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2c1b15dc
Commit
2c1b15dc
authored
Feb 11, 2016
by
Ralf Jung
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algebra/auth.v
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algebra/auth.v
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2c1b15dc
...
...
@@ 147,12 +147,8 @@ Proof. done. Qed.
Lemma
auth_both_op
a
b
:
Auth
(
Excl
a
)
b
≡
●
a
⋅
◯
b
.
Proof
.
by
rewrite
/
op
/
auth_op
/=
left_id
.
Qed
.
(* FIXME tentative name. Or maybe remove this notion entirely. *)
Definition
auth_step
(
a
a'
b
b'
:
A
)
:
Prop
:
=
∀
n
af
,
✓
{
n
}
a
→
a
≡
{
n
}
≡
a'
⋅
af
→
b
≡
{
n
}
≡
b'
⋅
af
∧
✓
{
n
}
b
.
Lemma
auth_update
a
a'
b
b'
:
auth_step
a
a'
b
b'
→
(
∀
n
af
,
✓
{
n
}
a
→
a
≡
{
n
}
≡
a'
⋅
af
→
b
≡
{
n
}
≡
b'
⋅
af
∧
✓
{
n
}
b
)
→
●
a
⋅
◯
a'
~~>
●
b
⋅
◯
b'
.
Proof
.
move
=>
Hab
[[?
]
bf1
]
n
//
=>[[
bf2
Ha
]
?]
;
do
2
red
;
simpl
in
*.
...
...
@@ 161,20 +157,11 @@ Proof.
split
;
[
by
rewrite
Ha'
left_id
associative
;
apply
cmra_includedN_l

done
].
Qed
.
(* FIXME: are the following lemmas derivable from each other? *)
Lemma
auth_local_update_l
f
`
{!
LocalUpdate
P
f
}
a
a'
:
P
a
→
✓
(
f
a
⋅
a'
)
→
●
(
a
⋅
a'
)
⋅
◯
a
~~>
●
(
f
a
⋅
a'
)
⋅
◯
f
a
.
Proof
.
intros
;
apply
auth_update
=>
n
af
?
EQ
;
split
;
last
done
.
by
rewrite
(
local_updateN
f
)
//
EQ
(
local_updateN
f
)
//

EQ
.
Qed
.
Lemma
auth_local_update
f
`
{!
LocalUpdate
P
f
}
a
a'
:
P
a
→
✓
(
f
a'
)
→
●
a'
⋅
◯
a
~~>
●
f
a'
⋅
◯
f
a
.
Proof
.
intros
;
apply
auth_update
=>
n
af
?
EQ
;
split
;
last
done
.
intros
.
apply
auth_update
=>
n
af
?
EQ
;
split
;
last
done
.
by
rewrite
EQ
(
local_updateN
f
)
//

EQ
.
Qed
.
...
...
@@ 185,6 +172,18 @@ Lemma auth_update_op_r a a' b :
✓
(
a
⋅
b
)
→
●
a
⋅
◯
a'
~~>
●
(
a
⋅
b
)
⋅
◯
(
a'
⋅
b
).
Proof
.
rewrite
!(
commutative
_
b
)
;
apply
auth_update_op_l
.
Qed
.
(* This does not seem to follow from auth_local_update.
The trouble is that given ✓ (f a ⋅ a'), P a
we need ✓ (a ⋅ a'). I think this should hold for every local update,
but adding an extra axiom to local updates just for this is silly. *)
Lemma
auth_local_update_l
f
`
{!
LocalUpdate
P
f
}
a
a'
:
P
a
→
✓
(
f
a
⋅
a'
)
→
●
(
a
⋅
a'
)
⋅
◯
a
~~>
●
(
f
a
⋅
a'
)
⋅
◯
f
a
.
Proof
.
intros
.
apply
auth_update
=>
n
af
?
EQ
;
split
;
last
done
.
by
rewrite
(
local_updateN
f
)
//
EQ
(
local_updateN
f
)
//

EQ
.
Qed
.
End
cmra
.
Arguments
authRA
:
clear
implicits
.
...
...
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