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Simon Spies
stdpp
Commits
eecf7526
Commit
eecf7526
authored
Sep 28, 2017
by
Hai Dang
Committed by
Robbert Krebbers
Sep 29, 2017
Browse files
simplify proofs of gmap filter
parent
c809b3b5
Changes
1
Hide whitespace changes
Inline
Sidebyside
theories/gmap.v
View file @
eecf7526
...
...
@@ 256,15 +256,8 @@ Section filter.
∀
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
.
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
:
...
...
@@ 274,51 +267,37 @@ Section filter.
destruct
1
as
[?[
Eq
_
]%
gmap_filter_lookup_Some
].
by
eexists
.
Qed
.
Lemma
gmap_filter_lookup_equiv
`
{
Equiv
A
}
`
{
Reflexive
A
(
≡
)}
m1
m2
:
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
.
→
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
.
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
`
{
Equiv
A
}
`
{
Reflexive
A
(
≡
)}
m
k
v
:
P
(
k
,
v
)
→
<[
k
:
=
v
]>
(
filter
P
m
)
≡
filter
P
(<[
k
:
=
v
]>
m
).
Lemma
gmap_filter_lookup_insert
m
k
v
:
P
(
k
,
v
)
→
<[
k
:
=
v
]>
(
filter
P
m
)
=
filter
P
(<[
k
:
=
v
]>
m
).
Proof
.
intros
HP
k'
.
intros
HP
.
apply
map_eq
.
intros
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
_
)).

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

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
.
[
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
.
intro
l
.
rewrite
lookup_empty
.
constructor
.
Qed
.
Lemma
gmap_filter_empty
`
{
Equiv
A
}
:
filter
P
(
∅
:
gmap
K
A
)
=
∅
.
Proof
.
apply
map_fold_empty
.
Qed
.
End
filter
.
...
...
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