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82853b40
Commit
82853b40
authored
Sep 28, 2017
by
Hai Dang
Committed by
Robbert Krebbers
Sep 29, 2017
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generalize filter from gmap to fin_map
parent
eecf7526
Changes
3
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3 changed files
with
71 additions
and
75 deletions
+71
75
theories/fin_map_dom.v
theories/fin_map_dom.v
+7
0
theories/fin_maps.v
theories/fin_maps.v
+64
0
theories/gmap.v
theories/gmap.v
+0
75
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theories/fin_map_dom.v
View file @
82853b40
...
...
@@ 19,6 +19,13 @@ Class FinMapDom K M D `{∀ A, Dom (M A) D, FMap M,
Section
fin_map_dom
.
Context
`
{
FinMapDom
K
M
D
}.
Lemma
dom_map_filter
{
A
}
(
P
:
K
*
A
→
Prop
)
`
{!
∀
x
,
Decision
(
P
x
)}
(
m
:
M
A
)
:
dom
D
(
filter
P
m
)
⊆
dom
D
m
.
Proof
.
intros
?.
rewrite
2
!
elem_of_dom
.
destruct
1
as
[?[
Eq
_
]%
map_filter_lookup_Some
].
by
eexists
.
Qed
.
Lemma
elem_of_dom_2
{
A
}
(
m
:
M
A
)
i
x
:
m
!!
i
=
Some
x
→
i
∈
dom
D
m
.
Proof
.
rewrite
elem_of_dom
;
eauto
.
Qed
.
Lemma
not_elem_of_dom
{
A
}
(
m
:
M
A
)
i
:
i
∉
dom
D
m
↔
m
!!
i
=
None
.
...
...
theories/fin_maps.v
View file @
82853b40
...
...
@@ 130,6 +130,9 @@ is unspecified. *)
Definition
map_fold
`
{
FinMapToList
K
A
M
}
{
B
}
(
f
:
K
→
A
→
B
→
B
)
(
b
:
B
)
:
M
→
B
:
=
foldr
(
curry
f
)
b
∘
map_to_list
.
Instance
map_filter
`
{
FinMap
K
M
}
{
A
}
:
Filter
(
K
*
A
)
(
M
A
)
:
=
λ
P
_
,
map_fold
(
λ
k
v
m
,
if
decide
(
P
(
k
,
v
))
then
<[
k
:
=
v
]>
m
else
m
)
∅
.
(** * Theorems *)
Section
theorems
.
Context
`
{
FinMap
K
M
}.
...
...
@@ 1002,6 +1005,67 @@ Proof.
assert
(
m
!!
j
=
Some
y
)
by
(
apply
Hm
;
by
right
).
naive_solver
.
Qed
.
(** ** The filter operation *)
Section
map_Filter
.
Context
{
A
}
(
P
:
K
*
A
→
Prop
)
`
{!
∀
x
,
Decision
(
P
x
)}.
Lemma
map_filter_lookup_Some
:
∀
m
k
v
,
filter
P
m
!!
k
=
Some
v
↔
m
!!
k
=
Some
v
∧
P
(
k
,
v
).
Proof
.
apply
(
map_fold_ind
(
λ
m1
m2
,
∀
k
v
,
m1
!!
k
=
Some
v
↔
m2
!!
k
=
Some
v
∧
P
_
)).

setoid_rewrite
lookup_empty
.
naive_solver
.

intros
k
v
m
m'
Hm
Eq
k'
v'
.
case_match
;
case
(
decide
(
k'
=
k
))
as
[>?].
+
rewrite
2
!
lookup_insert
.
naive_solver
.
+
do
2
(
rewrite
lookup_insert_ne
;
[
auto
]).
by
apply
Eq
.
+
rewrite
Eq
,
Hm
,
lookup_insert
.
split
;
[
naive_solver
].
destruct
1
as
[
Eq'
].
inversion
Eq'
.
by
subst
.
+
by
rewrite
lookup_insert_ne
.
Qed
.
Lemma
map_filter_lookup_None
:
∀
m
k
,
filter
P
m
!!
k
=
None
↔
m
!!
k
=
None
∨
∀
v
,
m
!!
k
=
Some
v
→
¬
P
(
k
,
v
).
Proof
.
intros
m
k
.
rewrite
eq_None_not_Some
.
unfold
is_Some
.
setoid_rewrite
map_filter_lookup_Some
.
naive_solver
.
Qed
.
Lemma
map_filter_lookup_equiv
m1
m2
:
(
∀
k
v
,
P
(
k
,
v
)
→
m1
!!
k
=
Some
v
↔
m2
!!
k
=
Some
v
)
→
filter
P
m1
=
filter
P
m2
.
Proof
.
intros
HP
.
apply
map_eq
.
intros
k
.
destruct
(
filter
P
m2
!!
k
)
as
[
v2
]
eqn
:
Hv2
;
[
apply
map_filter_lookup_Some
in
Hv2
as
[
Hv2
HP2
]
;
specialize
(
HP
k
v2
HP2
)

apply
map_filter_lookup_None
;
right
;
intros
v
EqS
ISP
;
apply
map_filter_lookup_None
in
Hv2
as
[
Hv2

Hv2
]].

apply
map_filter_lookup_Some
.
by
rewrite
HP
.

specialize
(
HP
_
_
ISP
).
rewrite
HP
,
Hv2
in
EqS
.
naive_solver
.

apply
(
Hv2
v
)
;
[
by
apply
HP

done
].
Qed
.
Lemma
map_filter_lookup_insert
m
k
v
:
P
(
k
,
v
)
→
<[
k
:
=
v
]>
(
filter
P
m
)
=
filter
P
(<[
k
:
=
v
]>
m
).
Proof
.
intros
HP
.
apply
map_eq
.
intros
k'
.
case
(
decide
(
k'
=
k
))
as
[>?]
;
[
rewrite
lookup_insert

rewrite
lookup_insert_ne
;
[
auto
]].

symmetry
.
apply
map_filter_lookup_Some
.
by
rewrite
lookup_insert
.

destruct
(
filter
P
(<[
k
:
=
v
]>
m
)
!!
k'
)
eqn
:
Hk
;
revert
Hk
;
[
rewrite
map_filter_lookup_Some
,
lookup_insert_ne
;
[
by
auto
]
;
by
rewrite
<
map_filter_lookup_Some

rewrite
map_filter_lookup_None
,
lookup_insert_ne
;
[
auto
]
;
by
rewrite
<
map_filter_lookup_None
].
Qed
.
Lemma
map_filter_empty
:
filter
P
∅
=
∅
.
Proof
.
apply
map_fold_empty
.
Qed
.
End
map_Filter
.
(** ** Properties of the [map_Forall] predicate *)
Section
map_Forall
.
Context
{
A
}
(
P
:
K
→
A
→
Prop
).
...
...
theories/gmap.v
View file @
82853b40
...
...
@@ 227,81 +227,6 @@ Proof.

by
rewrite
option_guard_False
by
(
rewrite
not_elem_of_dom
;
eauto
).
Qed
.
(** Filter *)
(* This filter creates a submap whose (key,value) pairs satisfy P *)
Instance
gmap_filter
`
{
Countable
K
}
{
A
}
:
Filter
(
K
*
A
)
(
gmap
K
A
)
:
=
λ
P
_
,
map_fold
(
λ
k
v
m
,
if
decide
(
P
(
k
,
v
))
then
<[
k
:
=
v
]>
m
else
m
)
∅
.
Section
filter
.
Context
`
{
Countable
K
}
{
A
}
(
P
:
K
*
A
→
Prop
)
`
{!
∀
x
,
Decision
(
P
x
)}.
Implicit
Type
(
m
:
gmap
K
A
)
(
k
:
K
)
(
v
:
A
).
Lemma
gmap_filter_lookup_Some
:
∀
m
k
v
,
filter
P
m
!!
k
=
Some
v
↔
m
!!
k
=
Some
v
∧
P
(
k
,
v
).
Proof
.
apply
(
map_fold_ind
(
λ
m1
m2
,
∀
k
v
,
m1
!!
k
=
Some
v
↔
m2
!!
k
=
Some
v
∧
P
_
)).

naive_solver
.

intros
k
v
m
m'
Hm
Eq
k'
v'
.
case_match
;
case
(
decide
(
k'
=
k
))
as
[>?].
+
rewrite
2
!
lookup_insert
.
naive_solver
.
+
do
2
(
rewrite
lookup_insert_ne
;
[
auto
]).
by
apply
Eq
.
+
rewrite
Eq
,
Hm
,
lookup_insert
.
split
;
[
naive_solver
].
destruct
1
as
[
Eq'
].
inversion
Eq'
.
by
subst
.
+
by
rewrite
lookup_insert_ne
.
Qed
.
Lemma
gmap_filter_lookup_None
:
∀
m
k
,
filter
P
m
!!
k
=
None
↔
m
!!
k
=
None
∨
∀
v
,
m
!!
k
=
Some
v
→
¬
P
(
k
,
v
).
Proof
.
intros
m
k
.
rewrite
eq_None_not_Some
.
unfold
is_Some
.
setoid_rewrite
gmap_filter_lookup_Some
.
naive_solver
.
Qed
.
Lemma
gmap_filter_dom
m
:
dom
(
gset
K
)
(
filter
P
m
)
⊆
dom
(
gset
K
)
m
.
Proof
.
intros
?.
rewrite
2
!
elem_of_dom
.
destruct
1
as
[?[
Eq
_
]%
gmap_filter_lookup_Some
].
by
eexists
.
Qed
.
Lemma
gmap_filter_lookup_equiv
m1
m2
:
(
∀
k
v
,
P
(
k
,
v
)
→
m1
!!
k
=
Some
v
↔
m2
!!
k
=
Some
v
)
→
filter
P
m1
=
filter
P
m2
.
Proof
.
intros
HP
.
apply
map_eq
.
intros
k
.
destruct
(
filter
P
m2
!!
k
)
as
[
v2
]
eqn
:
Hv2
;
[
apply
gmap_filter_lookup_Some
in
Hv2
as
[
Hv2
HP2
]
;
specialize
(
HP
k
v2
HP2
)

apply
gmap_filter_lookup_None
;
right
;
intros
v
EqS
ISP
;
apply
gmap_filter_lookup_None
in
Hv2
as
[
Hv2

Hv2
]].

apply
gmap_filter_lookup_Some
.
by
rewrite
HP
.

specialize
(
HP
_
_
ISP
).
rewrite
HP
,
Hv2
in
EqS
.
naive_solver
.

apply
(
Hv2
v
)
;
[
by
apply
HP

done
].
Qed
.
Lemma
gmap_filter_lookup_insert
m
k
v
:
P
(
k
,
v
)
→
<[
k
:
=
v
]>
(
filter
P
m
)
=
filter
P
(<[
k
:
=
v
]>
m
).
Proof
.
intros
HP
.
apply
map_eq
.
intros
k'
.
case
(
decide
(
k'
=
k
))
as
[>?]
;
[
rewrite
lookup_insert

rewrite
lookup_insert_ne
;
[
auto
]].

symmetry
.
apply
gmap_filter_lookup_Some
.
by
rewrite
lookup_insert
.

destruct
(
filter
P
(<[
k
:
=
v
]>
m
)
!!
k'
)
eqn
:
Hk
;
revert
Hk
;
[
rewrite
gmap_filter_lookup_Some
,
lookup_insert_ne
;
[
by
auto
]
;
by
rewrite
<
gmap_filter_lookup_Some

rewrite
gmap_filter_lookup_None
,
lookup_insert_ne
;
[
auto
]
;
by
rewrite
<
gmap_filter_lookup_None
].
Qed
.
Lemma
gmap_filter_empty
`
{
Equiv
A
}
:
filter
P
(
∅
:
gmap
K
A
)
=
∅
.
Proof
.
apply
map_fold_empty
.
Qed
.
End
filter
.
(** * Fresh elements *)
(* This is pretty adhoc and just for the case of [gset positive]. We need a
notion of countable nonfinite types to generalize this. *)
...
...
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