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Iris
Iris
Commits
51b04b25
Commit
51b04b25
authored
Nov 16, 2015
by
Robbert Krebbers
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RAs with empty (global unit) element.
parent
8dc73363
Changes
4
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4 changed files
with
30 additions
and
9 deletions
+30
-9
iris/auth.v
iris/auth.v
+5
-2
iris/cmra.v
iris/cmra.v
+1
-1
iris/excl.v
iris/excl.v
+8
-4
iris/ra.v
iris/ra.v
+16
-2
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iris/auth.v
View file @
51b04b25
...
...
@@ -9,6 +9,7 @@ Arguments own {_} _.
Notation
"∘ x"
:
=
(
Auth
ExclUnit
x
)
(
at
level
20
).
Notation
"∙ x"
:
=
(
Auth
(
Excl
x
)
∅
)
(
at
level
20
).
Instance
auth_empty
`
{
Empty
A
}
:
Empty
(
auth
A
)
:
=
Auth
∅
∅
.
Instance
auth_valid
`
{
Valid
A
,
Included
A
}
:
Valid
(
auth
A
)
:
=
λ
x
,
valid
(
authorative
x
)
∧
excl_above
(
own
x
)
(
authorative
x
).
Instance
auth_equiv
`
{
Equiv
A
}
:
Equiv
(
auth
A
)
:
=
λ
x
y
,
...
...
@@ -44,7 +45,9 @@ Proof.
by
apply
excl_above_weaken
with
(
own
x
⋅
own
y
)
(
authorative
x
⋅
authorative
y
)
;
try
apply
ra_included_l
.
*
split
;
simpl
;
apply
ra_included_l
.
*
by
intros
??
[??]
;
split
;
simpl
;
apply
ra_op_
difference
.
*
by
intros
??
[??]
;
split
;
simpl
;
apply
ra_op_
minus
.
Qed
.
Instance
auth_ra_empty
`
{
RA
A
,
Empty
A
,
!
RAEmpty
A
}
:
RAEmpty
(
auth
A
).
Proof
.
split
.
done
.
by
intros
x
;
constructor
;
simpl
;
rewrite
(
left_id
_
_
).
Qed
.
Lemma
auth_frag_op
`
{
RA
A
}
a
b
:
∘
(
a
⋅
b
)
≡
∘
a
⋅
∘
b
.
Proof
.
done
.
Qed
.
\ No newline at end of file
Proof
.
done
.
Qed
.
iris/cmra.v
View file @
51b04b25
...
...
@@ -24,7 +24,7 @@ Class CMRA A `{Equiv A, Compl A,
cmra_unit_weaken
x
y
:
unit
x
≼
unit
(
x
⋅
y
)
;
cmra_valid_op_l
n
x
y
:
validN
n
(
x
⋅
y
)
→
validN
n
x
;
cmra_included_l
x
y
:
x
≼
x
⋅
y
;
cmra_op_
difference
x
y
:
x
≼
y
→
x
⋅
y
⩪
x
≡
y
cmra_op_
minus
x
y
:
x
≼
y
→
x
⋅
y
⩪
x
≡
y
}.
Class
CMRAExtend
A
`
{
Equiv
A
,
Dist
A
,
Op
A
,
ValidN
A
}
:
=
cmra_extend_op
x
y1
y2
n
:
...
...
iris/excl.v
View file @
51b04b25
...
...
@@ -4,20 +4,22 @@ Local Arguments included _ _ !_ !_ /.
Inductive
excl
(
A
:
Type
)
:
=
|
Excl
:
A
→
excl
A
|
ExclUnit
:
excl
A
|
ExclUnit
:
Empty
(
excl
A
)
|
ExclBot
:
excl
A
.
Arguments
Excl
{
_
}
_
.
Arguments
ExclUnit
{
_
}.
Arguments
ExclBot
{
_
}.
Existing
Instance
ExclUnit
.
Inductive
excl_equiv
`
{
Equiv
A
}
:
Equiv
(
excl
A
)
:
=
|
Excl_equiv
(
x
y
:
A
)
:
x
≡
y
→
Excl
x
≡
Excl
y
|
ExclUnit_equiv
:
ExclUnit
≡
ExclUnit
|
ExclUnit_equiv
:
∅
≡
∅
|
ExclBot_equiv
:
ExclBot
≡
ExclBot
.
Existing
Instance
excl_equiv
.
Instance
excl_valid
{
A
}
:
Valid
(
excl
A
)
:
=
λ
x
,
match
x
with
Excl
_
|
ExclUnit
=>
True
|
ExclBot
=>
False
end
.
Instance
excl_unit
{
A
}
:
Unit
(
excl
A
)
:
=
λ
_
,
ExclUnit
.
Instance
excl_empty
{
A
}
:
Empty
(
excl
A
)
:
=
ExclUnit
.
Instance
excl_unit
{
A
}
:
Unit
(
excl
A
)
:
=
λ
_
,
∅
.
Instance
excl_op
{
A
}
:
Op
(
excl
A
)
:
=
λ
x
y
,
match
x
,
y
with
|
Excl
x
,
ExclUnit
|
ExclUnit
,
Excl
x
=>
Excl
x
...
...
@@ -60,6 +62,8 @@ Proof.
*
by
intros
[?|
|]
[?|
|]
;
simpl
;
try
constructor
.
*
by
intros
[?|
|]
[?|
|]
?
;
try
constructor
.
Qed
.
Instance
excl_empty_ra
`
{
Equiv
A
,
!
Equivalence
(@
equiv
A
_
)}
:
RAEmpty
(
excl
A
).
Proof
.
split
.
done
.
by
intros
[].
Qed
.
Lemma
excl_update
{
A
}
(
x
:
A
)
y
:
valid
y
→
Excl
x
⇝
y
.
Proof
.
by
destruct
y
;
intros
?
[?|
|].
Qed
.
...
...
@@ -73,4 +77,4 @@ Section excl_above.
destruct
x
as
[
a'
|
|],
y
as
[
b'
|
|]
;
simpl
;
intros
??
;
try
done
.
by
intros
Hab
;
rewrite
Hab
;
transitivity
b
.
Qed
.
End
excl_above
.
\ No newline at end of file
End
excl_above
.
iris/ra.v
View file @
51b04b25
...
...
@@ -37,7 +37,11 @@ Class RA A `{Equiv A, Valid A, Unit A, Op A, Included A, Minus A} : Prop := {
ra_unit_weaken
x
y
:
unit
x
≼
unit
(
x
⋅
y
)
;
ra_valid_op_l
x
y
:
valid
(
x
⋅
y
)
→
valid
x
;
ra_included_l
x
y
:
x
≼
x
⋅
y
;
ra_op_difference
x
y
:
x
≼
y
→
x
⋅
y
⩪
x
≡
y
ra_op_minus
x
y
:
x
≼
y
→
x
⋅
y
⩪
x
≡
y
}.
Class
RAEmpty
A
`
{
Equiv
A
,
Valid
A
,
Op
A
,
Empty
A
}
:
Prop
:
=
{
ra_empty_valid
:
valid
∅
;
ra_empty_l
:
>
LeftId
(
≡
)
∅
(
⋅
)
}.
(** Updates *)
...
...
@@ -72,7 +76,7 @@ Proof. by rewrite (commutative _ x), ra_unit_l. Qed.
(** ** Order *)
Lemma
ra_included_spec
x
y
:
x
≼
y
↔
∃
z
,
y
≡
x
⋅
z
.
Proof
.
split
;
[
by
exists
(
y
⩪
x
)
;
rewrite
ra_op_
difference
|].
split
;
[
by
exists
(
y
⩪
x
)
;
rewrite
ra_op_
minus
|].
intros
[
z
Hz
]
;
rewrite
Hz
;
apply
ra_included_l
.
Qed
.
Global
Instance
ra_included_proper'
:
Proper
((
≡
)
==>
(
≡
)
==>
iff
)
(
≼
).
...
...
@@ -106,4 +110,14 @@ Qed.
(** ** Properties of [(⇝)] relation *)
Global
Instance
ra_update_preorder
:
PreOrder
ra_update
.
Proof
.
split
.
by
intros
x
y
.
intros
x
y
y'
??
z
?
;
naive_solver
.
Qed
.
(** ** RAs with empty element *)
Context
`
{
Empty
A
,
!
RAEmpty
A
}.
Global
Instance
ra_empty_r
:
RightId
(
≡
)
∅
(
⋅
).
Proof
.
by
intros
x
;
rewrite
(
commutative
op
),
(
left_id
_
_
).
Qed
.
Lemma
ra_unit_empty
x
:
unit
∅
≡
∅
.
Proof
.
by
rewrite
<-(
ra_unit_l
∅
)
at
2
;
rewrite
(
right_id
_
_
).
Qed
.
Lemma
ra_empty_least
x
:
∅
≼
x
.
Proof
.
by
rewrite
ra_included_spec
;
exists
x
;
rewrite
(
left_id
_
_
).
Qed
.
End
ra
.
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