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iris
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Tej Chajed
iris
Commits
6fddc211
Commit
6fddc211
authored
8 years ago
by
Ralf Jung
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Merge branch 'master' of gitlab.mpi-sws.org:FP/iris-coq
parents
ab8063db
9c5a95d3
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algebra/sts.v
+19
-24
19 additions, 24 deletions
algebra/sts.v
with
19 additions
and
24 deletions
algebra/sts.v
+
19
−
24
View file @
6fddc211
...
...
@@ -122,6 +122,8 @@ Lemma elem_of_up s T : s ∈ up s T.
Proof
.
constructor
.
Qed
.
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
elem_of_up_set
S
T
s
:
s
∈
S
→
s
∈
up_set
S
T
.
Proof
.
apply
subseteq_up_set
.
Qed
.
Lemma
up_up_set
s
T
:
up
s
T
≡
up_set
{[
s
]}
T
.
Proof
.
by
rewrite
/
up_set
collection_bind_singleton
.
Qed
.
Lemma
closed_up_set
S
T
:
(
∀
s
,
s
∈
S
→
tok
s
⊥
T
)
→
closed
(
up_set
S
T
)
T
.
...
...
@@ -143,12 +145,6 @@ Lemma closed_up_set_empty S : closed (up_set S ∅) ∅.
Proof
.
eauto
using
closed_up_set
with
sts
.
Qed
.
Lemma
closed_up_empty
s
:
closed
(
up
s
∅
)
∅.
Proof
.
eauto
using
closed_up
with
sts
.
Qed
.
Lemma
up_set_empty
S
T
:
up_set
S
T
≡
∅
→
S
≡
∅.
Proof
.
move
:(
subseteq_up_set
S
T
)
.
set_solver
.
Qed
.
Lemma
up_set_non_empty
S
T
:
S
≢
∅
→
up_set
S
T
≢
∅.
Proof
.
by
move
=>?
/
up_set_empty
.
Qed
.
Lemma
up_non_empty
s
T
:
up
s
T
≢
∅.
Proof
.
eapply
non_empty_inhabited
,
elem_of_up
.
Qed
.
Lemma
up_closed
S
T
:
closed
S
T
→
up_set
S
T
≡
S
.
Proof
.
intros
?;
apply
collection_equiv_spec
;
split
;
auto
using
subseteq_up_set
.
...
...
@@ -190,7 +186,7 @@ Existing Instance sts_equiv.
Instance
sts_valid
:
Valid
(
car
sts
)
:=
λ
x
,
match
x
with
|
auth
s
T
=>
tok
s
⊥
T
|
frag
S'
T
=>
closed
S'
T
∧
S'
≢
∅
|
frag
S'
T
=>
closed
S'
T
∧
∃
s
,
s
∈
S'
end
.
Instance
sts_core
:
Core
(
car
sts
)
:=
λ
x
,
match
x
with
...
...
@@ -199,7 +195,7 @@ Instance sts_core : Core (car sts) := λ x,
end
.
Inductive
sts_disjoint
:
Disjoint
(
car
sts
)
:=
|
frag_frag_disjoint
S1
S2
T1
T2
:
S1
∩
S2
≢
∅
→
T1
⊥
T2
→
frag
S1
T1
⊥
frag
S2
T2
(
∃
s
,
s
∈
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
.
Existing
Instance
sts_disjoint
.
...
...
@@ -212,7 +208,7 @@ Instance sts_op : Op (car sts) := λ x1 x2,
end
.
Hint
Extern
50
(
equiv
(
A
:=
set
_)
_
_)
=>
set_solver
:
sts
.
Hint
Extern
50
(
¬
equiv
(
A
:=
set
_)
_
_)
=>
set_solver
:
sts
.
Hint
Extern
50
(
∃
s
:
state
sts
,
_)
=>
set_solver
:
sts
.
Hint
Extern
50
(_
∈
_)
=>
set_solver
:
sts
.
Hint
Extern
50
(_
⊆
_)
=>
set_solver
:
sts
.
Hint
Extern
50
(_
⊥
_)
=>
set_solver
:
sts
.
...
...
@@ -236,27 +232,26 @@ Proof.
-
by
do
2
destruct
1
;
constructor
;
setoid_subst
.
-
by
destruct
1
;
constructor
;
setoid_subst
.
-
by
destruct
1
;
simpl
;
intros
?;
setoid_subst
.
-
by
intros
?
[|];
destruct
1
;
inversion_clear
1
;
constructor
;
setoid_subst
.
-
by
intros
?
[|];
destruct
1
;
inversion_clear
1
;
e
constructor
;
setoid_subst
.
-
destruct
3
;
simpl
in
*
;
destruct_and
?;
eauto
using
closed_op
;
match
goal
with
H
:
closed
_
_
|
-
_
=>
destruct
H
end
;
set_solver
.
-
intros
[];
simpl
;
intros
;
destruct_and
?;
split
;
e
auto
using
closed_up
,
up_non_empty
,
closed_up_set
,
up_set_empty
with
sts
.
-
intros
[];
naive_solver
eauto
using
closed_up
,
closed_up_set
,
e
lem_of_up
,
elem_of_up_set
with
sts
.
-
intros
[]
[]
[];
constructor
;
rewrite
?assoc
;
auto
with
sts
.
-
destruct
4
;
inversion_clear
1
;
constructor
;
auto
with
sts
.
-
destruct
4
;
inversion_clear
1
;
constructor
;
auto
with
sts
.
-
destruct
1
;
constructor
;
auto
with
sts
.
-
destruct
3
;
constructor
;
auto
with
sts
.
-
intros
[
|
S
T
];
constructor
;
auto
using
elem_of_up
with
sts
.
-
intros
[
|
S
T
];
constructor
;
auto
with
sts
.
-
intros
[];
constructor
;
e
auto
with
sts
.
-
intros
[];
constructor
;
auto
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
_
_));
eauto
using
closed_up_set
with
sts
.
-
intros
x
y
.
exists
(
core
(
x
⋅
y
))=>
??
Hxy
;
split_and
?
.
+
destruct
Hxy
;
constructor
;
unfold
up_set
;
set_solver
.
+
destruct
Hxy
;
simpl
;
split_and
?;
auto
using
closed_up_set_empty
,
closed_up_empty
,
up_non_empty
;
[]
.
apply
up_set_non_empty
.
set_solver
.
+
destruct
Hxy
;
constructor
;
+
destruct
Hxy
;
simpl
;
eauto
using
closed_up_set_empty
,
closed_up_empty
with
sts
.
+
destruct
Hxy
;
econstructor
;
repeat
match
goal
with
|
|
-
context
[
up_set
?S
?T
]
=>
unless
(
S
⊆
up_set
S
T
)
by
done
;
pose
proof
(
subseteq_up_set
S
T
)
...
...
@@ -304,10 +299,10 @@ Proof. solve_proper. Qed.
(** Validity *)
Lemma
sts_auth_valid
s
T
:
✓
sts_auth
s
T
↔
tok
s
⊥
T
.
Proof
.
done
.
Qed
.
Lemma
sts_frag_valid
S
T
:
✓
sts_frag
S
T
↔
closed
S
T
∧
S
≢
∅
.
Lemma
sts_frag_valid
S
T
:
✓
sts_frag
S
T
↔
closed
S
T
∧
∃
s
,
s
∈
S
.
Proof
.
done
.
Qed
.
Lemma
sts_frag_up_valid
s
T
:
tok
s
⊥
T
→
✓
sts_frag_up
s
T
.
Proof
.
intros
.
by
apply
sts_frag_valid
;
auto
using
closed_up
,
up_non_empty
.
Qed
.
Proof
.
intros
.
apply
sts_frag_valid
;
split
.
by
apply
closed_up
.
set_solver
.
Qed
.
Lemma
sts_auth_frag_valid_inv
s
S
T1
T2
:
✓
(
sts_auth
s
T1
⋅
sts_frag
S
T2
)
→
s
∈
S
.
...
...
@@ -329,7 +324,7 @@ Proof.
-
intros
;
split_and
?
.
+
set_solver
+.
+
by
apply
closed_up
.
+
apply
up_non_empty
.
+
exists
s
.
set_solver
.
+
constructor
;
last
set_solver
.
apply
elem_of_up
.
Qed
.
...
...
@@ -338,7 +333,7 @@ Lemma sts_op_frag S1 S2 T1 T2 :
sts_frag
(
S1
∩
S2
)
(
T1
∪
T2
)
≡
sts_frag
S1
T1
⋅
sts_frag
S2
T2
.
Proof
.
intros
HT
HS1
HS2
.
rewrite
/
sts_frag
-
to_validity_op
//.
move
=>
/=
[?
?
]
.
split_and
!
;
[
auto
;
set_solver
..|
by
constructor
]
.
move
=>
/=
[?
[?
?]
]
.
split_and
!
;
[
set_solver
..|
constructor
;
set_solver
]
.
Qed
.
(** Frame preserving updates *)
...
...
@@ -356,8 +351,8 @@ Lemma sts_update_frag S1 S2 T1 T2 :
Proof
.
rewrite
/
sts_frag
=>
?
HS
HT
.
apply
validity_update
.
inversion
3
as
[|?
S
?
Tf
|];
simplify_eq
/=.
-
split_and
!
;
first
done
;
first
set_solver
.
constructor
;
set_solver
.
-
split_and
!
;
first
done
;
first
set_solver
.
constructor
;
set_solver
.
-
split_and
!
.
done
.
set_solver
.
constructor
;
set_solver
.
-
split_and
!
.
done
.
set_solver
.
constructor
;
set_solver
.
Qed
.
Lemma
sts_update_frag_up
s1
S2
T1
T2
:
...
...
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