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Tej Chajed
iris
Commits
3e72f509
Commit
3e72f509
authored
Jan 14, 2016
by
Robbert Krebbers
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Iris namespaces.
parent
a89818de
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3e72f509
Require
Export
modures
.
base
prelude
.
countable
prelude
.
co_pset
.
Definition
namespace
:
=
list
positive
.
Definition
nnil
:
namespace
:
=
nil
.
Definition
ndot
`
{
Countable
A
}
(
I
:
namespace
)
(
x
:
A
)
:
namespace
:
=
encode
x
::
I
.
Definition
nclose
(
I
:
namespace
)
:
coPset
:
=
coPset_suffixes
(
encode
I
).
Instance
ndot_injective
`
{
Countable
A
}
:
Injective2
(=)
(=)
(=)
(@
ndot
A
_
_
).
Proof
.
by
intros
I1
x1
I2
x2
?
;
simplify_equality
.
Qed
.
Lemma
nclose_nnil
:
nclose
nnil
=
coPset_all
.
Proof
.
by
apply
(
sig_eq_pi
_
).
Qed
.
Lemma
encode_nclose
I
:
encode
I
∈
nclose
I
.
Proof
.
by
apply
elem_coPset_suffixes
;
exists
xH
;
rewrite
(
left_id_L
_
_
).
Qed
.
Lemma
nclose_subseteq
`
{
Countable
A
}
I
x
:
nclose
(
ndot
I
x
)
⊆
nclose
I
.
Proof
.
intros
p
;
rewrite
/
nclose
!
elem_coPset_suffixes
;
intros
[
q
->].
destruct
(
list_encode_suffix
I
(
ndot
I
x
))
as
[
q'
?]
;
[
by
exists
[
encode
x
]|].
by
exists
(
q
++
q'
)%
positive
;
rewrite
<-(
associative_L
_
)
;
f_equal
.
Qed
.
Lemma
ndot_nclose
`
{
Countable
A
}
I
x
:
encode
(
ndot
I
x
)
∈
nclose
I
.
Proof
.
apply
nclose_subseteq
with
x
,
encode_nclose
.
Qed
.
Lemma
nclose_disjoint
`
{
Countable
A
}
I
(
x
y
:
A
)
:
x
≠
y
→
nclose
(
ndot
I
x
)
∩
nclose
(
ndot
I
y
)
=
∅
.
Proof
.
intros
Hxy
;
apply
elem_of_equiv_empty_L
=>
p
;
unfold
nclose
,
ndot
.
rewrite
elem_of_intersection
!
elem_coPset_suffixes
;
intros
[[
q
->]
[
q'
Hq
]].
apply
Hxy
,
(
injective
encode
),
(
injective
encode_nat
)
;
revert
Hq
.
rewrite
!(
list_encode_cons
(
encode
_
)).
rewrite
!(
associative_L
_
)
(
injective_iff
(++
_
)%
positive
)
/=.
generalize
(
encode_nat
(
encode
y
)).
induction
(
encode_nat
(
encode
x
))
;
intros
[|?]
?
;
f_equal'
;
naive_solver
.
Qed
.
\ No newline at end of file
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