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ad561840
Commit
ad561840
authored
Feb 08, 2016
by
Robbert Krebbers
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Frame preserving updates related to ∅.
parent
34dcb910
Changes
2
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2 changed files
with
18 additions
and
2 deletions
+18
-2
algebra/cmra.v
algebra/cmra.v
+13
-0
algebra/option.v
algebra/option.v
+5
-2
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algebra/cmra.v
View file @
ad561840
...
...
@@ -336,6 +336,11 @@ Lemma cmra_updateP_compose (P Q : A → Prop) x :
Proof
.
intros
Hx
Hy
z
n
?.
destruct
(
Hx
z
n
)
as
(
y
&?&?)
;
auto
.
by
apply
(
Hy
y
).
Qed
.
Lemma
cmra_updateP_compose_l
(
Q
:
A
→
Prop
)
x
y
:
x
~~>
y
→
y
~~>
:
Q
→
x
~~>
:
Q
.
Proof
.
rewrite
cmra_update_updateP
.
intros
;
apply
cmra_updateP_compose
with
(
y
=)
;
intros
;
subst
;
auto
.
Qed
.
Lemma
cmra_updateP_weaken
(
P
Q
:
A
→
Prop
)
x
:
x
~~>
:
P
→
(
∀
y
,
P
y
→
Q
y
)
→
x
~~>
:
Q
.
Proof
.
eauto
using
cmra_updateP_compose
,
cmra_updateP_id
.
Qed
.
...
...
@@ -355,6 +360,14 @@ Lemma cmra_update_op x1 x2 y1 y2 : x1 ~~> y1 → x2 ~~> y2 → x1 ⋅ x2 ~~> y1
Proof
.
rewrite
!
cmra_update_updateP
;
eauto
using
cmra_updateP_op
with
congruence
.
Qed
.
Section
identity_updates
.
Context
`
{
Empty
A
,
!
CMRAIdentity
A
}.
Lemma
cmra_update_empty
x
:
x
~~>
∅
.
Proof
.
intros
z
n
;
rewrite
left_id
;
apply
cmra_validN_op_r
.
Qed
.
Lemma
cmra_update_empty_alt
y
:
∅
~~>
y
↔
∀
x
,
x
~~>
y
.
Proof
.
split
;
[
intros
;
transitivity
∅
|]
;
auto
using
cmra_update_empty
.
Qed
.
End
identity_updates
.
End
cmra
.
Hint
Extern
0
(
_
≼
{
0
}
_
)
=>
apply
cmra_includedN_0
.
...
...
algebra/option.v
View file @
ad561840
...
...
@@ -154,8 +154,12 @@ Lemma option_update x y : x ~~> y → Some x ~~> Some y.
Proof
.
rewrite
!
cmra_update_updateP
;
eauto
using
option_updateP
with
congruence
.
Qed
.
Lemma
option_update_None
`
{
Empty
A
,
!
CMRAIdentity
A
}
:
∅
~~>
Some
∅
.
Proof
.
intros
[
x
|]
n
?
;
rewrite
/
op
/
cmra_op
/
validN
/
cmra_validN
/=
?left_id
;
auto
using
cmra_empty_valid
.
Qed
.
End
cmra
.
Arguments
optionRA
:
clear
implicits
.
(** Functor *)
...
...
@@ -189,4 +193,3 @@ Next Obligation.
intros
Σ
A
B
C
f
g
x
.
rewrite
/=
-
option_fmap_compose
.
apply
option_fmap_setoid_ext
=>
y
;
apply
ifunctor_map_compose
.
Qed
.
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