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William Mansky
Iris
Commits
9ff3a924
Commit
9ff3a924
authored
4 years ago
by
Simon Friis Vindum
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Update proofs
parent
4d2f128e
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1 changed file
theories/algebra/auth.v
+15
-18
15 additions, 18 deletions
theories/algebra/auth.v
with
15 additions
and
18 deletions
theories/algebra/auth.v
+
15
−
18
View file @
9ff3a924
...
...
@@ -366,14 +366,13 @@ Lemma auth_update a b a' b' :
(
a
,
b
)
~l
~>
(
a'
,
b'
)
→
●
a
⋅
◯
b
~~>
●
a'
⋅
◯
b'
.
Proof
.
intros
Hup
;
apply
cmra_total_update
.
move
=>
n
[[[??]|]
bf1
]
[
/=
VL
[
a0
[
Eq
[[
bf2
Ha
]
VL2
]]]];
do
2
red
;
simpl
in
*.
+
exfalso
.
move
:
VL
=>
/
frac_valid'
.
rewrite
frac_op'
.
by
apply
Qp_not_plus_q_ge_1
.
+
split
;
[
done
|]
.
apply
(
inj
to_agree
)
in
Eq
.
move
:
Ha
;
rewrite
!
left_id
-
assoc
=>
Ha
.
destruct
(
Hup
n
(
Some
(
bf1
⋅
bf2
)));
[
by
rewrite
Eq
..|]
.
simpl
in
*.
exists
a'
.
split
;
[
done
|]
.
split
;
[|
done
]
.
exists
bf2
.
by
rewrite
left_id
-
assoc
.
move
=>
n
[[
qa
|]
bf1
]
[
/=
Hq
[
a0
[
Haa0
[[
bf2
Ha
]
Ha0
]]]];
do
2
red
;
simpl
in
*.
{
by
apply
(
exclusiveN_l
_
_)
in
Hq
.
}
split
;
[
done
|]
.
apply
(
inj
to_agree
)
in
Haa0
.
move
:
Ha
;
rewrite
!
left_id
-
assoc
=>
Ha
.
destruct
(
Hup
n
(
Some
(
bf1
⋅
bf2
)));
[
by
rewrite
Haa0
..|];
simpl
in
*.
exists
a'
.
split_and
!
;
[
done
|
|
done
]
.
exists
bf2
.
by
rewrite
left_id
-
assoc
.
Qed
.
Lemma
auth_update_alloc
a
a'
b'
:
(
a
,
ε
)
~l
~>
(
a'
,
b'
)
→
●
a
~~>
●
a'
⋅
◯
b'
.
...
...
@@ -388,16 +387,14 @@ Qed.
Lemma
auth_update_core_id
q
a
b
`{
!
CoreId
b
}
:
b
≼
a
→
●
{
q
}
a
~~>
●
{
q
}
a
⋅
◯
b
.
Proof
.
intros
Heq
%
core_id_extract
;
last
done
.
apply
cmra_total_update
.
move
=>
n
[[[
q'
ag
]|]
bf1
]
[
/=
VL
[
a0
[
Eq
[[
bf2
Ha
]
VL2
]]]];
do
2
red
;
simpl
in
*.
+
split
;
first
done
.
exists
a0
.
split
;
last
split
;
try
done
.
exists
bf2
.
assert
(
a
≡
{
n
}
≡
a0
)
as
Eq2
.
{
apply
to_agree_includedN
.
exists
ag
.
done
.
}
by
rewrite
left_id
-
Eq2
-
assoc
-
comm
Heq
Eq2
Ha
left_id
.
+
split
;
first
done
.
exists
a0
.
split
;
first
done
.
split
;
last
done
.
exists
bf2
.
apply
(
inj
to_agree
)
in
Eq
.
by
rewrite
left_id
-
Eq
-
assoc
-
comm
Heq
Eq
Ha
left_id
.
intros
Ha
%
(
core_id_extract
_
_)
.
apply
cmra_total_update
.
move
=>
n
[[[
q'
ag
]|]
bf1
]
[
Hq
[
a0
[
Haa0
[
Hbf1
Ha0
]]]];
do
2
red
;
simpl
in
*.
+
split
;
[
done
|]
.
exists
a0
.
split_and
!
;
[
done
|
|
done
]
.
assert
(
a
≡
{
n
}
≡
a0
)
as
Haa0'
by
(
apply
to_agree_includedN
;
by
exists
ag
)
.
rewrite
-
Haa0'
Ha
Haa0'
-
assoc
(
comm
_
b
)
assoc
.
by
apply
cmra_monoN_r
.
+
split
;
[
done
|]
.
exists
a0
.
split_and
!
;
[
done
|
|
done
]
.
apply
(
inj
to_agree
)
in
Haa0
.
rewrite
-
Haa0
Ha
Haa0
-
assoc
(
comm
_
b
)
assoc
.
by
apply
cmra_monoN_r
.
Qed
.
Lemma
auth_update_dealloc
a
b
a'
:
(
a
,
b
)
~l
~>
(
a'
,
ε
)
→
●
a
⋅
◯
b
~~>
●
a'
.
...
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