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Jonas Kastberg
iris
Commits
3034d8ef
Commit
3034d8ef
authored
Feb 17, 2016
by
Robbert Krebbers
Browse files
Use Disjoint type class to get nice notation for ndisjoint.
Also, put stuff in a section.
parent
b07dd0b5
Changes
2
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barrier/barrier.v
View file @
3034d8ef
...
...
@@ -102,7 +102,7 @@ Section proof.
(* TODO We could alternatively construct the namespaces to be disjoint.
But that would take a lot of flexibility from the client, who probably
wants to also use the heap_ctx elsewhere. *)
Context
(
HeapN_disj
:
ndisj
HeapN
N
).
Context
(
HeapN_disj
:
HeapN
⊥
N
).
Notation
iProp
:
=
(
iPropG
heap_lang
Σ
).
...
...
program_logic/invariants.v
View file @
3034d8ef
...
...
@@ -9,7 +9,6 @@ Local Hint Extern 100 (@subseteq coPset _ _) => solve_elem_of.
Local
Hint
Extern
100
(
_
∉
_
)
=>
solve_elem_of
.
Local
Hint
Extern
99
({[
_
]}
⊆
_
)
=>
apply
elem_of_subseteq_singleton
.
Definition
namespace
:
=
list
positive
.
Definition
nnil
:
namespace
:
=
nil
.
Definition
ndot
`
{
Countable
A
}
(
N
:
namespace
)
(
x
:
A
)
:
namespace
:
=
...
...
@@ -31,44 +30,38 @@ Qed.
Lemma
ndot_nclose
`
{
Countable
A
}
N
x
:
encode
(
ndot
N
x
)
∈
nclose
N
.
Proof
.
apply
nclose_subseteq
with
x
,
encode_nclose
.
Qed
.
Definition
ndisj
(
N1
N2
:
namespace
)
:
=
Instance
ndisjoint
:
Disjoint
namespace
:
=
λ
N1
N2
,
∃
N1'
N2'
,
N1'
`
suffix_of
`
N1
∧
N2'
`
suffix_of
`
N2
∧
length
N1'
=
length
N2'
∧
N1'
≠
N2'
.
Global
Instance
ndisj_comm
:
Comm
iff
ndisj
.
Proof
.
intros
N1
N2
.
rewrite
/
ndisj
;
naive_solver
.
Qed
.
Section
ndisjoint
.
Context
`
{
Countable
A
}.
Implicit
Types
x
y
:
A
.
Lemma
ndot_ne_disj
`
{
Countable
A
}
N
(
x
y
:
A
)
:
x
≠
y
→
ndisj
(
ndot
N
x
)
(
ndot
N
y
).
Proof
.
intros
Hxy
.
exists
(
ndot
N
x
),
(
ndot
N
y
).
split_ands
;
try
done
;
[].
by
apply
not_inj2_2
.
Qed
.
Global
Instance
ndisjoint_comm
:
Comm
iff
ndisjoint
.
Proof
.
intros
N1
N2
.
rewrite
/
disjoint
/
ndisjoint
;
naive_solver
.
Qed
.
Lemma
ndot_preserve_disj_l
`
{
Countable
A
}
N1
N2
(
x
:
A
)
:
ndisj
N1
N2
→
ndisj
(
ndot
N1
x
)
N2
.
Proof
.
intros
(
N1'
&
N2'
&
Hpr1
&
Hpr2
&
Hl
&
Hne
).
exists
N1'
,
N2'
.
split_ands
;
try
done
;
[].
by
apply
suffix_of_cons_r
.
Qed
.
Lemma
ndot_ne_disjoint
N
(
x
y
:
A
)
:
x
≠
y
→
ndot
N
x
⊥
ndot
N
y
.
Proof
.
intros
Hxy
.
exists
(
ndot
N
x
),
(
ndot
N
y
)
;
naive_solver
.
Qed
.
Lemma
ndot_preserve_disj
_r
`
{
Countable
A
}
N1
N2
(
x
:
A
)
:
ndisj
N1
N2
→
ndisj
N1
(
ndot
N2
x
)
.
Proof
.
rewrite
![
ndisj
N1
_
]
comm
.
apply
ndot_preserve_disj_l
.
Qed
.
Lemma
ndot_preserve_disj
oint_l
N1
N2
x
:
N1
⊥
N2
→
ndot
N1
x
⊥
N2
.
Proof
.
intros
(
N1'
&
N2'
&
Hpr1
&
Hpr2
&
Hl
&
Hne
).
exists
N1'
,
N2'
.
split_ands
;
try
done
;
[].
by
apply
suffix_of_cons_r
.
Qed
.
Lemma
ndisj_disjoint
N1
N2
:
ndisj
N1
N2
→
nclose
N1
∩
nclose
N2
=
∅
.
Proof
.
intros
(
N1'
&
N2'
&
[
N1''
Hpr1
]
&
[
N2''
Hpr2
]
&
Hl
&
Hne
).
subst
N1
N2
.
apply
elem_of_equiv_empty_L
=>
p
;
unfold
nclose
.
rewrite
elem_of_intersection
!
elem_coPset_suffixes
;
intros
[[
q
->]
[
q'
Hq
]].
rewrite
!
list_encode_app
!
assoc
in
Hq
.
apply
Hne
.
eapply
list_encode_suffix_eq
;
done
.
Qed
.
Lemma
ndot_preserve_disjoint_r
N1
N2
x
:
N1
⊥
N2
→
N1
⊥
ndot
N2
x
.
Proof
.
rewrite
![
N1
⊥
_
]
comm
.
apply
ndot_preserve_disjoint_l
.
Qed
.
Local
Hint
Resolve
nclose_subseteq
ndot_nclose
.
Lemma
ndisj_disjoint
N1
N2
:
N1
⊥
N2
→
nclose
N1
∩
nclose
N2
=
∅
.
Proof
.
intros
(
N1'
&
N2'
&
[
N1''
->]
&
[
N2''
->]
&
Hl
&
Hne
).
apply
elem_of_equiv_empty_L
=>
p
;
unfold
nclose
.
rewrite
elem_of_intersection
!
elem_coPset_suffixes
;
intros
[[
q
->]
[
q'
Hq
]].
rewrite
!
list_encode_app
!
assoc
in
Hq
.
by
eapply
Hne
,
list_encode_suffix_eq
.
Qed
.
End
ndisjoint
.
(** Derived forms and lemmas about them. *)
Definition
inv
{
Λ
Σ
}
(
N
:
namespace
)
(
P
:
iProp
Λ
Σ
)
:
iProp
Λ
Σ
:
=
...
...
@@ -128,7 +121,7 @@ Proof. intros. by apply: (inv_fsa pvs_fsa). Qed.
Lemma
wp_open_close
E
e
N
P
(
Q
:
val
Λ
→
iProp
Λ
Σ
)
R
:
atomic
e
→
nclose
N
⊆
E
→
R
⊑
inv
N
P
→
R
⊑
(
▷
P
-
★
wp
(
E
∖
nclose
N
)
e
(
λ
v
,
▷
P
★
Q
v
))
→
R
⊑
(
▷
P
-
★
wp
(
E
∖
nclose
N
)
e
(
λ
v
,
▷
P
★
Q
v
))
→
R
⊑
wp
E
e
Q
.
Proof
.
intros
.
by
apply
:
(
inv_fsa
(
wp_fsa
e
)).
Qed
.
...
...
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