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David Swasey
coqstdpp
Commits
eecf7526
Commit
eecf7526
authored
Sep 28, 2017
by
Hai Dang
Committed by
Robbert Krebbers
Sep 29, 2017
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simplify proofs of gmap filter
parent
c809b3b5
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theories/gmap.v
theories/gmap.v
+23
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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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