
Q
:
A
→
Prop
)
x
:
x
⇝
:
P
→
(
∀
y
,
P
y
→
y
⇝
:
Q
)
→
x
⇝
:
Q
.
Proof
.
intros
Hx
Hy
z
n
?.
destruct
(
Hx
z
n
)
as
(
y
&?&?)
;
auto
.
by
apply
(
Hy
y
).
Qed
.
Lemma
ra_updateP_weaken
(
P
Q
:
A
→
Prop
)
x
:
x
⇝
:
P
→
(
∀
y
,
P
y
→
Q
y
)
→
x
⇝
:
Q
.
Proof
.
eauto
using
ra_updateP_compose
,
ra_updateP_id
.
Qed
.
End
cmra
.
Hint
Extern
0
(
_
≼
{
0
}
_
)
=>
apply
cmra_includedN_0
.
...
...
@@ -384,14 +393,14 @@ Section discrete.
Qed
.
Definition
discreteRA
:
cmraT
:
=
CMRAT
(
cofe_mixin
A
)
discrete_cmra_mixin
discrete_extend_mixin
.
Lemma
discrete_updateP
(
x
:
A
)
(
P
:
A
→
Prop
)
`
{!
Inhabited
(
sig
P
)}
:
(
∀
z
,
✓
(
x
⋅
z
)
→
∃
y
,
P
y
∧
✓
(
y
⋅
z
))
→
(
x
:
discreteRA
)
⇝
:
P
.
Lemma
discrete_updateP
(
x
:
discreteR
A
)
(
P
:
A
→
Prop
)
`
{!
Inhabited
(
sig
P
)}
:
(
∀
z
,
✓
(
x
⋅
z
)
→
∃
y
,
P
y
∧
✓
(
y
⋅
z
))
→
x
⇝
:
P
.
Proof
.
intros
Hvalid
z
[|
n
]
;
[|
apply
Hvalid
].
by
destruct
(
_
:
Inhabited
(
sig
P
))
as
[[
y
?]]
;
exists
y
.
Qed
.
Lemma
discrete_update
(
x
y
:
A
)
:
(
∀
z
,
✓
(
x
⋅
z
)
→
✓
(
y
⋅
z
))
→
(
x
:
discreteRA
)
⇝
y
.
Lemma
discrete_update
(
x
y
:
discreteR
A
)
:
(
∀
z
,
✓
(
x
⋅
z
)
→
✓
(
y
⋅
z
))
→
x
⇝
y
.
Proof
.
intros
Hvalid
z
[|
n
]
;
[
done
|
apply
Hvalid
].
Qed
.
End
discrete
.
...
...
@@ -465,6 +474,15 @@ Section prod.
*
by
split
;
rewrite
/=
left_id
.
*
by
intros
?
[??]
;
split
;
apply
(
timeless
_
).
Qed
.
Lemma
prod_update
x
y
:
x
.
1
⇝
y
.
1
→
x
.
2
⇝
y
.
2
→
x
⇝
y
.
Proof
.
intros
??
z
n
[??]
;
split
;
simpl
in
*
;
auto
.
Qed
.
Lemma
prod_updateP
(
P
:
A
*
B
→
Prop
)
P1
P2
x
:
x
.
1
⇝
:
P1
→
x
.
2
⇝
:
P2
→
(
∀
y
,
P1
(
y
.
1
)
→
P2
(
y
.
2
)
→
P
y
)
→
x
⇝
:
P
.
Proof
.
intros
Hx1
Hx2
HP
z
n
[??]
;
simpl
in
*.
destruct
(
Hx1
(
z
.
1
)
n
)
as
(
a
&?&?),
(
Hx2
(
z
.
2
)
n
)
as
(
b
&?&?)
;
auto
.
exists
(
a
,
b
)
;
repeat
split
;
auto
.
Qed
.
End
prod
.
Arguments
prodRA
:
clear
implicits
.
...
...
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