Skip to content
GitLab
Menu
Projects
Groups
Snippets
Loading...
Help
Help
Support
Community forum
Keyboard shortcuts
?
Submit feedback
Contribute to GitLab
Sign in / Register
Toggle navigation
Menu
Open sidebar
David Swasey
coqstdpp
Commits
c809b3b5
Commit
c809b3b5
authored
Sep 28, 2017
by
Hai Dang
Committed by
Robbert Krebbers
Sep 29, 2017
Browse files
add filter for gmap
parent
2175e39f
Changes
1
Hide whitespace changes
Inline
Sidebyside
Showing
1 changed file
with
96 additions
and
0 deletions
+96
0
theories/gmap.v
theories/gmap.v
+96
0
No files found.
theories/gmap.v
View file @
c809b3b5
...
...
@@ 227,6 +227,102 @@ 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
.
apply
(
map_fold_ind
(
λ
m1
m2
,
∀
k
,
m1
!!
k
=
None
↔
(
m2
!!
k
=
None
∨
∀
v
,
m2
!!
k
=
Some
v
→
¬
P
_
))).

naive_solver
.

intros
k
v
m
m'
Hm
Eq
k'
.
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
.
naive_solver
.
+
by
rewrite
lookup_insert_ne
.
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
`
{
Equiv
A
}
`
{
Reflexive
A
(
≡
)}
m1
m2
:
(
∀
k
v
,
P
(
k
,
v
)
→
m1
!!
k
=
Some
v
↔
m2
!!
k
=
Some
v
)
→
filter
P
m1
≡
filter
P
m2
.
Proof
.
intros
HP
k
.
destruct
(
filter
P
m1
!!
k
)
as
[
v1
]
eqn
:
Hv1
;
[
apply
gmap_filter_lookup_Some
in
Hv1
as
[
Hv1
HP1
]
;
specialize
(
HP
k
v1
HP1
)]
;
destruct
(
filter
P
m2
!!
k
)
as
[
v2
]
eqn
:
Hv2
.

apply
gmap_filter_lookup_Some
in
Hv2
as
[
Hv2
_
].
rewrite
Hv1
,
Hv2
in
HP
.
destruct
HP
as
[
HP
_
].
specialize
(
HP
(
eq_refl
_
))
as
[].
by
apply
option_Forall2_refl
.

apply
gmap_filter_lookup_None
in
Hv2
as
[
Hv2

Hv2
]
;
[
naive_solver

by
apply
HP
,
Hv2
in
Hv1
].

apply
gmap_filter_lookup_Some
in
Hv2
as
[
Hv2
HP2
].
specialize
(
HP
k
v2
HP2
).
apply
gmap_filter_lookup_None
in
Hv1
as
[
Hv1

Hv1
].
+
rewrite
Hv1
in
HP
.
naive_solver
.
+
by
apply
HP
,
Hv1
in
Hv2
.

by
apply
option_Forall2_refl
.
Qed
.
Lemma
gmap_filter_lookup_insert
`
{
Equiv
A
}
`
{
Reflexive
A
(
≡
)}
m
k
v
:
P
(
k
,
v
)
→
<[
k
:
=
v
]>
(
filter
P
m
)
≡
filter
P
(<[
k
:
=
v
]>
m
).
Proof
.
intros
HP
k'
.
case
(
decide
(
k'
=
k
))
as
[>?]
;
[
rewrite
lookup_insert

rewrite
lookup_insert_ne
;
[
auto
]].

destruct
(
filter
P
(<[
k
:
=
v
]>
m
)
!!
k
)
eqn
:
Hk
.
+
apply
gmap_filter_lookup_Some
in
Hk
.
rewrite
lookup_insert
in
Hk
.
destruct
Hk
as
[
Hk
_
].
inversion
Hk
.
by
apply
option_Forall2_refl
.
+
apply
gmap_filter_lookup_None
in
Hk
.
rewrite
lookup_insert
in
Hk
.
destruct
Hk
as
[>
HNP
].
by
apply
option_Forall2_refl
.
by
specialize
(
HNP
v
(
eq_refl
_
)).

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

rewrite
gmap_filter_lookup_None
]
;
(
rewrite
lookup_insert_ne
;
[
by
auto
])
;
[
rewrite
<
gmap_filter_lookup_Some

rewrite
<
gmap_filter_lookup_None
]
;
intros
Hk
;
rewrite
Hk
;
by
apply
option_Forall2_refl
.
Qed
.
Lemma
gmap_filter_empty
`
{
Equiv
A
}
:
filter
P
(
∅
:
gmap
K
A
)
≡
∅
.
Proof
.
intro
l
.
rewrite
lookup_empty
.
constructor
.
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. *)
...
...
Write
Preview
Markdown
is supported
0%
Try again
or
attach a new file
.
Attach a file
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Cancel
Please
register
or
sign in
to comment