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Rice Wine
Iris
Commits
64b1650c
Commit
64b1650c
authored
Jun 08, 2017
by
Robbert Krebbers
Browse files
Some stepindexed properties for CMRAs that were missing.
parent
84227fa4
Changes
1
Show whitespace changes
Inline
Sidebyside
theories/algebra/cmra.v
View file @
64b1650c
...
...
@@ 1251,6 +1251,21 @@ Section option.
+
exists
(
Some
z
)
;
by
constructor
.
Qed
.
Lemma
option_includedN
n
(
mx
my
:
option
A
)
:
mx
≼
{
n
}
my
↔
mx
=
None
∨
∃
x
y
,
mx
=
Some
x
∧
my
=
Some
y
∧
(
x
≡
{
n
}
≡
y
∨
x
≼
{
n
}
y
).
Proof
.
split
.

intros
[
mz
Hmz
].
destruct
mx
as
[
x
]
;
[
right

by
left
].
destruct
my
as
[
y
]
;
[
exists
x
,
y

destruct
mz
;
inversion_clear
Hmz
].
destruct
mz
as
[
z
]
;
inversion_clear
Hmz
;
split_and
?
;
auto
;
ofe_subst
;
eauto
using
cmra_includedN_l
.

intros
[>(
x
&
y
&>&>&[
Hz
[
z
Hz
]])].
+
exists
my
.
by
destruct
my
.
+
exists
None
;
by
constructor
.
+
exists
(
Some
z
)
;
by
constructor
.
Qed
.
Lemma
option_cmra_mixin
:
CMRAMixin
(
option
A
).
Proof
.
apply
cmra_total_mixin
.
...
...
@@ 1317,16 +1332,27 @@ Section option.
Lemma
exclusive_Some_r
x
`
{!
Exclusive
x
}
my
:
✓
(
my
⋅
Some
x
)
→
my
=
None
.
Proof
.
rewrite
comm
.
by
apply
exclusive_Some_l
.
Qed
.
Lemma
Some_includedN
n
x
y
:
Some
x
≼
{
n
}
Some
y
↔
x
≡
{
n
}
≡
y
∨
x
≼
{
n
}
y
.
Proof
.
rewrite
option_includedN
;
naive_solver
.
Qed
.
Lemma
Some_included
x
y
:
Some
x
≼
Some
y
↔
x
≡
y
∨
x
≼
y
.
Proof
.
rewrite
option_included
;
naive_solver
.
Qed
.
Lemma
Some_included_2
x
y
:
x
≼
y
→
Some
x
≼
Some
y
.
Proof
.
rewrite
Some_included
;
eauto
.
Qed
.
Lemma
Some_includedN_total
`
{
CMRATotal
A
}
n
x
y
:
Some
x
≼
{
n
}
Some
y
↔
x
≼
{
n
}
y
.
Proof
.
rewrite
Some_includedN
.
split
.
by
intros
[>?].
eauto
.
Qed
.
Lemma
Some_included_total
`
{
CMRATotal
A
}
x
y
:
Some
x
≼
Some
y
↔
x
≼
y
.
Proof
.
rewrite
Some_included
.
split
.
by
intros
[>?].
eauto
.
Qed
.
Lemma
Some_includedN_exclusive
n
x
`
{!
Exclusive
x
}
y
:
Some
x
≼
{
n
}
Some
y
→
✓
{
n
}
y
→
x
≡
{
n
}
≡
y
.
Proof
.
move
=>
/
Some_includedN
[///
exclusive_includedN
]
;
tauto
.
Qed
.
Lemma
Some_included_exclusive
x
`
{!
Exclusive
x
}
y
:
Some
x
≼
Some
y
→
✓
y
→
x
≡
y
.
Proof
.
move
=>
/
Some_included
[///
exclusive_included
]
;
tauto
.
Qed
.
Lemma
is_Some_includedN
n
mx
my
:
mx
≼
{
n
}
my
→
is_Some
mx
→
is_Some
my
.
Proof
.
rewrite
!
not_eq_None_Some
option_includedN
.
naive_solver
.
Qed
.
Lemma
is_Some_included
mx
my
:
mx
≼
my
→
is_Some
mx
→
is_Some
my
.
Proof
.
rewrite
!
not_eq_None_Some
option_included
.
naive_solver
.
Qed
.
...
...
@@ 1348,6 +1374,17 @@ Arguments optionUR : clear implicits.
Section
option_prod
.
Context
{
A
B
:
cmraT
}.
Lemma
Some_pair_includedN
n
(
x1
x2
:
A
)
(
y1
y2
:
B
)
:
Some
(
x1
,
y1
)
≼
{
n
}
Some
(
x2
,
y2
)
→
Some
x1
≼
{
n
}
Some
x2
∧
Some
y1
≼
{
n
}
Some
y2
.
Proof
.
rewrite
!
Some_includedN
.
intros
[[??][??]%
prod_includedN
]
;
eauto
.
Qed
.
Lemma
Some_pair_includedN_total_1
`
{
CMRATotal
A
}
n
(
x1
x2
:
A
)
(
y1
y2
:
B
)
:
Some
(
x1
,
y1
)
≼
{
n
}
Some
(
x2
,
y2
)
→
x1
≼
{
n
}
x2
∧
Some
y1
≼
{
n
}
Some
y2
.
Proof
.
intros
?%
Some_pair_includedN
.
by
rewrite
(
Some_includedN_total
_
x1
).
Qed
.
Lemma
Some_pair_includedN_total_2
`
{
CMRATotal
B
}
n
(
x1
x2
:
A
)
(
y1
y2
:
B
)
:
Some
(
x1
,
y1
)
≼
{
n
}
Some
(
x2
,
y2
)
→
Some
x1
≼
{
n
}
Some
x2
∧
y1
≼
{
n
}
y2
.
Proof
.
intros
?%
Some_pair_includedN
.
by
rewrite
(
Some_includedN_total
_
y1
).
Qed
.
Lemma
Some_pair_included
(
x1
x2
:
A
)
(
y1
y2
:
B
)
:
Some
(
x1
,
y1
)
≼
Some
(
x2
,
y2
)
→
Some
x1
≼
Some
x2
∧
Some
y1
≼
Some
y2
.
Proof
.
rewrite
!
Some_included
.
intros
[[??][??]%
prod_included
]
;
eauto
.
Qed
.
...
...
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