Skip to content
GitLab
Projects
Groups
Snippets
Help
Loading...
Help
Help
Support
Community forum
Keyboard shortcuts
?
Submit feedback
Contribute to GitLab
Sign in / Register
Toggle navigation
C
coqstdpp
Project overview
Project overview
Details
Activity
Releases
Repository
Repository
Files
Commits
Branches
Tags
Contributors
Graph
Compare
Issues
0
Issues
0
List
Boards
Labels
Service Desk
Milestones
Merge Requests
0
Merge Requests
0
CI / CD
CI / CD
Pipelines
Jobs
Schedules
Operations
Operations
Incidents
Environments
Analytics
Analytics
CI / CD
Repository
Value Stream
Wiki
Wiki
Snippets
Snippets
Members
Members
Collapse sidebar
Close sidebar
Activity
Graph
Create a new issue
Jobs
Commits
Issue Boards
Open sidebar
David Swasey
coqstdpp
Commits
82853b40
Commit
82853b40
authored
Sep 28, 2017
by
Hai Dang
Committed by
Robbert Krebbers
Sep 29, 2017
Browse files
Options
Browse Files
Download
Email Patches
Plain Diff
generalize filter from gmap to fin_map
parent
eecf7526
Changes
3
Hide whitespace changes
Inline
Sidebyside
Showing
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
No files found.
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. *)
...
...
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