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b13debed
Commit
b13debed
authored
Jan 15, 2016
by
Robbert Krebbers
Browse files
Cleanup agree and prove some additional properties.
parent
69407372
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modures/agree.v
modures/agree.v
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modures/agree.v
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b13debed
...
...
@@ 51,6 +51,11 @@ Proof.
Qed
.
Canonical
Structure
agreeC
:
=
CofeT
agree_cofe_mixin
.
Lemma
agree_car_ne
(
x
y
:
agree
A
)
n
:
✓
{
n
}
x
→
x
={
n
}=
y
→
x
n
={
n
}=
y
n
.
Proof
.
by
intros
[??]
Hxy
;
apply
Hxy
.
Qed
.
Lemma
agree_cauchy
(
x
:
agree
A
)
n
i
:
✓
{
n
}
x
→
i
≤
n
→
x
i
={
i
}=
x
n
.
Proof
.
by
intros
[?
Hx
]
;
apply
Hx
.
Qed
.
Program
Instance
agree_op
:
Op
(
agree
A
)
:
=
λ
x
y
,
{
agree_car
:
=
x
;
agree_is_valid
n
:
=
agree_is_valid
x
n
∧
agree_is_valid
y
n
∧
x
={
n
}=
y
}.
...
...
@@ 62,6 +67,12 @@ Instance: Commutative (≡) (@op (agree A) _).
Proof
.
intros
x
y
;
split
;
[
naive_solver

by
intros
n
(?&?&
Hxy
)
;
apply
Hxy
].
Qed
.
Definition
agree_idempotent
(
x
:
agree
A
)
:
x
⋅
x
≡
x
.
Proof
.
split
;
naive_solver
.
Qed
.
Instance
:
∀
n
:
nat
,
Proper
(
dist
n
==>
impl
)
(@
validN
(
agree
A
)
_
n
).
Proof
.
intros
n
x
y
Hxy
[?
Hx
]
;
split
;
[
by
apply
Hxy

intros
n'
?].
rewrite
(
proj2
Hxy
n'
)
1
?(
Hx
n'
)
;
eauto
using
agree_valid_le
.
by
apply
dist_le
with
n
;
try
apply
Hxy
.
Qed
.
Instance
:
∀
x
:
agree
A
,
Proper
(
dist
n
==>
dist
n
)
(
op
x
).
Proof
.
intros
n
x
y1
y2
[
Hy'
Hy
]
;
split
;
[
done
].
...
...
@@ 88,9 +99,6 @@ Qed.
Definition
agree_cmra_mixin
:
CMRAMixin
(
agree
A
).
Proof
.
split
;
try
(
apply
_

done
).
*
intros
n
x
y
Hxy
[?
Hx
]
;
split
;
[
by
apply
Hxy

intros
n'
?].
rewrite
(
proj2
Hxy
n'
)
1
?(
Hx
n'
)
;
eauto
using
agree_valid_le
.
by
apply
dist_le
with
n
;
try
apply
Hxy
.
*
by
intros
n
x1
x2
Hx
y1
y2
Hy
.
*
intros
x
;
split
;
[
apply
agree_valid_0
].
by
intros
n'
;
rewrite
Nat
.
le_0_r
;
intros
>.
...
...
@@ 101,14 +109,18 @@ Proof.
*
by
intros
x
y
n
[(?&?&?)
?].
*
by
intros
x
y
n
;
rewrite
agree_includedN
.
Qed
.
Lemma
agree_op_inv
(
x
y1
y2
:
agree
A
)
n
:
✓
{
n
}
x
→
x
={
n
}=
y1
⋅
y2
→
y1
={
n
}=
y2
.
Proof
.
by
intros
[??]
Hxy
;
apply
Hxy
.
Qed
.
Lemma
agree_op_inv
(
x1
x2
:
agree
A
)
n
:
✓
{
n
}
(
x1
⋅
x2
)
→
x1
={
n
}=
x2
.
Proof
.
intros
Hxy
;
apply
Hxy
.
Qed
.
Lemma
agree_valid_includedN
(
x
y
:
agree
A
)
n
:
✓
{
n
}
y
→
x
≼
{
n
}
y
→
x
={
n
}=
y
.
Proof
.
move
=>
Hval
[
z
Hy
]
;
move
:
Hval
;
rewrite
Hy
.
by
move
=>
/
agree_op_inv
>
;
rewrite
agree_idempotent
.
Qed
.
Definition
agree_cmra_extend_mixin
:
CMRAExtendMixin
(
agree
A
).
Proof
.
intros
n
x
y1
y2
?
Hx
;
exists
(
x
,
x
)
;
simpl
;
split
.
intros
n
x
y1
y2
Hval
Hx
;
exists
(
x
,
x
)
;
simpl
;
split
.
*
by
rewrite
agree_idempotent
.
*
by
rewrite
Hx
(
agree_op_inv
x
y1
y2
)
//
agree_idempotent
.
*
by
move
:
Hval
;
rewrite
Hx
;
move
=>
/
agree_op_inv
>
;
rewrite
agree_idempotent
.
Qed
.
Canonical
Structure
agreeRA
:
cmraT
:
=
CMRAT
agree_cofe_mixin
agree_cmra_mixin
agree_cmra_extend_mixin
.
...
...
@@ 118,15 +130,9 @@ Program Definition to_agree (x : A) : agree A :=
Solve
Obligations
with
done
.
Global
Instance
to_agree_ne
n
:
Proper
(
dist
n
==>
dist
n
)
to_agree
.
Proof
.
intros
x1
x2
Hx
;
split
;
naive_solver
eauto
using
@
dist_le
.
Qed
.
Lemma
agree_car_ne
(
x
y
:
agree
A
)
n
:
✓
{
n
}
x
→
x
={
n
}=
y
→
x
n
={
n
}=
y
n
.
Proof
.
by
intros
[??]
Hxy
;
apply
Hxy
.
Qed
.
Lemma
agree_cauchy
(
x
:
agree
A
)
n
i
:
n
≤
i
→
✓
{
i
}
x
→
x
n
={
n
}=
x
i
.
Proof
.
by
intros
?
[?
Hx
]
;
apply
Hx
.
Qed
.
Lemma
agree_to_agree_inj
(
x
y
:
agree
A
)
a
n
:
✓
{
n
}
x
→
x
={
n
}=
to_agree
a
⋅
y
→
x
n
={
n
}=
a
.
Proof
.
by
intros
;
transitivity
((
to_agree
a
⋅
y
)
n
)
;
first
apply
agree_car_ne
.
Qed
.
Global
Instance
to_agree_proper
:
Proper
((
≡
)
==>
(
≡
))
to_agree
:
=
ne_proper
_
.
Global
Instance
to_agree_inj
n
:
Injective
(
dist
n
)
(
dist
n
)
(
to_agree
).
Proof
.
by
intros
x
y
[
_
Hxy
]
;
apply
Hxy
.
Qed
.
End
agree
.
Arguments
agreeC
:
clear
implicits
.
...
...
@@ 137,8 +143,8 @@ Program Definition agree_map {A B} (f : A → B) (x : agree A) : agree B :=
Solve
Obligations
with
auto
using
agree_valid_0
,
agree_valid_S
.
Lemma
agree_map_id
{
A
}
(
x
:
agree
A
)
:
agree_map
id
x
=
x
.
Proof
.
by
destruct
x
.
Qed
.
Lemma
agree_map_compose
{
A
B
C
}
(
f
:
A
→
B
)
(
g
:
B
→
C
)
(
x
:
agree
A
)
:
agree_map
(
g
∘
f
)
x
=
agree_map
g
(
agree_map
f
x
).
Lemma
agree_map_compose
{
A
B
C
}
(
f
:
A
→
B
)
(
g
:
B
→
C
)
(
x
:
agree
A
)
:
agree_map
(
g
∘
f
)
x
=
agree_map
g
(
agree_map
f
x
).
Proof
.
done
.
Qed
.
Section
agree_map
.
...
...
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