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Iris
Iris
Commits
d011f232
Commit
d011f232
authored
Dec 08, 2015
by
Robbert Krebbers
Browse files
Valid STS elements should be nonempty.
parent
4847b5c1
Changes
1
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iris/sts.v
iris/sts.v
+46
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iris/sts.v
View file @
d011f232
...
...
@@ 29,6 +29,7 @@ Inductive frame_step (T : set B) (s1 s2 : A) : Prop :=
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
{
closed_ne
:
S
≢
∅
;
closed_disjoint
s
:
s
∈
S
→
tok
s
∩
T
≡
∅
;
closed_step
s1
s2
:
s1
∈
S
→
frame_step
T
s1
s2
→
s2
∈
S
}.
...
...
@@ 44,7 +45,8 @@ Global Instance sts_unit : Unit (t R tok) := λ x,

frag
S'
_
=>
frag
(
up_set
∅
S'
)
∅

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

frag_frag_disjoint
S1
S2
T1
T2
:
T1
∩
T2
≡
∅
→
frag
S1
T1
⊥
frag
S2
T2

frag_frag_disjoint
S1
S2
T1
T2
:
S1
∩
S2
≢
∅
→
T1
∩
T2
≡
∅
→
frag
S1
T1
⊥
frag
S2
T2

auth_frag_disjoint
s
S
T1
T2
:
s
∈
S
→
T1
∩
T2
≡
∅
→
auth
s
T1
⊥
frag
S
T2

frag_auth_disjoint
s
S
T1
T2
:
s
∈
S
→
T1
∩
T2
≡
∅
→
frag
S
T1
⊥
auth
s
T2
.
Global
Existing
Instance
sts_disjoint
.
...
...
@@ 64,6 +66,7 @@ Global Instance sts_minus : Minus (t R tok) := λ x1 x2,
end
.
Hint
Extern
10
(
equiv
(
A
:
=
set
_
)
_
_
)
=>
esolve_elem_of
:
sts
.
Hint
Extern
10
(
¬
(
equiv
(
A
:
=
set
_
)
_
_
))
=>
esolve_elem_of
:
sts
.
Hint
Extern
10
(
_
∈
_
)
=>
esolve_elem_of
:
sts
.
Hint
Extern
10
(
_
⊆
_
)
=>
esolve_elem_of
:
sts
.
Instance
:
Equivalence
((
≡
)
:
relation
(
t
R
tok
)).
...
...
@@ 83,16 +86,14 @@ Qed.
Instance
closed_proper
:
Proper
((
≡
)
==>
(
≡
)
==>
iff
)
closed
.
Proof
.
by
split
;
apply
closed_proper'
.
Qed
.
Lemma
closed_op
T1
T2
S1
S2
:
closed
T1
S1
→
closed
T2
S2
→
T1
∩
T2
≡
∅
→
closed
(
T1
∪
T2
)
(
S1
∩
S2
).
closed
T1
S1
→
closed
T2
S2
→
T1
∩
T2
≡
∅
→
S1
∩
S2
≢
∅
→
closed
(
T1
∪
T2
)
(
S1
∩
S2
).
Proof
.
intros
[?
Hstep1
]
[?
Hstep2
]
?
;
split
;
[
esolve_elem_of
].
intros
s3
s4
;
rewrite
!
elem_of_intersection
;
intros
[??]
[
T
?
?]
;
split
.
*
apply
Hstep1
with
s3
;
e
auto
with
sts
.
*
apply
Hstep2
with
s3
;
e
auto
with
sts
.
intros
[
_
?
Hstep1
]
[
_
?
Hstep2
]
?
;
split
;
[
done

esolve_elem_of
].
intros
s3
s4
;
rewrite
!
elem_of_intersection
;
intros
[??]
[
T
3
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
.
Lemma
closed_all
:
closed
∅
set_all
.
Proof
.
split
;
auto
with
sts
.
Qed
.
Hint
Resolve
closed_all
:
sts
.
Instance
up_preserving
:
Proper
(
flip
(
⊆
)
==>
(=)
==>
(
⊆
))
up
.
Proof
.
intros
T
T'
HT
s
?
<
;
apply
elem_of_subseteq
.
...
...
@@ 105,30 +106,32 @@ Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set.
Proof
.
by
intros
T1
T2
HT
S1
S2
HS
;
unfold
up_set
;
rewrite
HS
,
HT
.
Qed
.
Lemma
elem_of_up
s
T
:
s
∈
up
T
s
.
Proof
.
constructor
.
Qed
.
Lemma
suseteq_up_set
S
T
:
S
⊆
up_set
T
S
.
Lemma
su
b
seteq_up_set
S
T
:
S
⊆
up_set
T
S
.
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
≡
∅
)
→
closed
T
(
up_set
T
S
).
Lemma
closed_up_set
S
T
:
(
∀
s
,
s
∈
S
→
tok
s
∩
T
≡
∅
)
→
S
≢
∅
→
closed
T
(
up_set
T
S
).
Proof
.
intros
HS
;
unfold
up_set
;
split
.
intros
HS
Hne
;
unfold
up_set
;
split
.
*
assert
(
∀
s
,
s
∈
up
T
s
)
by
eauto
using
elem_of_up
.
esolve_elem_of
.
*
intros
s
;
rewrite
!
elem_of_bind
;
intros
(
s'
&
Hstep
&
Hs'
).
specialize
(
HS
s'
Hs'
)
;
clear
Hs'
S
.
specialize
(
HS
s'
Hs'
)
;
clear
Hs'
Hne
S
.
induction
Hstep
as
[
s

s1
s2
s3
[
T1
T2
?
Hstep
]
?
IH
]
;
auto
.
inversion_clear
Hstep
;
apply
IH
;
clear
IH
;
auto
with
sts
.
*
intros
s1
s2
;
rewrite
!
elem_of_bind
;
intros
(
s
&?&?)
?
;
exists
s
.
split
;
[
eapply
rtc_r
]
;
eauto
.
Qed
.
Lemma
closed_up_set_empty
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
).
Proof
.
intros
;
rewrite
<(
collection_bind_singleton
(
up
T
)
s
).
apply
closed_up_set
;
auto
with
sts
.
apply
closed_up_set
;
esolve_elem_of
.
Qed
.
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
.
Proof
.
intros
;
split
;
auto
using
suseteq_up_set
;
intros
s
.
intros
;
split
;
auto
using
su
b
seteq_up_set
;
intros
s
.
unfold
up_set
;
rewrite
elem_of_bind
;
intros
(
s'
&
Hstep
&?).
induction
Hstep
;
eauto
using
closed_step
.
Qed
.
...
...
@@ 144,7 +147,7 @@ Proof.
closed
T
S
→
s
∈
S
→
tok
s
∩
T'
≡
∅
→
tok
s
∩
(
T
∪
T'
)
≡
∅
).
{
intros
S
T
T'
s
[??]
;
esolve_elem_of
.
}
destruct
3
;
simpl
in
*
;
auto
using
closed_op
with
sts
.
*
intros
[]
;
simpl
;
eauto
using
closed_up
,
closed_up_set
with
sts
.
*
intros
[]
;
simpl
;
eauto
using
closed_up
,
closed_up_set
,
closed_ne
with
sts
.
*
intros
????
(
z
&
Hy
&?&
Hxz
)
;
destruct
Hxz
;
inversion
Hy
;
clear
Hy
;
setoid_subst
;
rewrite
?disjoint_union_difference
;
auto
using
closed_up
with
sts
.
eapply
closed_up_set
;
eauto
2
using
closed_disjoint
with
sts
.
...
...
@@ 153,22 +156,37 @@ Proof.
*
destruct
4
;
inversion_clear
1
;
constructor
;
auto
with
sts
.
*
destruct
1
;
constructor
;
auto
with
sts
.
*
destruct
3
;
constructor
;
auto
with
sts
.
*
intros
[]
;
constructor
;
auto
using
elem_of_up
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
.
esolve_elem_of
.
*
intros
[
S
T
]
;
constructor
;
auto
with
sts
.
assert
(
S
⊆
up_set
∅
S
)
;
auto
using
suseteq_up_set
with
sts
.
assert
(
S
⊆
up_set
∅
S
)
;
auto
using
su
b
seteq_up_set
with
sts
.
*
intros
[
s
T

S
T
]
;
constructor
;
auto
with
sts
.
+
by
rewrite
(
up_closed
(
up
_
_
))
by
auto
using
closed_up
with
sts
.
+
by
rewrite
(
up_closed
(
up_set
_
_
))
by
auto
using
closed_up_set
with
sts
.
*
intros
x
y
??
(
z
&
Hy
&?&
Hxz
)
;
exists
(
unit
(
x
⋅
y
)).
destruct
Hxz
;
inversion_clear
Hy
;
simpl
;
split_ands
;
auto
using
closed_up_set_empty
,
closed_up_empty
;
constructor
;
unfold
up_set
;
auto
with
sts
.
*
intros
x
y
??
(
z
&
Hy
&
_
&
Hxz
)
;
destruct
Hxz
;
inversion_clear
Hy
;
constructor
;
eauto
using
elem_of_up
;
auto
with
sts
.
+
by
rewrite
(
up_closed
(
up_set
_
_
))
by
eauto
using
closed_up_set
,
closed_ne
with
sts
.
*
intros
x
y
??
(
z
&
Hy
&?&
Hxz
)
;
exists
(
unit
(
x
⋅
y
))
;
split_ands
.
+
destruct
Hxz
;
inversion_clear
Hy
;
constructor
;
unfold
up_set
;
esolve_elem_of
.
+
destruct
Hxz
;
inversion_clear
Hy
;
simpl
;
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
)
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
)
end
;
auto
with
sts
.
*
intros
x
y
??
(
z
&
Hy
&?&
Hxz
)
;
destruct
Hxz
as
[
S1
S2
T1
T2

]
;
inversion
Hy
;
clear
Hy
;
constructor
;
setoid_subst
;
rewrite
?disjoint_union_difference
by
done
;
auto
.
split
;
[
apply
intersection_greatest
;
auto
using
suseteq_up_set
with
sts
].
split
;
[
apply
intersection_greatest
;
auto
using
su
b
seteq_up_set
with
sts
].
apply
intersection_greatest
;
[
auto
with
sts
].
intros
s2
;
rewrite
elem_of_intersection
.
unfold
up_set
;
rewrite
elem_of_bind
;
intros
(?&
s1
&?&?&?).
...
...
@@ 178,7 +196,7 @@ Lemma step_closed s1 s2 T1 T2 S Tf :
step
(
s1
,
T1
)
(
s2
,
T2
)
→
closed
Tf
S
→
s1
∈
S
→
T1
∩
Tf
≡
∅
→
s2
∈
S
∧
T2
∩
Tf
≡
∅
∧
tok
s2
∩
T2
≡
∅
.
Proof
.
inversion_clear
1
as
[????
HR
Hs1
Hs2
]
;
intros
[?
Hstep
]
??
;
split_ands
;
auto
.
inversion_clear
1
as
[????
HR
Hs1
Hs2
]
;
intros
[?
?
Hstep
]??
;
split_ands
;
auto
.
*
eapply
Hstep
with
s1
,
Frame_step
with
T1
T2
;
auto
with
sts
.
*
clear
Hstep
Hs1
Hs2
;
esolve_elem_of
.
Qed
.
...
...
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