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Joshua Yanovski
iriscoq
Commits
f8f48666
Commit
f8f48666
authored
Feb 15, 2016
by
Ralf Jung
Browse files
sts: change order of S and T to consistently be "S T" (also for consistency with onpaper def.)
parent
bbb33095
Changes
1
Hide whitespace changes
Inline
Sidebyside
algebra/sts.v
View file @
f8f48666
From
prelude
Require
Export
sets
.
From
algebra
Require
Export
cmra
.
From
prelude
Require
Import
sets
.
From
algebra
Require
Import
dra
.
Local
Arguments
valid
_
_
!
_
/
.
Local
Arguments
op
_
_
!
_
!
_
/
.
...
...
@@ 29,21 +29,21 @@ Inductive frame_step (T : set B) (s1 s2 : A) : Prop :=

Frame_step
T1
T2
:
T1
∩
(
tok
s1
∪
T
)
≡
∅
→
step
(
s1
,
T1
)
(
s2
,
T2
)
→
frame_step
T
s1
s2
.
Hint
Resolve
Frame_step
.
Record
closed
(
T
:
set
B
)
(
S
:
set
A
)
:
Prop
:=
Closed
{
Record
closed
(
S
:
set
A
)
(
T
:
set
B
)
:
Prop
:=
Closed
{
closed_ne
:
S
≢
∅
;
closed_disjoint
s
:
s
∈
S
→
tok
s
∩
T
≡
∅
;
closed_step
s1
s2
:
s1
∈
S
→
frame_step
T
s1
s2
→
s2
∈
S
}
.
Lemma
closed_steps
S
T
s1
s2
:
closed
T
S
→
s1
∈
S
→
rtc
(
frame_step
T
)
s1
s2
→
s2
∈
S
.
closed
S
T
→
s1
∈
S
→
rtc
(
frame_step
T
)
s1
s2
→
s2
∈
S
.
Proof
.
induction
3
;
eauto
using
closed_step
.
Qed
.
Global
Instance
sts_valid
:
Valid
(
sts
R
tok
)
:=
λ
x
,
match
x
with
auth
s
T
=>
tok
s
∩
T
≡
∅

frag
S
'
T
=>
closed
T
S
'
end
.
Definition
up
(
T
:
set
B
)
(
s
:
A
)
:
set
A
:=
mkSet
(
rtc
(
frame_step
T
)
s
).
Definition
up_set
(
T
:
set
B
)
(
S
:
set
A
)
:
set
A
:=
S
≫
=
up
T
.
match
x
with
auth
s
T
=>
tok
s
∩
T
≡
∅

frag
S
'
T
=>
closed
S
'
T
end
.
Definition
up
(
s
:
A
)
(
T
:
set
B
)
:
set
A
:=
mkSet
(
rtc
(
frame_step
T
)
s
).
Definition
up_set
(
S
:
set
A
)
(
T
:
set
B
)
:
set
A
:=
S
≫
=
λ
s
,
up
s
T
.
Global
Instance
sts_unit
:
Unit
(
sts
R
tok
)
:=
λ
x
,
match
x
with

frag
S
'
_
=>
frag
(
up_set
∅
S
'
)
∅

auth
s
_
=>
frag
(
up
∅
s
)
∅

frag
S
'
_
=>
frag
(
up_set
S
'
∅
)
∅

auth
s
_
=>
frag
(
up
s
∅
)
∅
end
.
Inductive
sts_disjoint
:
Disjoint
(
sts
R
tok
)
:=

frag_frag_disjoint
S1
S2
T1
T2
:
...
...
@@ 60,10 +60,10 @@ Global Instance sts_op : Op (sts R tok) := λ x1 x2,
end
.
Global
Instance
sts_minus
:
Minus
(
sts
R
tok
)
:=
λ
x1
x2
,
match
x1
,
x2
with

frag
S1
T1
,
frag
S2
T2
=>
frag
(
up_set
(
T1
∖
T2
)
S1
)
(
T1
∖
T2
)

frag
S1
T1
,
frag
S2
T2
=>
frag
(
up_set
S1
(
T1
∖
T2
))
(
T1
∖
T2
)

auth
s
T1
,
frag
_
T2
=>
auth
s
(
T1
∖
T2
)

frag
_
T2
,
auth
s
T1
=>
auth
s
(
T1
∖
T2
)
(
*
never
happens
*
)

auth
s
T1
,
auth
_
T2
=>
frag
(
up
(
T1
∖
T2
)
s
)
(
T1
∖
T2
)

auth
s
T1
,
auth
_
T2
=>
frag
(
up
s
(
T1
∖
T2
))
(
T1
∖
T2
)
end
.
Hint
Extern
10
(
equiv
(
A
:=
set
_
)
_
_
)
=>
solve_elem_of
:
sts
.
...
...
@@ 87,33 +87,36 @@ Qed.
Instance
closed_proper
:
Proper
((
≡
)
==>
(
≡
)
==>
iff
)
closed
.
Proof
.
by
split
;
apply
closed_proper
'
.
Qed
.
Lemma
closed_op
T1
T2
S1
S2
:
closed
T
1
S
1
→
closed
T
2
S
2
→
T1
∩
T2
≡
∅
→
S1
∩
S2
≢
∅
→
closed
(
T
1
∪
T
2
)
(
S
1
∩
S
2
).
closed
S
1
T
1
→
closed
S
2
T
2
→
T1
∩
T2
≡
∅
→
S1
∩
S2
≢
∅
→
closed
(
S
1
∩
S
2
)
(
T
1
∪
T
2
).
Proof
.
intros
[
_
?
Hstep1
]
[
_
?
Hstep2
]
?
;
split
;
[
done

solve_elem_of

].
intros
s3
s4
;
rewrite
!
elem_of_intersection
;
intros
[
??
]
[
T3
T4
?
];
split
.
*
apply
Hstep1
with
s3
,
Frame_step
with
T3
T4
;
auto
with
sts
.
*
apply
Hstep2
with
s3
,
Frame_step
with
T3
T4
;
auto
with
sts
.
Qed
.
Instance
up_preserving
:
Proper
(
flip
(
⊆
)
==>
(
=
)
==>
(
⊆
))
up
.
Instance
up_preserving
:
Proper
(
(
=
)
==>
flip
(
⊆
)
==>
(
⊆
))
up
.
Proof
.
intros
T
T
'
HT
s
?
<
;
apply
elem_of_subseteq
.
intros
s
?
<
T
T
'
HT
;
apply
elem_of_subseteq
.
induction
1
as
[

s1
s2
s3
[
T1
T2
]];
[
constructor

].
eapply
rtc_l
;
[
eapply
Frame_step
with
T1
T2

];
eauto
with
sts
.
Qed
.
Instance
up_proper
:
Proper
((
≡
)
==>
(
=
)
==>
(
≡
))
up
.
Proof
.
by
intros
??
[
??
]
???
;
split
;
apply
up_preserving
.
Qed
.
Instance
up_proper
:
Proper
((
=
)
==>
(
≡
)
==>
(
≡
))
up
.
Proof
.
by
intros
???
??
[
??
];
split
;
apply
up_preserving
.
Qed
.
Instance
up_set_proper
:
Proper
((
≡
)
==>
(
≡
)
==>
(
≡
))
up_set
.
Proof
.
by
intros
T1
T2
HT
S1
S2
HS
;
rewrite
/
up_set
HS
HT
.
Qed
.
Lemma
elem_of_up
s
T
:
s
∈
up
T
s
.
Proof
.
intros
S1
S2
HS
T1
T2
HT
.
rewrite
/
up_set
HS
.
f_equiv
=>
s1
s2
Hs
.
by
rewrite
Hs
HT
.
Qed
.
Lemma
elem_of_up
s
T
:
s
∈
up
s
T
.
Proof
.
constructor
.
Qed
.
Lemma
subseteq_up_set
S
T
:
S
⊆
up_set
T
S
.
Lemma
subseteq_up_set
S
T
:
S
⊆
up_set
S
T
.
Proof
.
intros
s
?
;
apply
elem_of_bind
;
eauto
using
elem_of_up
.
Qed
.
Lemma
closed_up_set
S
T
:
(
∀
s
,
s
∈
S
→
tok
s
∩
T
≡
∅
)
→
S
≢
∅
→
closed
T
(
up_set
T
S
)
.
(
∀
s
,
s
∈
S
→
tok
s
∩
T
≡
∅
)
→
S
≢
∅
→
closed
(
up_set
S
T
)
T
.
Proof
.
intros
HS
Hne
;
unfold
up_set
;
split
.
*
assert
(
∀
s
,
s
∈
up
T
s
)
by
eauto
using
elem_of_up
.
solve_elem_of
.
*
assert
(
∀
s
,
s
∈
up
s
T
)
by
eauto
using
elem_of_up
.
solve_elem_of
.
*
intros
s
;
rewrite
!
elem_of_bind
;
intros
(
s
'
&
Hstep
&
Hs
'
).
specialize
(
HS
s
'
Hs
'
);
clear
Hs
'
Hne
S
.
induction
Hstep
as
[
s

s1
s2
s3
[
T1
T2
?
Hstep
]
?
IH
];
auto
.
...
...
@@ 121,16 +124,16 @@ Proof.
*
intros
s1
s2
;
rewrite
!
elem_of_bind
;
intros
(
s
&?&?
)
?
;
exists
s
.
split
;
[
eapply
rtc_r

];
eauto
.
Qed
.
Lemma
closed_up_set_empty
S
:
S
≢
∅
→
closed
∅
(
up_set
∅
S
)
.
Lemma
closed_up_set_empty
S
:
S
≢
∅
→
closed
(
up_set
S
∅
)
∅
.
Proof
.
eauto
using
closed_up_set
with
sts
.
Qed
.
Lemma
closed_up
s
T
:
tok
s
∩
T
≡
∅
→
closed
T
(
up
T
s
)
.
Lemma
closed_up
s
T
:
tok
s
∩
T
≡
∅
→
closed
(
up
s
T
)
T
.
Proof
.
intros
;
rewrite

(
collection_bind_singleton
(
up
T
)
s
).
intros
;
rewrite

(
collection_bind_singleton
(
λ
s
,
up
s
T
)
s
).
apply
closed_up_set
;
solve_elem_of
.
Qed
.
Lemma
closed_up_empty
s
:
closed
∅
(
up
∅
s
)
.
Lemma
closed_up_empty
s
:
closed
(
up
s
∅
)
∅
.
Proof
.
eauto
using
closed_up
with
sts
.
Qed
.
Lemma
up_closed
S
T
:
closed
T
S
→
up_set
T
S
≡
S
.
Lemma
up_closed
S
T
:
closed
S
T
→
up_set
S
T
≡
S
.
Proof
.
intros
;
split
;
auto
using
subseteq_up_set
;
intros
s
.
unfold
up_set
;
rewrite
elem_of_bind
;
intros
(
s
'
&
Hstep
&?
).
...
...
@@ 145,7 +148,7 @@ Proof.
*
by
intros
?
[

];
destruct
1
;
inversion_clear
1
;
constructor
;
setoid_subst
.
*
by
do
2
destruct
1
;
constructor
;
setoid_subst
.
*
assert
(
∀
T
T
'
S
s
,
closed
T
S
→
s
∈
S
→
tok
s
∩
T
'
≡
∅
→
tok
s
∩
(
T
∪
T
'
)
≡
∅
).
closed
S
T
→
s
∈
S
→
tok
s
∩
T
'
≡
∅
→
tok
s
∩
(
T
∪
T
'
)
≡
∅
).
{
intros
S
T
T
'
s
[
??
];
solve_elem_of
.
}
destruct
3
;
simpl
in
*
;
auto
using
closed_op
with
sts
.
*
intros
[];
simpl
;
eauto
using
closed_up
,
closed_up_set
,
closed_ne
with
sts
.
...
...
@@ 158,10 +161,10 @@ Proof.
*
destruct
1
;
constructor
;
auto
with
sts
.
*
destruct
3
;
constructor
;
auto
with
sts
.
*
intros
[

S
T
];
constructor
;
auto
using
elem_of_up
with
sts
.
assert
(
S
⊆
up_set
∅
S
∧
S
≢
∅
)
by
eauto
using
subseteq_up_set
,
closed_ne
.
assert
(
S
⊆
up_set
S
∅
∧
S
≢
∅
)
by
eauto
using
subseteq_up_set
,
closed_ne
.
solve_elem_of
.
*
intros
[

S
T
];
constructor
;
auto
with
sts
.
assert
(
S
⊆
up_set
∅
S
);
auto
using
subseteq_up_set
with
sts
.
assert
(
S
⊆
up_set
S
∅
);
auto
using
subseteq_up_set
with
sts
.
*
intros
[
s
T

S
T
];
constructor
;
auto
with
sts
.
+
rewrite
(
up_closed
(
up
_
_
));
auto
using
closed_up
with
sts
.
+
rewrite
(
up_closed
(
up_set
_
_
));
...
...
@@ 172,17 +175,17 @@ Proof.
auto
using
closed_up_set_empty
,
closed_up_empty
with
sts
.
+
destruct
Hxz
;
inversion_clear
Hy
;
constructor
;
repeat
match
goal
with


context
[
up_set
?
T
?
S
]
=>
unless
(
S
⊆
up_set
T
S
)
by
done
;
pose
proof
(
subseteq_up_set
S
T
)


context
[
up
?
T
?
s
]
=>
unless
(
s
∈
up
T
s
)
by
done
;
pose
proof
(
elem_of_up
s
T
)


context
[
up_set
?
S
?
T
]
=>
unless
(
S
⊆
up_set
S
T
)
by
done
;
pose
proof
(
subseteq_up_set
S
T
)


context
[
up
?
s
?
T
]
=>
unless
(
s
∈
up
s
T
)
by
done
;
pose
proof
(
elem_of_up
s
T
)
end
;
auto
with
sts
.
*
intros
x
y
??
(
z
&
Hy
&
_
&
Hxz
);
destruct
Hxz
;
inversion_clear
Hy
;
constructor
;
repeat
match
goal
with


context
[
up_set
?
T
?
S
]
=>
unless
(
S
⊆
up_set
T
S
)
by
done
;
pose
proof
(
subseteq_up_set
S
T
)


context
[
up
?
T
?
s
]
=>
unless
(
s
∈
up
T
s
)
by
done
;
pose
proof
(
elem_of_up
s
T
)


context
[
up_set
?
S
?
T
]
=>
unless
(
S
⊆
up_set
S
T
)
by
done
;
pose
proof
(
subseteq_up_set
S
T
)


context
[
up
?
s
?
T
]
=>
unless
(
s
∈
up
s
T
)
by
done
;
pose
proof
(
elem_of_up
s
T
)
end
;
auto
with
sts
.
*
intros
x
y
??
(
z
&
Hy
&?&
Hxz
);
destruct
Hxz
as
[
S1
S2
T1
T2


];
inversion
Hy
;
clear
Hy
;
constructor
;
setoid_subst
;
...
...
@@ 194,7 +197,7 @@ Proof.
apply
closed_steps
with
T2
s1
;
auto
with
sts
.
Qed
.
Lemma
step_closed
s1
s2
T1
T2
S
Tf
:
step
(
s1
,
T1
)
(
s2
,
T2
)
→
closed
Tf
S
→
s1
∈
S
→
T1
∩
Tf
≡
∅
→
step
(
s1
,
T1
)
(
s2
,
T2
)
→
closed
S
Tf
→
s1
∈
S
→
T1
∩
Tf
≡
∅
→
s2
∈
S
∧
T2
∩
Tf
≡
∅
∧
tok
s2
∩
T2
≡
∅
.
Proof
.
inversion_clear
1
as
[
????
HR
Hs1
Hs2
];
intros
[
??
Hstep
]
??
;
split_ands
;
auto
.
...
...
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