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David Swasey
coqstdpp
Commits
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
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3 changed files
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71 additions
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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 [Hv2Hv2]].
 apply map_filter_lookup_Some. by rewrite HP.
 specialize (HP _ _ ISP). rewrite HP, Hv2 in EqS. naive_solver.
 apply (Hv2 v); [by apply HPdone].
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_insertrewrite 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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